What is the integral of x^2(x^2-25)^{3/2} ?

The integral of x^2(x^2-25)^{3/2} is

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integral of x^2(x^2-25)^{3/2}

\int x^{2} (x^{2} - 25)^{\frac{3}{2}} d x

\frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 (\frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{15000} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{400} x (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}})) + C

Solution steps

\int x^{2} (x^{2} - 25)^{\frac{3}{2}} d x

Simplify

x^{2} : xx

= \int xx (x^{2} - 25)^{\frac{3}{2}} d x

Apply Integration By Parts

: \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - \int x^{4} (x^{2} - 25)^{\frac{1}{2}} d x

= \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - \int x^{4} (x^{2} - 25)^{\frac{1}{2}} d x

\int x^{4} (x^{2} - 25)^{\frac{1}{2}} d x = 15625 - \frac{1}{2500} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{200} x (x^{2} - 25)^{\frac{1}{2}} + 25 ln \frac{( x ^{2} - 25 ) ^{\frac{1}{2}} + x}{5} + \frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{3000} x^{3} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{80} x (x^{2} - 25)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{5} (x^{2} - 25)^{\frac{1}{2}} + x)

= \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 - \frac{1}{2500} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{200} x (x^{2} - 25)^{\frac{1}{2}} + 25 ln \frac{( x ^{2} - 25 ) ^{\frac{1}{2}} + x}{5} + \frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{3000} x^{3} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{80} x (x^{2} - 25)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{5} (x^{2} - 25)^{\frac{1}{2}} + x)

Simplify

\frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 - \frac{1}{2500} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{200} x (x^{2} - 25)^{\frac{1}{2}} + 25 ln \frac{( x ^{2} - 25 ) ^{\frac{1}{2}} + x}{5} + \frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{3000} x^{3} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{80} x (x^{2} - 25)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{5} (x^{2} - 25)^{\frac{1}{2}} + x) : \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 (\frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{15000} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{400} x (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}))

= \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 (\frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{15000} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{400} x (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}))

Add a constant to the solution

= \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 (\frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{15000} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{400} x (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}})) + C

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What is the integral of x^2(x^2-25)^{3/2} ?
The integral of x^2(x^2-25)^{3/2} is