What is the integral of (x^3+2x)(x^4+4x^2+4)^7 ?

The integral of (x^3+2x)(x^4+4x^2+4)^7 is

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integral of (x^3+2x)(x^4+4x^2+4)^7

\int (x^{3} + 2 x) (x^{4} + 4 x^{2} + 4)^{7} d x

\frac{x ^{32}}{32} + x^{30} + 15 x^{28} + 140 x^{26} + 910 x^{24} + 4368 x^{22} + 16016 x^{20} + 45760 x^{18} + 102960 x^{16} + 183040 x^{14} + 256256 x^{12} + 279552 x^{10} + 232960 x^{8} + 143360 x^{6} + 61440 x^{4} + 16384 x^{2} + C

Solution steps

\int (x^{3} + 2 x) (x^{4} + 4 x^{2} + 4)^{7} d x

Expand

(x^{3} + 2 x) (x^{4} + 4 x^{2} + 4)^{7} : x^{31} + 30 x^{29} + 420 x^{27} + 3640 x^{25} + 21840 x^{23} + 96096 x^{21} + 320320 x^{19} + 823680 x^{17} + 1647360 x^{15} + 2562560 x^{13} + 3075072 x^{11} + 2795520 x^{9} + 1863680 x^{7} + 860160 x^{5} + 245760 x^{3} + 32768 x

= \int x^{31} + 30 x^{29} + 420 x^{27} + 3640 x^{25} + 21840 x^{23} + 96096 x^{21} + 320320 x^{19} + 823680 x^{17} + 1647360 x^{15} + 2562560 x^{13} + 3075072 x^{11} + 2795520 x^{9} + 1863680 x^{7} + 860160 x^{5} + 245760 x^{3} + 32768 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{31} d x + \int 30 x^{29} d x + \int 420 x^{27} d x + \int 3640 x^{25} d x + \int 21840 x^{23} d x + \int 96096 x^{21} d x + \int 320320 x^{19} d x + \int 823680 x^{17} d x + \int 1647360 x^{15} d x + \int 2562560 x^{13} d x + \int 3075072 x^{11} d x + \int 2795520 x^{9} d x + \int 1863680 x^{7} d x + \int 860160 x^{5} d x + \int 245760 x^{3} d x + \int 32768 x d x

\int x^{31} d x = \frac{x ^{32}}{32}

\int 30 x^{29} d x = x^{30}

\int 420 x^{27} d x = 15 x^{28}

\int 3640 x^{25} d x = 140 x^{26}

\int 21840 x^{23} d x = 910 x^{24}

\int 96096 x^{21} d x = 4368 x^{22}

\int 320320 x^{19} d x = 16016 x^{20}

\int 823680 x^{17} d x = 45760 x^{18}

\int 1647360 x^{15} d x = 102960 x^{16}

\int 2562560 x^{13} d x = 183040 x^{14}

\int 3075072 x^{11} d x = 256256 x^{12}

\int 2795520 x^{9} d x = 279552 x^{10}

\int 1863680 x^{7} d x = 232960 x^{8}

\int 860160 x^{5} d x = 143360 x^{6}

\int 245760 x^{3} d x = 61440 x^{4}

\int 32768 x d x = 16384 x^{2}

= \frac{x ^{32}}{32} + x^{30} + 15 x^{28} + 140 x^{26} + 910 x^{24} + 4368 x^{22} + 16016 x^{20} + 45760 x^{18} + 102960 x^{16} + 183040 x^{14} + 256256 x^{12} + 279552 x^{10} + 232960 x^{8} + 143360 x^{6} + 61440 x^{4} + 16384 x^{2}

Add a constant to the solution

= \frac{x ^{32}}{32} + x^{30} + 15 x^{28} + 140 x^{26} + 910 x^{24} + 4368 x^{22} + 16016 x^{20} + 45760 x^{18} + 102960 x^{16} + 183040 x^{14} + 256256 x^{12} + 279552 x^{10} + 232960 x^{8} + 143360 x^{6} + 61440 x^{4} + 16384 x^{2} + C

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What is the integral of (x^3+2x)(x^4+4x^2+4)^7 ?
The integral of (x^3+2x)(x^4+4x^2+4)^7 is