Integrand size = 22, antiderivative size = 58 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\frac {x^2 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {5}{3}+2 p,\frac {5}{3},-\frac {b x^3}{a}\right )}{2 a} \]
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Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1370, 371} \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\frac {1}{2} x^2 \left (\frac {b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-2 p,\frac {5}{3},-\frac {b x^3}{a}\right ) \]
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Rule 371
Rule 1370
Rubi steps \begin{align*} \text {integral}& = \left (\left (1+\frac {b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p\right ) \int x \left (1+\frac {b x^3}{a}\right )^{2 p} \, dx \\ & = \frac {1}{2} x^2 \left (1+\frac {b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac {2}{3},-2 p;\frac {5}{3};-\frac {b x^3}{a}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.88 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\frac {1}{2} x^2 \left (\left (a+b x^3\right )^2\right )^p \left (1+\frac {b x^3}{a}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-2 p,\frac {5}{3},-\frac {b x^3}{a}\right ) \]
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\[\int x \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p}d x\]
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\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int { {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x \,d x } \]
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\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int x \left (\left (a + b x^{3}\right )^{2}\right )^{p}\, dx \]
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\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int { {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x \,d x } \]
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\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int { {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x \,d x } \]
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Timed out. \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int x\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^p \,d x \]
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