Optimal. Leaf size=130 \[ \frac{\left (a+b x^3\right )^3 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^3 (2 p+3)}-\frac{a \left (a+b x^3\right )^2 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^3 (p+1)}+\frac{a^2 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^3 (2 p+1)} \]
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Rubi [A] time = 0.0788709, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1356, 266, 43} \[ \frac{\left (a+b x^3\right )^3 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^3 (2 p+3)}-\frac{a \left (a+b x^3\right )^2 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^3 (p+1)}+\frac{a^2 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^3 (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 1356
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^8 \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx &=\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^8 \left (1+\frac{b x^3}{a}\right )^{2 p} \, dx\\ &=\frac{1}{3} \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 ) \operatorname{Subst}\left (\int x^2 \left (1+\frac{b x}{a}\right )^{2 p} \, dx,x,x^3\right )\\ &=\frac{1}{3} \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 ) \operatorname{Subst}\left (\int \left (\frac{a^2 \left (1+\frac{b x}{a}\right )^{2 p}}{b^2}-\frac{2 a^2 \left (1+\frac{b x}{a}\right )^{1+2 p}}{b^2}+\frac{a^2 \left (1+\frac{b x}{a}\right )^{2+2 p}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^3 (1+2 p)}-\frac{a \left (a+b x^3\right )^2 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^3 (1+p)}+\frac{\left (a+b x^3\right )^3 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^3 (3+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0329366, size = 77, normalized size = 0.59 \[ \frac{\left (a+b x^3\right ) \left (\left (a+b x^3\right )^2\right )^p \left (a^2-a b (2 p+1) x^3+b^2 \left (2 p^2+3 p+1\right ) x^6\right )}{3 b^3 (p+1) (2 p+1) (2 p+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 96, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,{b}^{2}{p}^{2}{x}^{6}+3\,{b}^{2}p{x}^{6}+{b}^{2}{x}^{6}-2\,abp{x}^{3}-ab{x}^{3}+{a}^{2} \right ) \left ( b{x}^{3}+a \right ) \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}}{3\,{b}^{3} \left ( 4\,{p}^{3}+12\,{p}^{2}+11\,p+3 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02904, size = 107, normalized size = 0.82 \begin{align*} \frac{{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{9} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{6} - 2 \, a^{2} b p x^{3} + a^{3}\right )}{\left (b x^{3} + a\right )}^{2 \, p}}{3 \,{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62213, size = 223, normalized size = 1.72 \begin{align*} \frac{{\left ({\left (2 \, b^{3} p^{2} + 3 \, b^{3} p + b^{3}\right )} x^{9} - 2 \, a^{2} b p x^{3} +{\left (2 \, a b^{2} p^{2} + a b^{2} p\right )} x^{6} + a^{3}\right )}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{3 \,{\left (4 \, b^{3} p^{3} + 12 \, b^{3} p^{2} + 11 \, b^{3} p + 3 \, b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12567, size = 317, normalized size = 2.44 \begin{align*} \frac{2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} b^{3} p^{2} x^{9} + 3 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} b^{3} p x^{9} +{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} b^{3} x^{9} + 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} a b^{2} p^{2} x^{6} +{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} a b^{2} p x^{6} - 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} a^{2} b p x^{3} +{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} a^{3}}{3 \,{\left (4 \, b^{3} p^{3} + 12 \, b^{3} p^{2} + 11 \, b^{3} p + 3 \, b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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