Optimal. Leaf size=128 \[ \frac{\left (a+b x^2\right )^4 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (p+2)}-\frac{a \left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{b^3 (2 p+3)}+\frac{a^2 \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (p+1)} \]
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Rubi [A] time = 0.128993, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1249, 770, 21, 43} \[ \frac{\left (a+b x^2\right )^4 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (p+2)}-\frac{a \left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{b^3 (2 p+3)}+\frac{a^2 \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (p+1)} \]
Antiderivative was successfully verified.
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Rule 1249
Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx,x,x^2\right )\\ &=\frac{1}{2} \left (\left (b \left (a+b x^2\right )\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \operatorname{Subst}\left (\int x^2 (a+b x) \left (a b+b^2 x\right )^{2 p} \, dx,x,x^2\right )\\ &=\frac{\left (\left (b \left (a+b x^2\right )\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \operatorname{Subst}\left (\int x^2 \left (a b+b^2 x\right )^{1+2 p} \, dx,x,x^2\right )}{2 b}\\ &=\frac{\left (\left (b \left (a+b x^2\right )\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \operatorname{Subst}\left (\int \left (\frac{a^2 \left (a b+b^2 x\right )^{1+2 p}}{b^2}-\frac{2 a \left (a b+b^2 x\right )^{2+2 p}}{b^3}+\frac{\left (a b+b^2 x\right )^{3+2 p}}{b^4}\right ) \, dx,x,x^2\right )}{2 b}\\ &=\frac{a^2 \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (1+p)}-\frac{a \left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{b^3 (3+2 p)}+\frac{\left (a+b x^2\right )^4 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0368035, size = 68, normalized size = 0.53 \[ \frac{\left (\left (a+b x^2\right )^2\right )^{p+1} \left (a^2-2 a b (p+1) x^2+b^2 \left (2 p^2+5 p+3\right ) x^4\right )}{4 b^3 (p+1) (p+2) (2 p+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 99, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,{b}^{2}{p}^{2}{x}^{4}+5\,{b}^{2}p{x}^{4}+3\,{b}^{2}{x}^{4}-2\,abp{x}^{2}-2\,ab{x}^{2}+{a}^{2} \right ) \left ( b{x}^{2}+a \right ) ^{2} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p}}{4\,{b}^{3} \left ( 2\,{p}^{3}+9\,{p}^{2}+13\,p+6 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01826, size = 265, normalized size = 2.07 \begin{align*} \frac{{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{6} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{4} - 2 \, a^{2} b p x^{2} + a^{3}\right )}{\left (b x^{2} + a\right )}^{2 \, p} a}{2 \,{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac{{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{8} + 2 \,{\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{6} - 3 \,{\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{4} + 6 \, a^{3} b p x^{2} - 3 \, a^{4}\right )}{\left (b x^{2} + a\right )}^{2 \, p}}{4 \,{\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56729, size = 284, normalized size = 2.22 \begin{align*} \frac{{\left ({\left (2 \, b^{4} p^{2} + 5 \, b^{4} p + 3 \, b^{4}\right )} x^{8} - 2 \, a^{3} b p x^{2} + 4 \,{\left (a b^{3} p^{2} + 2 \, a b^{3} p + a b^{3}\right )} x^{6} +{\left (2 \, a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x^{4} + a^{4}\right )}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{4 \,{\left (2 \, b^{3} p^{3} + 9 \, b^{3} p^{2} + 13 \, b^{3} p + 6 \, 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 [B] time = 1.13106, size = 447, normalized size = 3.49 \begin{align*} \frac{2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} b^{4} p^{2} x^{8} + 5 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} b^{4} p x^{8} + 4 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a b^{3} p^{2} x^{6} + 3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} b^{4} x^{8} + 8 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a b^{3} p x^{6} + 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{2} b^{2} p^{2} x^{4} + 4 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a b^{3} x^{6} +{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{2} b^{2} p x^{4} - 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{3} b p x^{2} +{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{4}}{4 \,{\left (2 \, b^{3} p^{3} + 9 \, b^{3} p^{2} + 13 \, b^{3} p + 6 \, b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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