Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^2\right )^5}{10 b^3}+\frac{\left (a+b x^2\right )^7}{14 b^3}-\frac{a \left (a+b x^2\right )^6}{6 b^3} \]
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Rubi [A] time = 0.070219, antiderivative size = 53, 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 = {28, 266, 43} \[ \frac{a^2 \left (a+b x^2\right )^5}{10 b^3}+\frac{\left (a+b x^2\right )^7}{14 b^3}-\frac{a \left (a+b x^2\right )^6}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac{\int x^5 \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \left (a b+b^2 x\right )^4 \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2 \left (a b+b^2 x\right )^4}{b^2}-\frac{2 a \left (a b+b^2 x\right )^5}{b^3}+\frac{\left (a b+b^2 x\right )^6}{b^4}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{a^2 \left (a+b x^2\right )^5}{10 b^3}-\frac{a \left (a+b x^2\right )^6}{6 b^3}+\frac{\left (a+b x^2\right )^7}{14 b^3}\\ \end{align*}
Mathematica [A] time = 0.002353, size = 56, normalized size = 1.06 \[ \frac{3}{5} a^2 b^2 x^{10}+\frac{1}{2} a^3 b x^8+\frac{a^4 x^6}{6}+\frac{1}{3} a b^3 x^{12}+\frac{b^4 x^{14}}{14} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 47, normalized size = 0.9 \begin{align*}{\frac{{b}^{4}{x}^{14}}{14}}+{\frac{a{b}^{3}{x}^{12}}{3}}+{\frac{3\,{a}^{2}{b}^{2}{x}^{10}}{5}}+{\frac{{a}^{3}b{x}^{8}}{2}}+{\frac{{a}^{4}{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986442, size = 62, normalized size = 1.17 \begin{align*} \frac{1}{14} \, b^{4} x^{14} + \frac{1}{3} \, a b^{3} x^{12} + \frac{3}{5} \, a^{2} b^{2} x^{10} + \frac{1}{2} \, a^{3} b x^{8} + \frac{1}{6} \, a^{4} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24881, size = 109, normalized size = 2.06 \begin{align*} \frac{1}{14} x^{14} b^{4} + \frac{1}{3} x^{12} b^{3} a + \frac{3}{5} x^{10} b^{2} a^{2} + \frac{1}{2} x^{8} b a^{3} + \frac{1}{6} x^{6} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.073926, size = 49, normalized size = 0.92 \begin{align*} \frac{a^{4} x^{6}}{6} + \frac{a^{3} b x^{8}}{2} + \frac{3 a^{2} b^{2} x^{10}}{5} + \frac{a b^{3} x^{12}}{3} + \frac{b^{4} x^{14}}{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15813, size = 62, normalized size = 1.17 \begin{align*} \frac{1}{14} \, b^{4} x^{14} + \frac{1}{3} \, a b^{3} x^{12} + \frac{3}{5} \, a^{2} b^{2} x^{10} + \frac{1}{2} \, a^{3} b x^{8} + \frac{1}{6} \, a^{4} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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