Optimal. Leaf size=34 \[ \frac{\left (a+b x^2\right )^6}{12 b^2}-\frac{a \left (a+b x^2\right )^5}{10 b^2} \]
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Rubi [A] time = 0.0395182, antiderivative size = 34, 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} \[ \frac{\left (a+b x^2\right )^6}{12 b^2}-\frac{a \left (a+b x^2\right )^5}{10 b^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac{\int x^3 \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int x \left (a b+b^2 x\right )^4 \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a \left (a b+b^2 x\right )^4}{b}+\frac{\left (a b+b^2 x\right )^5}{b^2}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac{a \left (a+b x^2\right )^5}{10 b^2}+\frac{\left (a+b x^2\right )^6}{12 b^2}\\ \end{align*}
Mathematica [A] time = 0.0021975, size = 56, normalized size = 1.65 \[ \frac{3}{4} a^2 b^2 x^8+\frac{2}{3} a^3 b x^6+\frac{a^4 x^4}{4}+\frac{2}{5} a b^3 x^{10}+\frac{b^4 x^{12}}{12} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 47, normalized size = 1.4 \begin{align*}{\frac{{b}^{4}{x}^{12}}{12}}+{\frac{2\,a{b}^{3}{x}^{10}}{5}}+{\frac{3\,{a}^{2}{b}^{2}{x}^{8}}{4}}+{\frac{2\,{a}^{3}b{x}^{6}}{3}}+{\frac{{a}^{4}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994285, size = 62, normalized size = 1.82 \begin{align*} \frac{1}{12} \, b^{4} x^{12} + \frac{2}{5} \, a b^{3} x^{10} + \frac{3}{4} \, a^{2} b^{2} x^{8} + \frac{2}{3} \, a^{3} b x^{6} + \frac{1}{4} \, a^{4} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46452, size = 108, normalized size = 3.18 \begin{align*} \frac{1}{12} x^{12} b^{4} + \frac{2}{5} x^{10} b^{3} a + \frac{3}{4} x^{8} b^{2} a^{2} + \frac{2}{3} x^{6} b a^{3} + \frac{1}{4} x^{4} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.074655, size = 53, normalized size = 1.56 \begin{align*} \frac{a^{4} x^{4}}{4} + \frac{2 a^{3} b x^{6}}{3} + \frac{3 a^{2} b^{2} x^{8}}{4} + \frac{2 a b^{3} x^{10}}{5} + \frac{b^{4} x^{12}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13718, size = 62, normalized size = 1.82 \begin{align*} \frac{1}{12} \, b^{4} x^{12} + \frac{2}{5} \, a b^{3} x^{10} + \frac{3}{4} \, a^{2} b^{2} x^{8} + \frac{2}{3} \, a^{3} b x^{6} + \frac{1}{4} \, a^{4} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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