Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^5}{10 b} \]
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Rubi [A] time = 0.0048408, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {28, 261} \[ \frac{\left (a+b x^2\right )^5}{10 b} \]
Antiderivative was successfully verified.
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Rule 28
Rule 261
Rubi steps
\begin{align*} \int x \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac{\int x \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac{\left (a+b x^2\right )^5}{10 b}\\ \end{align*}
Mathematica [A] time = 0.0022879, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^5}{10 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 45, normalized size = 2.8 \begin{align*}{\frac{{b}^{4}{x}^{10}}{10}}+{\frac{a{b}^{3}{x}^{8}}{2}}+{a}^{2}{b}^{2}{x}^{6}+{a}^{3}b{x}^{4}+{\frac{{a}^{4}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.982767, size = 59, normalized size = 3.69 \begin{align*} \frac{1}{10} \, b^{4} x^{10} + \frac{1}{2} \, a b^{3} x^{8} + a^{2} b^{2} x^{6} + a^{3} b x^{4} + \frac{1}{2} \, a^{4} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.466, size = 96, normalized size = 6. \begin{align*} \frac{1}{10} x^{10} b^{4} + \frac{1}{2} x^{8} b^{3} a + x^{6} b^{2} a^{2} + x^{4} b a^{3} + \frac{1}{2} x^{2} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.072371, size = 44, normalized size = 2.75 \begin{align*} \frac{a^{4} x^{2}}{2} + a^{3} b x^{4} + a^{2} b^{2} x^{6} + \frac{a b^{3} x^{8}}{2} + \frac{b^{4} x^{10}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14335, size = 59, normalized size = 3.69 \begin{align*} \frac{1}{10} \, b^{4} x^{10} + \frac{1}{2} \, a b^{3} x^{8} + a^{2} b^{2} x^{6} + a^{3} b x^{4} + \frac{1}{2} \, a^{4} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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