Optimal. Leaf size=33 \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{6} b B x^6 \]
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Rubi [A] time = 0.0316592, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {444, 43} \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{6} b B x^6 \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x) (A+B x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a A+(A b+a B) x+b B x^2\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} a A x^2+\frac{1}{4} (A b+a B) x^4+\frac{1}{6} b B x^6\\ \end{align*}
Mathematica [A] time = 0.0071658, size = 33, normalized size = 1. \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{6} b B x^6 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 28, normalized size = 0.9 \begin{align*}{\frac{aA{x}^{2}}{2}}+{\frac{ \left ( Ab+Ba \right ){x}^{4}}{4}}+{\frac{bB{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975434, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{6} \, B b x^{6} + \frac{1}{4} \,{\left (B a + A b\right )} x^{4} + \frac{1}{2} \, A a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27144, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{6} x^{6} b B + \frac{1}{4} x^{4} a B + \frac{1}{4} x^{4} b A + \frac{1}{2} x^{2} a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.056516, size = 29, normalized size = 0.88 \begin{align*} \frac{A a x^{2}}{2} + \frac{B b x^{6}}{6} + x^{4} \left (\frac{A b}{4} + \frac{B a}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13427, size = 39, normalized size = 1.18 \begin{align*} \frac{1}{6} \, B b x^{6} + \frac{1}{4} \, B a x^{4} + \frac{1}{4} \, A b x^{4} + \frac{1}{2} \, A a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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