Optimal. Leaf size=55 \[ \frac{1}{4} A b^2 x^4+\frac{1}{8} c x^8 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{10} B c^2 x^{10} \]
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Rubi [A] time = 0.0676805, antiderivative size = 55, 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 = {1584, 446, 76} \[ \frac{1}{4} A b^2 x^4+\frac{1}{8} c x^8 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{10} B c^2 x^{10} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 446
Rule 76
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x} \, dx &=\int x^3 \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x (A+B x) (b+c x)^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (A b^2 x+b (b B+2 A c) x^2+c (2 b B+A c) x^3+B c^2 x^4\right ) \, dx,x,x^2\right )\\ &=\frac{1}{4} A b^2 x^4+\frac{1}{6} b (b B+2 A c) x^6+\frac{1}{8} c (2 b B+A c) x^8+\frac{1}{10} B c^2 x^{10}\\ \end{align*}
Mathematica [A] time = 0.0079181, size = 55, normalized size = 1. \[ \frac{1}{4} A b^2 x^4+\frac{1}{8} c x^8 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{10} B c^2 x^{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{10}}{10}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{8}}{8}}+{\frac{ \left ( 2\,Abc+B{b}^{2} \right ){x}^{6}}{6}}+{\frac{A{b}^{2}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11603, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{10} \, B c^{2} x^{10} + \frac{1}{8} \,{\left (2 \, B b c + A c^{2}\right )} x^{8} + \frac{1}{4} \, A b^{2} x^{4} + \frac{1}{6} \,{\left (B b^{2} + 2 \, A b c\right )} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.461019, size = 120, normalized size = 2.18 \begin{align*} \frac{1}{10} \, B c^{2} x^{10} + \frac{1}{8} \,{\left (2 \, B b c + A c^{2}\right )} x^{8} + \frac{1}{4} \, A b^{2} x^{4} + \frac{1}{6} \,{\left (B b^{2} + 2 \, A b c\right )} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.070706, size = 53, normalized size = 0.96 \begin{align*} \frac{A b^{2} x^{4}}{4} + \frac{B c^{2} x^{10}}{10} + x^{8} \left (\frac{A c^{2}}{8} + \frac{B b c}{4}\right ) + x^{6} \left (\frac{A b c}{3} + \frac{B b^{2}}{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18926, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{10} \, B c^{2} x^{10} + \frac{1}{4} \, B b c x^{8} + \frac{1}{8} \, A c^{2} x^{8} + \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{3} \, A b c x^{6} + \frac{1}{4} \, A b^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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