Optimal. Leaf size=28 \[ \frac{1}{3} x^3 (A c+b B)+A b x+\frac{1}{5} B c x^5 \]
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Rubi [A] time = 0.018882, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1584, 373} \[ \frac{1}{3} x^3 (A c+b B)+A b x+\frac{1}{5} B c x^5 \]
Antiderivative was successfully verified.
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Rule 1584
Rule 373
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^2} \, dx &=\int \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b+(b B+A c) x^2+B c x^4\right ) \, dx\\ &=A b x+\frac{1}{3} (b B+A c) x^3+\frac{1}{5} B c x^5\\ \end{align*}
Mathematica [A] time = 0.0051632, size = 28, normalized size = 1. \[ \frac{1}{3} x^3 (A c+b B)+A b x+\frac{1}{5} B c x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 0.9 \begin{align*} Abx+{\frac{ \left ( Ac+Bb \right ){x}^{3}}{3}}+{\frac{Bc{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11203, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{5} \, B c x^{5} + \frac{1}{3} \,{\left (B b + A c\right )} x^{3} + A b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.476049, size = 58, normalized size = 2.07 \begin{align*} \frac{1}{5} \, B c x^{5} + \frac{1}{3} \,{\left (B b + A c\right )} x^{3} + A b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.05726, size = 26, normalized size = 0.93 \begin{align*} A b x + \frac{B c x^{5}}{5} + x^{3} \left (\frac{A c}{3} + \frac{B b}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12314, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{5} \, B c x^{5} + \frac{1}{3} \, B b x^{3} + \frac{1}{3} \, A c x^{3} + A b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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