Optimal. Leaf size=26 \[ x (a B+A b)-\frac{a A}{x}+\frac{1}{3} b B x^3 \]
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Rubi [A] time = 0.0164823, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {448} \[ x (a B+A b)-\frac{a A}{x}+\frac{1}{3} b B x^3 \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (A+B x^2\right )}{x^2} \, dx &=\int \left (A b \left (1+\frac{a B}{A b}\right )+\frac{a A}{x^2}+b B x^2\right ) \, dx\\ &=-\frac{a A}{x}+(A b+a B) x+\frac{1}{3} b B x^3\\ \end{align*}
Mathematica [A] time = 0.0088594, size = 26, normalized size = 1. \[ x (a B+A b)-\frac{a A}{x}+\frac{1}{3} b B x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 0.9 \begin{align*}{\frac{bB{x}^{3}}{3}}+Abx+Bax-{\frac{Aa}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00229, size = 32, normalized size = 1.23 \begin{align*} \frac{1}{3} \, B b x^{3} +{\left (B a + A b\right )} x - \frac{A a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43145, size = 61, normalized size = 2.35 \begin{align*} \frac{B b x^{4} + 3 \,{\left (B a + A b\right )} x^{2} - 3 \, A a}{3 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.245351, size = 20, normalized size = 0.77 \begin{align*} - \frac{A a}{x} + \frac{B b x^{3}}{3} + x \left (A b + B a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18391, size = 31, normalized size = 1.19 \begin{align*} \frac{1}{3} \, B b x^{3} + B a x + A b x - \frac{A a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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