Optimal. Leaf size=48 \[ -\frac{a^2 A}{x}+\frac{1}{3} b x^3 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]
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Rubi [A] time = 0.026731, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ -\frac{a^2 A}{x}+\frac{1}{3} b x^3 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^2} \, dx &=\int \left (a (2 A b+a B)+\frac{a^2 A}{x^2}+b (A b+2 a B) x^2+b^2 B x^4\right ) \, dx\\ &=-\frac{a^2 A}{x}+a (2 A b+a B) x+\frac{1}{3} b (A b+2 a B) x^3+\frac{1}{5} b^2 B x^5\\ \end{align*}
Mathematica [A] time = 0.0154579, size = 48, normalized size = 1. \[ -\frac{a^2 A}{x}+\frac{1}{3} b x^3 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 49, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{5}}{5}}+{\frac{A{x}^{3}{b}^{2}}{3}}+{\frac{2\,B{x}^{3}ab}{3}}+2\,abAx+{a}^{2}Bx-{\frac{A{a}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975647, size = 65, normalized size = 1.35 \begin{align*} \frac{1}{5} \, B b^{2} x^{5} + \frac{1}{3} \,{\left (2 \, B a b + A b^{2}\right )} x^{3} - \frac{A a^{2}}{x} +{\left (B a^{2} + 2 \, A a b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42518, size = 116, normalized size = 2.42 \begin{align*} \frac{3 \, B b^{2} x^{6} + 5 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 15 \, A a^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{15 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.275702, size = 48, normalized size = 1. \begin{align*} - \frac{A a^{2}}{x} + \frac{B b^{2} x^{5}}{5} + x^{3} \left (\frac{A b^{2}}{3} + \frac{2 B a b}{3}\right ) + x \left (2 A a b + B a^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13251, size = 65, normalized size = 1.35 \begin{align*} \frac{1}{5} \, B b^{2} x^{5} + \frac{2}{3} \, B a b x^{3} + \frac{1}{3} \, A b^{2} x^{3} + B a^{2} x + 2 \, A a b x - \frac{A a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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