Optimal. Leaf size=48 \[ -\frac{a^2 A}{3 x^3}+b x (2 a B+A b)-\frac{a (a B+2 A b)}{x}+\frac{1}{3} b^2 B x^3 \]
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Rubi [A] time = 0.029933, 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} \[ -\frac{a^2 A}{3 x^3}+b x (2 a B+A b)-\frac{a (a B+2 A b)}{x}+\frac{1}{3} b^2 B x^3 \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^4} \, dx &=\int \left (b (A b+2 a B)+\frac{a^2 A}{x^4}+\frac{a (2 A b+a B)}{x^2}+b^2 B x^2\right ) \, dx\\ &=-\frac{a^2 A}{3 x^3}-\frac{a (2 A b+a B)}{x}+b (A b+2 a B) x+\frac{1}{3} b^2 B x^3\\ \end{align*}
Mathematica [A] time = 0.0177868, size = 50, normalized size = 1.04 \[ \frac{a^2 (-B)-2 a A b}{x}-\frac{a^2 A}{3 x^3}+b x (2 a B+A b)+\frac{1}{3} b^2 B x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 46, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{3}}{3}}+{b}^{2}Ax+2\,abBx-{\frac{A{a}^{2}}{3\,{x}^{3}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96608, size = 68, normalized size = 1.42 \begin{align*} \frac{1}{3} \, B b^{2} x^{3} +{\left (2 \, B a b + A b^{2}\right )} x - \frac{A a^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42695, size = 109, normalized size = 2.27 \begin{align*} \frac{B b^{2} x^{6} + 3 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - A a^{2} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.379725, size = 49, normalized size = 1.02 \begin{align*} \frac{B b^{2} x^{3}}{3} + x \left (A b^{2} + 2 B a b\right ) - \frac{A a^{2} + x^{2} \left (6 A a b + 3 B a^{2}\right )}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12623, size = 68, normalized size = 1.42 \begin{align*} \frac{1}{3} \, B b^{2} x^{3} + 2 \, B a b x + A b^{2} x - \frac{3 \, B a^{2} x^{2} + 6 \, A a b x^{2} + A a^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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