Optimal. Leaf size=26 \[ -\frac{a B+A b}{x}-\frac{a A}{3 x^3}+b B x \]
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Rubi [A] time = 0.015107, 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} \[ -\frac{a B+A b}{x}-\frac{a A}{3 x^3}+b B x \]
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^4} \, dx &=\int \left (b B+\frac{a A}{x^4}+\frac{A b+a B}{x^2}\right ) \, dx\\ &=-\frac{a A}{3 x^3}-\frac{A b+a B}{x}+b B x\\ \end{align*}
Mathematica [A] time = 0.0124445, size = 27, normalized size = 1.04 \[ \frac{-a B-A b}{x}-\frac{a A}{3 x^3}+b B x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 1. \begin{align*} bBx-{\frac{Aa}{3\,{x}^{3}}}-{\frac{Ab+Ba}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986408, size = 35, normalized size = 1.35 \begin{align*} B b x - \frac{3 \,{\left (B a + A b\right )} x^{2} + A a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43557, size = 63, normalized size = 2.42 \begin{align*} \frac{3 \, B b x^{4} - 3 \,{\left (B a + A b\right )} x^{2} - A a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.332068, size = 26, normalized size = 1. \begin{align*} B b x - \frac{A a + x^{2} \left (3 A b + 3 B a\right )}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09227, size = 38, normalized size = 1.46 \begin{align*} B b x - \frac{3 \, B a x^{2} + 3 \, A b x^{2} + A a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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