Optimal. Leaf size=32 \[ -\frac{\left (A+B x^2\right )^2}{4 \left (a+b x^2\right )^2 (A b-a B)} \]
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Rubi [A] time = 0.0195665, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {444, 37} \[ -\frac{\left (A+B x^2\right )^2}{4 \left (a+b x^2\right )^2 (A b-a B)} \]
Antiderivative was successfully verified.
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Rule 444
Rule 37
Rubi steps
\begin{align*} \int \frac{x \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{(a+b x)^3} \, dx,x,x^2\right )\\ &=-\frac{\left (A+B x^2\right )^2}{4 (A b-a B) \left (a+b x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0130987, size = 30, normalized size = 0.94 \[ -\frac{B \left (a+2 b x^2\right )+A b}{4 b^2 \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 39, normalized size = 1.2 \begin{align*} -{\frac{Ab-Ba}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{B}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997769, size = 57, normalized size = 1.78 \begin{align*} -\frac{2 \, B b x^{2} + B a + A b}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18997, size = 86, normalized size = 2.69 \begin{align*} -\frac{2 \, B b x^{2} + B a + A b}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.658565, size = 42, normalized size = 1.31 \begin{align*} - \frac{A b + B a + 2 B b x^{2}}{4 a^{2} b^{2} + 8 a b^{3} x^{2} + 4 b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1122, size = 38, normalized size = 1.19 \begin{align*} -\frac{2 \, B b x^{2} + B a + A b}{4 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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