Optimal. Leaf size=19 \[ \frac{x^4}{4 b \left (a x^2+b\right )^2} \]
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Rubi [A] time = 0.0054465, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 264} \[ \frac{x^4}{4 b \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{\left (\frac{b}{x}+a x\right )^3} \, dx &=\int \frac{x^3}{\left (b+a x^2\right )^3} \, dx\\ &=\frac{x^4}{4 b \left (b+a x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0084775, size = 24, normalized size = 1.26 \[ -\frac{2 a x^2+b}{4 a^2 \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 31, normalized size = 1.6 \begin{align*}{\frac{b}{4\,{a}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{1}{2\,{a}^{2} \left ( a{x}^{2}+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15915, size = 49, normalized size = 2.58 \begin{align*} -\frac{2 \, a x^{2} + b}{4 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.784184, size = 73, normalized size = 3.84 \begin{align*} -\frac{2 \, a x^{2} + b}{4 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.482015, size = 36, normalized size = 1.89 \begin{align*} - \frac{2 a x^{2} + b}{4 a^{4} x^{4} + 8 a^{3} b x^{2} + 4 a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15051, size = 30, normalized size = 1.58 \begin{align*} -\frac{2 \, a x^{2} + b}{4 \,{\left (a x^{2} + b\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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