Optimal. Leaf size=21 \[ -\frac{1}{3 a x^3 \left (a+\frac{b}{x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.0061545, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{1}{3 a x^3 \left (a+\frac{b}{x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^4} \, dx &=-\frac{1}{3 a \left (a+\frac{b}{x^2}\right )^{3/2} x^3}\\ \end{align*}
Mathematica [A] time = 0.0059178, size = 28, normalized size = 1.33 \[ -\frac{a x^2+b}{3 a x^5 \left (a+\frac{b}{x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 1.4 \begin{align*} -{\frac{a{x}^{2}+b}{3\,a{x}^{5}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968092, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70378, size = 84, normalized size = 4. \begin{align*} -\frac{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{3} x^{4} + 2 \, a^{2} b x^{2} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.03345, size = 48, normalized size = 2.29 \begin{align*} - \frac{1}{3 a^{2} \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1} + 3 a b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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