Optimal. Leaf size=21 \[ -\frac{1}{2 a x^2 \sqrt{a+\frac{b}{x^4}}} \]
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Rubi [A] time = 0.0066597, 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} \[ -\frac{1}{2 a x^2 \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^4}\right )^{3/2} x^3} \, dx &=-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}} x^2}\\ \end{align*}
Mathematica [A] time = 0.0065657, size = 21, normalized size = 1. \[ -\frac{1}{2 a x^2 \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 1.4 \begin{align*} -{\frac{a{x}^{4}+b}{2\,{x}^{6}a} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960122, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{2 \, \sqrt{a + \frac{b}{x^{4}}} a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46497, size = 65, normalized size = 3.1 \begin{align*} -\frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{2 \,{\left (a^{2} x^{4} + a b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.11494, size = 22, normalized size = 1.05 \begin{align*} - \frac{1}{2 a \sqrt{b} \sqrt{\frac{a x^{4}}{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^{4}}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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