Optimal. Leaf size=18 \[ \frac{2}{3 b \sqrt{a+\frac{b}{x^3}}} \]
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Rubi [A] time = 0.0063507, antiderivative size = 18, 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 = {261} \[ \frac{2}{3 b \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x^4} \, dx &=\frac{2}{3 b \sqrt{a+\frac{b}{x^3}}}\\ \end{align*}
Mathematica [A] time = 0.0067539, size = 18, normalized size = 1. \[ \frac{2}{3 b \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 1.6 \begin{align*}{\frac{2\,a{x}^{3}+2\,b}{3\,b{x}^{3}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982481, size = 19, normalized size = 1.06 \begin{align*} \frac{2}{3 \, \sqrt{a + \frac{b}{x^{3}}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.4839, size = 63, normalized size = 3.5 \begin{align*} \frac{2 \, x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{3 \,{\left (a b x^{3} + b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.84906, size = 27, normalized size = 1.5 \begin{align*} \begin{cases} \frac{2}{3 b \sqrt{a + \frac{b}{x^{3}}}} & \text{for}\: b \neq 0 \\- \frac{1}{3 a^{\frac{3}{2}} x^{3}} & \text{otherwise} \end{cases} \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^{3}}\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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