Optimal. Leaf size=47 \[ -\frac{\sqrt{a+b x^3}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
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Rubi [A] time = 0.0266854, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ -\frac{\sqrt{a+b x^3}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^3}}{x^4} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3}}{3 x^3}+\frac{1}{6} b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3}}{3 x^3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )\\ &=-\frac{\sqrt{a+b x^3}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0315483, size = 59, normalized size = 1.26 \[ -\frac{b x^3 \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+a+b x^3}{3 x^3 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 36, normalized size = 0.8 \begin{align*} -{\frac{b}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}-{\frac{1}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56332, size = 258, normalized size = 5.49 \begin{align*} \left [\frac{\sqrt{a} b x^{3} \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) - 2 \, \sqrt{b x^{3} + a} a}{6 \, a x^{3}}, \frac{\sqrt{-a} b x^{3} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) - \sqrt{b x^{3} + a} a}{3 \, a x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.17346, size = 49, normalized size = 1.04 \begin{align*} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11195, size = 58, normalized size = 1.23 \begin{align*} \frac{1}{3} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x^{3} + a}}{b x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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