Optimal. Leaf size=27 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
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Rubi [A] time = 0.0176971, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 208} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^3}} x} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^3}}\right )}{3 b}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 \sqrt{a}}\\ \end{align*}
Mathematica [B] time = 0.0155124, size = 59, normalized size = 2.19 \[ \frac{2 \sqrt{a x^3+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )}{3 \sqrt{a} x^{3/2} \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.008, size = 480, normalized size = 17.8 \begin{align*} -4\,{\frac{ \left ( a{x}^{3}+b \right ) \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) ^{2}}{{a}^{2}x\sqrt{x \left ( a{x}^{3}+b \right ) } \left ( i\sqrt{3}-3 \right ) }\sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-b{a}^{2}}+2\,ax+\sqrt [3]{-b{a}^{2}}}{ \left ( 1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-b{a}^{2}}-2\,ax-\sqrt [3]{-b{a}^{2}}}{ \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}} \left ({\it EllipticF} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( -1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }}} \right ) -{\it EllipticPi} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( -1+i\sqrt{3} \right ) \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) }}},{\frac{-1+i\sqrt{3}}{i\sqrt{3}-3}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( -1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }}} \right ) \right ){\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}{\frac{1}{\sqrt{{\frac{x \left ( -ax+\sqrt [3]{-b{a}^{2}} \right ) \left ( i\sqrt{3}\sqrt [3]{-b{a}^{2}}+2\,ax+\sqrt [3]{-b{a}^{2}} \right ) \left ( i\sqrt{3}\sqrt [3]{-b{a}^{2}}-2\,ax-\sqrt [3]{-b{a}^{2}} \right ) }{{a}^{2}}}}}}} \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 [B] time = 2.35428, size = 242, normalized size = 8.96 \begin{align*} \left [\frac{\log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \,{\left (2 \, a x^{6} + b x^{3}\right )} \sqrt{a} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right )}{6 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right )}{3 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.26425, size = 24, normalized size = 0.89 \begin{align*} \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{3 \sqrt{a}} \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}{\sqrt{a + \frac{b}{x^{3}}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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