Optimal. Leaf size=221 \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}} \]
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Rubi [A] time = 0.0624405, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {335, 218} \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 335
Rule 218
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^3}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}+\frac{b^{2/3}}{x^2}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.011577, size = 49, normalized size = 0.22 \[ \frac{2 \sqrt{\frac{a x^3}{b}+1} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{a x^3}{b}\right )}{x \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 437, normalized size = 2. \begin{align*} -4\,{\frac{ \left ( a{x}^{3}+b \right ) \left ( i\sqrt{3}{x}^{2}{a}^{2}-2\,i\sqrt [3]{-b{a}^{2}}\sqrt{3}xa+i \left ( -b{a}^{2} \right ) ^{2/3}\sqrt{3}-{a}^{2}{x}^{2}+2\,\sqrt [3]{-b{a}^{2}}xa- \left ( -b{a}^{2} \right ) ^{2/3} \right ) }{\sqrt [3]{-b{a}^{2}}xa\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 ) }}}{\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 ){\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x^{3}}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x \sqrt{\frac{a x^{3} + b}{x^{3}}}}{a x^{3} + b}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.41462, size = 37, normalized size = 0.17 \begin{align*} - \frac{\Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt{a} x \Gamma \left (\frac{4}{3}\right )} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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