Optimal. Leaf size=144 \[ \frac{4 a^{3/4} b^{3/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{\sqrt{a x+b \sqrt [3]{x}}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{x}+4 a \sqrt{a x+b \sqrt [3]{x}} \]
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Rubi [A] time = 0.20206, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2018, 2020, 2021, 2011, 329, 220} \[ \frac{4 a^{3/4} b^{3/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{\sqrt{a x+b \sqrt [3]{x}}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{x}+4 a \sqrt{a x+b \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 2020
Rule 2021
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (b x+a x^3\right )^{3/2}}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+(6 a) \operatorname{Subst}\left (\int \frac{\sqrt{b x+a x^3}}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=4 a \sqrt{b \sqrt [3]{x}+a x}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+(4 a b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=4 a \sqrt{b \sqrt [3]{x}+a x}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+\frac{\left (4 a b \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{b \sqrt [3]{x}+a x}}\\ &=4 a \sqrt{b \sqrt [3]{x}+a x}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+\frac{\left (8 a b \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{\sqrt{b \sqrt [3]{x}+a x}}\\ &=4 a \sqrt{b \sqrt [3]{x}+a x}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+\frac{4 a^{3/4} b^{3/4} \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{\frac{b+a x^{2/3}}{\left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{\sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0582228, size = 60, normalized size = 0.42 \[ -\frac{2 b \sqrt{a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{a x^{2/3}}{b}\right )}{x^{2/3} \sqrt{\frac{a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 130, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{\sqrt [3]{x}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }} \left ( 2\,\sqrt [3]{x}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}b+{x}^{4/3}{a}^{2}-{b}^{2} \right ) } \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{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{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{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}{x^{2}}\, dx \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{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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