Optimal. Leaf size=319 \[ \frac{4 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{8 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 \sqrt{a} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{4}{5} b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}+\frac{2}{3} \left (a x+b \sqrt [3]{x}\right )^{3/2} \]
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Rubi [A] time = 0.32934, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {2018, 2021, 2004, 2032, 329, 305, 220, 1196} \[ \frac{4 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 b^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{8 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 \sqrt{a} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{4}{5} b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}+\frac{2}{3} \left (a x+b \sqrt [3]{x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 2021
Rule 2004
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (b x+a x^3\right )^{3/2}}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{3} \left (b \sqrt [3]{x}+a x\right )^{3/2}+(2 b) \operatorname{Subst}\left (\int \sqrt{b x+a x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4}{5} b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{3} \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{1}{5} \left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4}{5} b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{3} \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{\left (4 b^2 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{4}{5} b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{3} \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{\left (8 b^2 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{4}{5} b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{3} \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{\left (8 b^{5/2} \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 )}{5 \sqrt{a} \sqrt{b \sqrt [3]{x}+a x}}-\frac{\left (8 b^{5/2} \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{a} x^2}{\sqrt{b}}}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 \sqrt{a} \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{8 b^2 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 \sqrt{a} \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{b \sqrt [3]{x}+a x}}+\frac{4}{5} b \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{3} \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac{8 b^{9/4} \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} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{b \sqrt [3]{x}+a x}}+\frac{4 b^{9/4} \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 )}{5 a^{3/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0457798, size = 60, normalized size = 0.19 \[ \frac{2 b \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{a x^{2/3}}{b}\right )}{\sqrt{\frac{a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 230, normalized size = 0.7 \begin{align*}{\frac{2}{15\,a} \left ( 12\,{b}^{3}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -6\,{b}^{3}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\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 ) +11\,{x}^{2/3}a{b}^{2}+16\,{x}^{4/3}{a}^{2}b+5\,{x}^{2}{a}^{3} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \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}\,{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}, 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}\, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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