Optimal. Leaf size=350 \[ \frac{4 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{8 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{5 b \sqrt [3]{x}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x^2}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{5 x} \]
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Rubi [A] time = 0.431339, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {2018, 2020, 2025, 2032, 329, 305, 220, 1196} \[ \frac{4 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{8 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{5 b \sqrt [3]{x}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x^2}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{5 x} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 2020
Rule 2025
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^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (b x+a x^3\right )^{3/2}}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+(2 a) \operatorname{Subst}\left (\int \frac{\sqrt{b x+a x^3}}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{5 x}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac{1}{5} \left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{5 x}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac{\left (4 a^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{5 b}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{5 x}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac{\left (4 a^3 \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 b \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{5 x}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac{\left (8 a^3 \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 b \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{5 x}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac{\left (8 a^{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{b} \sqrt{b \sqrt [3]{x}+a x}}-\frac{\left (8 a^{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{b} \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{8 a^{5/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 b \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{b \sqrt [3]{x}+a x}}-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{5 x}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}-\frac{8 a^{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 b^{3/4} \sqrt{b \sqrt [3]{x}+a x}}+\frac{4 a^{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 b^{3/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0529985, size = 62, normalized size = 0.18 \[ -\frac{2 b \sqrt{a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac{9}{4},-\frac{3}{2};-\frac{5}{4};-\frac{a x^{2/3}}{b}\right )}{3 x^{5/3} \sqrt{\frac{a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 339, normalized size = 1. \begin{align*} -{\frac{2}{15\,b{x}^{3}} \left ( -12\,{a}^{2}b\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}}}}{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +6\,{a}^{2}b\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}}}}{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +12\,{x}^{10/3}\sqrt{b\sqrt [3]{x}+ax}{a}^{3}+12\,{x}^{8/3}\sqrt{b\sqrt [3]{x}+ax}{a}^{2}b+16\,{x}^{2}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }a{b}^{2}+11\,{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{2}b+5\,{x}^{4/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{b}^{3} \right ) \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \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^{3}}\,{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^{3}}, 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^{3}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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