Optimal. Leaf size=239 \[ \frac{663 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{154 a^{21/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{1989 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^4}-\frac{663 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^5}-\frac{221 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a^3}+\frac{17 x^2 \sqrt{a x+b \sqrt [3]{x}}}{5 a^2}-\frac{3 x^3}{a \sqrt{a x+b \sqrt [3]{x}}} \]
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Rubi [A] time = 0.377479, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2018, 2022, 2024, 2011, 329, 220} \[ \frac{1989 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^4}+\frac{663 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{154 a^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{663 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^5}-\frac{221 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a^3}+\frac{17 x^2 \sqrt{a x+b \sqrt [3]{x}}}{5 a^2}-\frac{3 x^3}{a \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 2022
Rule 2024
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^{11}}{\left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 x^3}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{51 \operatorname{Subst}\left (\int \frac{x^8}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 a}\\ &=-\frac{3 x^3}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{17 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a^2}-\frac{(221 b) \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 a^2}\\ &=-\frac{3 x^3}{a \sqrt{b \sqrt [3]{x}+a x}}-\frac{221 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^3}+\frac{17 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a^2}+\frac{\left (1989 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{110 a^3}\\ &=-\frac{3 x^3}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{1989 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^4}-\frac{221 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^3}+\frac{17 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a^2}-\frac{\left (1989 b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 a^4}\\ &=-\frac{3 x^3}{a \sqrt{b \sqrt [3]{x}+a x}}-\frac{663 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^5}+\frac{1989 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^4}-\frac{221 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^3}+\frac{17 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a^2}+\frac{\left (663 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 a^5}\\ &=-\frac{3 x^3}{a \sqrt{b \sqrt [3]{x}+a x}}-\frac{663 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^5}+\frac{1989 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^4}-\frac{221 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^3}+\frac{17 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a^2}+\frac{\left (663 b^4 \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 )}{154 a^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{3 x^3}{a \sqrt{b \sqrt [3]{x}+a x}}-\frac{663 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^5}+\frac{1989 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^4}-\frac{221 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^3}+\frac{17 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a^2}+\frac{\left (663 b^4 \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 )}{77 a^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{3 x^3}{a \sqrt{b \sqrt [3]{x}+a x}}-\frac{663 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^5}+\frac{1989 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^4}-\frac{221 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^3}+\frac{17 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a^2}+\frac{663 b^{15/4} \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 )}{154 a^{21/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0826685, size = 124, normalized size = 0.52 \[ \frac{\sqrt{a x+b \sqrt [3]{x}} \left (442 a^2 b^2 x^{4/3}-238 a^3 b x^2+154 a^4 x^{8/3}+3315 b^4 \sqrt{\frac{a x^{2/3}}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{a x^{2/3}}{b}\right )-1326 a b^3 x^{2/3}-3315 b^4\right )}{385 a^5 \left (a x^{2/3}+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 261, normalized size = 1.1 \begin{align*}{\frac{1}{770\,{a}^{6}} \left ( 3315\,\sqrt{-ab}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }\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 ){b}^{4}+884\,{x}^{5/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{3}{b}^{2}-476\,{x}^{7/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{4}b-2652\,x\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{2}{b}^{3}+308\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{3}{a}^{5}-4320\,\sqrt [3]{x}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }a{b}^{4}-2310\,\sqrt [3]{x}\sqrt{b\sqrt [3]{x}+ax}a{b}^{4} \right ){\frac{1}{\sqrt [3]{x}}} \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{x^{3}}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{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^{4} x^{5} + 3 \, a^{2} b^{2} x^{\frac{11}{3}} - 2 \, a b^{3} x^{3} -{\left (2 \, a^{3} b x^{4} - b^{4} x^{2}\right )} x^{\frac{1}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}{a^{6} x^{4} + 2 \, a^{3} b^{3} x^{2} + b^{6}}, 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{x^{3}}{\left (a x + b \sqrt [3]{x}\right )^{\frac{3}{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{x^{3}}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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