Optimal. Leaf size=388 \[ -\frac{77 a^{13/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{65 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{154 a^{7/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{65 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{154 a^{13/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{154 a^3 \sqrt{a x+b \sqrt [3]{x}}}{65 b^4 \sqrt [3]{x}}-\frac{154 a^2 \sqrt{a x+b \sqrt [3]{x}}}{195 b^3 x}+\frac{22 a \sqrt{a x+b \sqrt [3]{x}}}{39 b^2 x^{5/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{13 b x^{7/3}} \]
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Rubi [A] time = 0.475911, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {2018, 2025, 2032, 329, 305, 220, 1196} \[ -\frac{154 a^{7/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{65 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{77 a^{13/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{154 a^{13/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{65 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{154 a^3 \sqrt{a x+b \sqrt [3]{x}}}{65 b^4 \sqrt [3]{x}}-\frac{154 a^2 \sqrt{a x+b \sqrt [3]{x}}}{195 b^3 x}+\frac{22 a \sqrt{a x+b \sqrt [3]{x}}}{39 b^2 x^{5/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{13 b x^{7/3}} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^7 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{13 b x^{7/3}}-\frac{(33 a) \operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{13 b}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac{22 a \sqrt{b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}+\frac{\left (77 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{39 b^2}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac{22 a \sqrt{b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac{154 a^2 \sqrt{b \sqrt [3]{x}+a x}}{195 b^3 x}-\frac{\left (77 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{65 b^3}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac{22 a \sqrt{b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac{154 a^2 \sqrt{b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac{154 a^3 \sqrt{b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}-\frac{\left (77 a^4\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{65 b^4}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac{22 a \sqrt{b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac{154 a^2 \sqrt{b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac{154 a^3 \sqrt{b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}-\frac{\left (77 a^4 \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 )}{65 b^4 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac{22 a \sqrt{b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac{154 a^2 \sqrt{b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac{154 a^3 \sqrt{b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}-\frac{\left (154 a^4 \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 )}{65 b^4 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac{22 a \sqrt{b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac{154 a^2 \sqrt{b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac{154 a^3 \sqrt{b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}-\frac{\left (154 a^{7/2} \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 )}{65 b^{7/2} \sqrt{b \sqrt [3]{x}+a x}}+\frac{\left (154 a^{7/2} \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 )}{65 b^{7/2} \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{154 a^{7/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{65 b^4 \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{b \sqrt [3]{x}+a x}}-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac{22 a \sqrt{b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac{154 a^2 \sqrt{b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac{154 a^3 \sqrt{b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}+\frac{154 a^{13/4} \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 )}{65 b^{15/4} \sqrt{b \sqrt [3]{x}+a x}}-\frac{77 a^{13/4} \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 )}{65 b^{15/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.049712, size = 59, normalized size = 0.15 \[ -\frac{6 \sqrt{\frac{a x^{2/3}}{b}+1} \, _2F_1\left (-\frac{13}{4},\frac{1}{2};-\frac{9}{4};-\frac{a x^{2/3}}{b}\right )}{13 x^2 \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 363, normalized size = 0.9 \begin{align*} -{\frac{1}{195\,{b}^{4}} \left ( 462\,{a}^{3}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}^{10/3}\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 ) -231\,{a}^{3}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}^{10/3}\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 ) -462\,{x}^{10/3}\sqrt{b\sqrt [3]{x}+ax}{a}^{3}b+44\,{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{2}{b}^{2}+154\,{x}^{10/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{3}b-20\,{x}^{2}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }a{b}^{3}-462\,{x}^{4}\sqrt{b\sqrt [3]{x}+ax}{a}^{4}+90\,{x}^{4/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{b}^{4} \right ){x}^{-{\frac{11}{3}}} \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{1}{\sqrt{a x + b x^{\frac{1}{3}}} 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^{2} x^{2} - a b x^{\frac{4}{3}} + b^{2} x^{\frac{2}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}{a^{3} x^{6} + b^{3} x^{4}}, 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{1}{x^{3} \sqrt{a x + b \sqrt [3]{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{1}{\sqrt{a x + b x^{\frac{1}{3}}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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