Optimal. Leaf size=216 \[ \frac{39 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{77 a^{17/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{234 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^3}-\frac{78 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^4}-\frac{26 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a^2}+\frac{2 x^2 \sqrt{a x+b \sqrt [3]{x}}}{5 a} \]
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Rubi [A] time = 0.318692, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2018, 2024, 2011, 329, 220} \[ \frac{234 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^3}+\frac{39 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 a^{17/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{78 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^4}-\frac{26 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a^2}+\frac{2 x^2 \sqrt{a x+b \sqrt [3]{x}}}{5 a} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 2024
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^8}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a}-\frac{(13 b) \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{5 a}\\ &=-\frac{26 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^2}+\frac{2 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a}+\frac{\left (117 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{55 a^2}\\ &=\frac{234 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^3}-\frac{26 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^2}+\frac{2 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a}-\frac{\left (117 b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 a^3}\\ &=-\frac{78 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^4}+\frac{234 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^3}-\frac{26 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^2}+\frac{2 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a}+\frac{\left (39 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 a^4}\\ &=-\frac{78 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^4}+\frac{234 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^3}-\frac{26 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^2}+\frac{2 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a}+\frac{\left (39 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 )}{77 a^4 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{78 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^4}+\frac{234 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^3}-\frac{26 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^2}+\frac{2 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a}+\frac{\left (78 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^4 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{78 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^4}+\frac{234 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a^3}-\frac{26 b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{55 a^2}+\frac{2 x^2 \sqrt{b \sqrt [3]{x}+a x}}{5 a}+\frac{39 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 )}{77 a^{17/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.066839, size = 124, normalized size = 0.57 \[ \frac{2 \sqrt{a x+b \sqrt [3]{x}} \left (26 a^2 b^2 x^{4/3}-14 a^3 b x^2+77 a^4 x^{8/3}+195 b^4 \sqrt{\frac{a x^{2/3}}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{a x^{2/3}}{b}\right )-78 a b^3 x^{2/3}-195 b^4\right )}{385 a^4 \left (a x^{2/3}+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 164, normalized size = 0.8 \begin{align*}{\frac{1}{385\,{a}^{5}} \left ( 195\,{b}^{4}\sqrt{-ab}\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 ) +52\,{x}^{5/3}{a}^{3}{b}^{2}-28\,{x}^{7/3}{a}^{4}b-156\,x{a}^{2}{b}^{3}+154\,{x}^{3}{a}^{5}-390\,\sqrt [3]{x}a{b}^{4} \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{x^{2}}{\sqrt{a x + b x^{\frac{1}{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^{3} - a b x^{\frac{7}{3}} + b^{2} x^{\frac{5}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}{a^{3} x^{2} + b^{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{x^{2}}{\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{x^{2}}{\sqrt{a x + b x^{\frac{1}{3}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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