Optimal. Leaf size=188 \[ \frac{10 a^{11/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{77 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{20 a^2 \sqrt{a x+b \sqrt [3]{x}}}{77 b^2 x^{2/3}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{77 b x^{4/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{11 x^2} \]
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Rubi [A] time = 0.243934, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2018, 2020, 2025, 2011, 329, 220} \[ \frac{20 a^2 \sqrt{a x+b \sqrt [3]{x}}}{77 b^2 x^{2/3}}+\frac{10 a^{11/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{77 b x^{4/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{11 x^2} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 2020
Rule 2025
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sqrt [3]{x}+a x}}{x^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{\sqrt{b x+a x^3}}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{11 x^2}+\frac{1}{11} (6 a) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{11 x^2}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{77 b x^{4/3}}-\frac{\left (30 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 b}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{11 x^2}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{77 b x^{4/3}}+\frac{20 a^2 \sqrt{b \sqrt [3]{x}+a x}}{77 b^2 x^{2/3}}+\frac{\left (10 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 b^2}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{11 x^2}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{77 b x^{4/3}}+\frac{20 a^2 \sqrt{b \sqrt [3]{x}+a x}}{77 b^2 x^{2/3}}+\frac{\left (10 a^3 \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 b^2 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{11 x^2}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{77 b x^{4/3}}+\frac{20 a^2 \sqrt{b \sqrt [3]{x}+a x}}{77 b^2 x^{2/3}}+\frac{\left (20 a^3 \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 b^2 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{11 x^2}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{77 b x^{4/3}}+\frac{20 a^2 \sqrt{b \sqrt [3]{x}+a x}}{77 b^2 x^{2/3}}+\frac{10 a^{11/4} \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 b^{9/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0501204, size = 59, normalized size = 0.31 \[ -\frac{6 \sqrt{a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac{11}{4},-\frac{1}{2};-\frac{7}{4};-\frac{a x^{2/3}}{b}\right )}{11 x^2 \sqrt{\frac{a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 179, normalized size = 1. \begin{align*} -{\frac{6}{11\,{x}^{2}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{12\,a}{77\,b}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{4}{3}}}}+{\frac{20\,{a}^{2}}{77\,{b}^{2}}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{2}{3}}}}+{\frac{10\,{a}^{2}}{77\,{b}^{2}}\sqrt{-ab}\sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b\sqrt [3]{x}+ax}}}} \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{\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{\sqrt{a x + b x^{\frac{1}{3}}}}{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{\sqrt{a x + b \sqrt [3]{x}}}{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{\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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