Optimal. Leaf size=213 \[ \frac{4 a^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{77 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{8 a^3 \sqrt{a x+b \sqrt [3]{x}}}{77 b^2 x^{2/3}}-\frac{24 a^2 \sqrt{a x+b \sqrt [3]{x}}}{385 b x^{4/3}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{55 x^2}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{5 x^3} \]
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Rubi [A] time = 0.308573, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{8 a^3 \sqrt{a x+b \sqrt [3]{x}}}{77 b^2 x^{2/3}}+\frac{4 a^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 b^{9/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{24 a^2 \sqrt{a x+b \sqrt [3]{x}}}{385 b x^{4/3}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{55 x^2}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{5 x^3} \]
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{\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (b x+a x^3\right )^{3/2}}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{5 x^3}+\frac{1}{5} (6 a) \operatorname{Subst}\left (\int \frac{\sqrt{b x+a x^3}}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{55 x^2}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{5 x^3}+\frac{1}{55} \left (12 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{55 x^2}-\frac{24 a^2 \sqrt{b \sqrt [3]{x}+a x}}{385 b x^{4/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{5 x^3}-\frac{\left (12 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 b}\\ &=-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{55 x^2}-\frac{24 a^2 \sqrt{b \sqrt [3]{x}+a x}}{385 b x^{4/3}}+\frac{8 a^3 \sqrt{b \sqrt [3]{x}+a x}}{77 b^2 x^{2/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{5 x^3}+\frac{\left (4 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 b^2}\\ &=-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{55 x^2}-\frac{24 a^2 \sqrt{b \sqrt [3]{x}+a x}}{385 b x^{4/3}}+\frac{8 a^3 \sqrt{b \sqrt [3]{x}+a x}}{77 b^2 x^{2/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{5 x^3}+\frac{\left (4 a^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 b^2 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{55 x^2}-\frac{24 a^2 \sqrt{b \sqrt [3]{x}+a x}}{385 b x^{4/3}}+\frac{8 a^3 \sqrt{b \sqrt [3]{x}+a x}}{77 b^2 x^{2/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{5 x^3}+\frac{\left (8 a^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 b^2 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{55 x^2}-\frac{24 a^2 \sqrt{b \sqrt [3]{x}+a x}}{385 b x^{4/3}}+\frac{8 a^3 \sqrt{b \sqrt [3]{x}+a x}}{77 b^2 x^{2/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{5 x^3}+\frac{4 a^{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 b^{9/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0661396, size = 62, normalized size = 0.29 \[ -\frac{2 b \sqrt{a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac{15}{4},-\frac{3}{2};-\frac{11}{4};-\frac{a x^{2/3}}{b}\right )}{5 x^{8/3} \sqrt{\frac{a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 168, normalized size = 0.8 \begin{align*}{\frac{2}{385\,{b}^{2}} \left ( 10\,{a}^{3}\sqrt{-ab}\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}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{14/3}-131\,{x}^{11/3}{a}^{2}{b}^{2}+8\,{x}^{13/3}{a}^{3}b-196\,a{b}^{3}{x}^{3}+20\,{x}^{5}{a}^{4}-77\,{x}^{7/3}{b}^{4} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}{x}^{-{\frac{14}{3}}}} \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^{4}}\,{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^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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