Optimal. Leaf size=208 \[ \frac{4 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{77 a^{9/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^2}+\frac{24 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a}+\frac{12}{55} b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}+\frac{2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2} \]
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Rubi [A] time = 0.270748, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {2004, 2018, 2021, 2024, 2011, 329, 220} \[ \frac{4 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 a^{9/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^2}+\frac{24 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a}+\frac{12}{55} b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}+\frac{2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2004
Rule 2018
Rule 2021
Rule 2024
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx &=\frac{2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{1}{5} (2 b) \int \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x} \, dx\\ &=\frac{2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{1}{5} (6 b) \operatorname{Subst}\left (\int x^3 \sqrt{b x+a x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{12}{55} b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{1}{55} \left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{24 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a}+\frac{12}{55} b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac{\left (12 b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 a}\\ &=-\frac{8 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^2}+\frac{24 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a}+\frac{12}{55} b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{\left (4 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 a^2}\\ &=-\frac{8 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^2}+\frac{24 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a}+\frac{12}{55} b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{\left (4 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^2 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{8 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^2}+\frac{24 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a}+\frac{12}{55} b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{\left (8 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^2 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{8 b^3 \sqrt{b \sqrt [3]{x}+a x}}{77 a^2}+\frac{24 b^2 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{385 a}+\frac{12}{55} b x^{4/3} \sqrt{b \sqrt [3]{x}+a x}+\frac{2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac{4 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^{9/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0809067, size = 106, normalized size = 0.51 \[ \frac{2 \sqrt{a x+b \sqrt [3]{x}} \left (5 b^3 \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{a x^{2/3}}{b}\right )-\left (5 b-11 a x^{2/3}\right ) \left (a x^{2/3}+b\right )^2 \sqrt{\frac{a x^{2/3}}{b}+1}\right )}{55 a^2 \sqrt{\frac{a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 164, normalized size = 0.8 \begin{align*}{\frac{2}{385\,{a}^{3}} \left ( 10\,{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 ) +131\,{x}^{5/3}{a}^{3}{b}^{2}+196\,{x}^{7/3}{a}^{4}b-8\,x{a}^{2}{b}^{3}+77\,{x}^{3}{a}^{5}-20\,\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{\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 ({\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}, 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 \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{\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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