Optimal. Leaf size=158 \[ -\frac{4 a^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{77 b^{5/4} \sqrt{a x+b x^3}}+\frac{8 a^2 \sqrt{a x+b x^3}}{77 b}+\frac{12}{77} a x^2 \sqrt{a x+b x^3}+\frac{2}{11} x \left (a x+b x^3\right )^{3/2} \]
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Rubi [A] time = 0.133583, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2004, 2021, 2024, 2011, 329, 220} \[ -\frac{4 a^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{77 b^{5/4} \sqrt{a x+b x^3}}+\frac{8 a^2 \sqrt{a x+b x^3}}{77 b}+\frac{12}{77} a x^2 \sqrt{a x+b x^3}+\frac{2}{11} x \left (a x+b x^3\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2004
Rule 2021
Rule 2024
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \left (a x+b x^3\right )^{3/2} \, dx &=\frac{2}{11} x \left (a x+b x^3\right )^{3/2}+\frac{1}{11} (6 a) \int x \sqrt{a x+b x^3} \, dx\\ &=\frac{12}{77} a x^2 \sqrt{a x+b x^3}+\frac{2}{11} x \left (a x+b x^3\right )^{3/2}+\frac{1}{77} \left (12 a^2\right ) \int \frac{x^2}{\sqrt{a x+b x^3}} \, dx\\ &=\frac{8 a^2 \sqrt{a x+b x^3}}{77 b}+\frac{12}{77} a x^2 \sqrt{a x+b x^3}+\frac{2}{11} x \left (a x+b x^3\right )^{3/2}-\frac{\left (4 a^3\right ) \int \frac{1}{\sqrt{a x+b x^3}} \, dx}{77 b}\\ &=\frac{8 a^2 \sqrt{a x+b x^3}}{77 b}+\frac{12}{77} a x^2 \sqrt{a x+b x^3}+\frac{2}{11} x \left (a x+b x^3\right )^{3/2}-\frac{\left (4 a^3 \sqrt{x} \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x^2}} \, dx}{77 b \sqrt{a x+b x^3}}\\ &=\frac{8 a^2 \sqrt{a x+b x^3}}{77 b}+\frac{12}{77} a x^2 \sqrt{a x+b x^3}+\frac{2}{11} x \left (a x+b x^3\right )^{3/2}-\frac{\left (8 a^3 \sqrt{x} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{77 b \sqrt{a x+b x^3}}\\ &=\frac{8 a^2 \sqrt{a x+b x^3}}{77 b}+\frac{12}{77} a x^2 \sqrt{a x+b x^3}+\frac{2}{11} x \left (a x+b x^3\right )^{3/2}-\frac{4 a^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{77 b^{5/4} \sqrt{a x+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.0354283, size = 83, normalized size = 0.53 \[ \frac{2 \sqrt{x \left (a+b x^2\right )} \left (\left (a+b x^2\right )^2 \sqrt{\frac{b x^2}{a}+1}-a^2 \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )\right )}{11 b \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 166, normalized size = 1.1 \begin{align*}{\frac{2\,b{x}^{4}}{11}\sqrt{b{x}^{3}+ax}}+{\frac{26\,a{x}^{2}}{77}\sqrt{b{x}^{3}+ax}}+{\frac{8\,{a}^{2}}{77\,b}\sqrt{b{x}^{3}+ax}}-{\frac{4\,{a}^{3}}{77\,{b}^{2}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b{x}^{3}+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{\left (b x^{3} + a x\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 (b x^{3} + a x\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 x^{3}\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 (b x^{3} + a x\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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