Optimal. Leaf size=134 \[ \frac{4 a^{7/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{a x+b x^3}}+\frac{4}{7} a \sqrt{a x+b x^3}+\frac{2 \left (a x+b x^3\right )^{3/2}}{7 x} \]
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Rubi [A] time = 0.131994, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2021, 2011, 329, 220} \[ \frac{4 a^{7/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{a x+b x^3}}+\frac{4}{7} a \sqrt{a x+b x^3}+\frac{2 \left (a x+b x^3\right )^{3/2}}{7 x} \]
Antiderivative was successfully verified.
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Rule 2021
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^3\right )^{3/2}}{x^2} \, dx &=\frac{2 \left (a x+b x^3\right )^{3/2}}{7 x}+\frac{1}{7} (6 a) \int \frac{\sqrt{a x+b x^3}}{x} \, dx\\ &=\frac{4}{7} a \sqrt{a x+b x^3}+\frac{2 \left (a x+b x^3\right )^{3/2}}{7 x}+\frac{1}{7} \left (4 a^2\right ) \int \frac{1}{\sqrt{a x+b x^3}} \, dx\\ &=\frac{4}{7} a \sqrt{a x+b x^3}+\frac{2 \left (a x+b x^3\right )^{3/2}}{7 x}+\frac{\left (4 a^2 \sqrt{x} \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x^2}} \, dx}{7 \sqrt{a x+b x^3}}\\ &=\frac{4}{7} a \sqrt{a x+b x^3}+\frac{2 \left (a x+b x^3\right )^{3/2}}{7 x}+\frac{\left (8 a^2 \sqrt{x} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{7 \sqrt{a x+b x^3}}\\ &=\frac{4}{7} a \sqrt{a x+b x^3}+\frac{2 \left (a x+b x^3\right )^{3/2}}{7 x}+\frac{4 a^{7/4} \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 )}{7 \sqrt [4]{b} \sqrt{a x+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.0144247, size = 49, normalized size = 0.37 \[ \frac{2 a \sqrt{x \left (a+b x^2\right )} \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )}{\sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 144, normalized size = 1.1 \begin{align*}{\frac{2\,b{x}^{2}}{7}\sqrt{b{x}^{3}+ax}}+{\frac{6\,a}{7}\sqrt{b{x}^{3}+ax}}+{\frac{4\,{a}^{2}}{7\,b}\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 \frac{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{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 (\frac{\sqrt{b x^{3} + a x}{\left (b x^{2} + a\right )}}{x}, 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{\left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}{x^{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 \frac{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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