Optimal. Leaf size=274 \[ \frac{12 a^{5/4} \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{5 \sqrt{a x+b x^3}}-\frac{24 a^{5/4} \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a x+b x^3}}-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}+\frac{12}{5} b x \sqrt{a x+b x^3}+\frac{24 a \sqrt{b} x \left (a+b x^2\right )}{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}} \]
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Rubi [A] time = 0.232554, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {2020, 2004, 2032, 329, 305, 220, 1196} \[ \frac{12 a^{5/4} \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a x+b x^3}}-\frac{24 a^{5/4} \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a x+b x^3}}-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}+\frac{12}{5} b x \sqrt{a x+b x^3}+\frac{24 a \sqrt{b} x \left (a+b x^2\right )}{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}} \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2004
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^3\right )^{3/2}}{x^3} \, dx &=-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}+(6 b) \int \sqrt{a x+b x^3} \, dx\\ &=\frac{12}{5} b x \sqrt{a x+b x^3}-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}+\frac{1}{5} (12 a b) \int \frac{x}{\sqrt{a x+b x^3}} \, dx\\ &=\frac{12}{5} b x \sqrt{a x+b x^3}-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}+\frac{\left (12 a b \sqrt{x} \sqrt{a+b x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x^2}} \, dx}{5 \sqrt{a x+b x^3}}\\ &=\frac{12}{5} b x \sqrt{a x+b x^3}-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}+\frac{\left (24 a b \sqrt{x} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{5 \sqrt{a x+b x^3}}\\ &=\frac{12}{5} b x \sqrt{a x+b x^3}-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}+\frac{\left (24 a^{3/2} \sqrt{b} \sqrt{x} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{5 \sqrt{a x+b x^3}}-\frac{\left (24 a^{3/2} \sqrt{b} \sqrt{x} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{5 \sqrt{a x+b x^3}}\\ &=\frac{24 a \sqrt{b} x \left (a+b x^2\right )}{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{12}{5} b x \sqrt{a x+b x^3}-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}-\frac{24 a^{5/4} \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a x+b x^3}}+\frac{12 a^{5/4} \sqrt [4]{b} \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 )}{5 \sqrt{a x+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.014335, size = 52, normalized size = 0.19 \[ -\frac{2 a \sqrt{x \left (a+b x^2\right )} \, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right )}{x \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 194, normalized size = 0.7 \begin{align*} -2\,{\frac{ \left ( b{x}^{2}+a \right ) a}{\sqrt{x \left ( b{x}^{2}+a \right ) }}}+{\frac{2\,bx}{5}\sqrt{b{x}^{3}+ax}}+{\frac{12\,a}{5}\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}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \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^{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{b x^{3} + a x}{\left (b x^{2} + a\right )}}{x^{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 \frac{\left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}{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{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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