Optimal. Leaf size=119 \[ -\frac{b^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{3 a^{5/4} \sqrt{a x+b x^3}}-\frac{2 \sqrt{a x+b x^3}}{3 a x^2} \]
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Rubi [A] time = 0.0893417, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2025, 2011, 329, 220} \[ -\frac{b^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 a^{5/4} \sqrt{a x+b x^3}}-\frac{2 \sqrt{a x+b x^3}}{3 a x^2} \]
Antiderivative was successfully verified.
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Rule 2025
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{a x+b x^3}} \, dx &=-\frac{2 \sqrt{a x+b x^3}}{3 a x^2}-\frac{b \int \frac{1}{\sqrt{a x+b x^3}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{a x+b x^3}}{3 a x^2}-\frac{\left (b \sqrt{x} \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x^2}} \, dx}{3 a \sqrt{a x+b x^3}}\\ &=-\frac{2 \sqrt{a x+b x^3}}{3 a x^2}-\frac{\left (2 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 )}{3 a \sqrt{a x+b x^3}}\\ &=-\frac{2 \sqrt{a x+b x^3}}{3 a x^2}-\frac{b^{3/4} \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 )}{3 a^{5/4} \sqrt{a x+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.0150918, size = 53, normalized size = 0.45 \[ -\frac{2 \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{b x^2}{a}\right )}{3 x \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 129, normalized size = 1.1 \begin{align*} -{\frac{2}{3\,a{x}^{2}}\sqrt{b{x}^{3}+ax}}-{\frac{1}{3\,a}\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{1}{\sqrt{b x^{3} + a x} 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}}{b x^{5} + a x^{3}}, 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{1}{x^{2} \sqrt{x \left (a + b x^{2}\right )}}\, 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{1}{\sqrt{b x^{3} + a x} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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