Optimal. Leaf size=30 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0105473, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2008, 206} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a x^2+b x^3}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x}{\sqrt{a x^2+b x^3}}\right )\right )\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0089751, size = 46, normalized size = 1.53 \[ -\frac{2 x \sqrt{a+b x} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 39, normalized size = 1.3 \begin{align*} -2\,{\frac{x\sqrt{bx+a}}{\sqrt{b{x}^{3}+a{x}^{2}}\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \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 \frac{1}{\sqrt{b x^{3} + a x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.803761, size = 171, normalized size = 5.7 \begin{align*} \left [\frac{\log \left (\frac{b x^{2} + 2 \, a x - 2 \, \sqrt{b x^{3} + a x^{2}} \sqrt{a}}{x^{2}}\right )}{\sqrt{a}}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}{a x}\right )}{a}\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}{\sqrt{a x^{2} + b x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16818, size = 61, normalized size = 2.03 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{-a}} + \frac{2 \, \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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