Optimal. Leaf size=32 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^5}}\right )}{3 \sqrt{a}} \]
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Rubi [A] time = 0.0109513, antiderivative size = 32, 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} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^5}}\right )}{3 \sqrt{a}} \]
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^5}} \, dx &=-\left (\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x}{\sqrt{a x^2+b x^5}}\right )\right )\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^5}}\right )}{3 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0103286, size = 54, normalized size = 1.69 \[ -\frac{2 x \sqrt{a+b x^3} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a} \sqrt{x^2 \left (a+b x^3\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 43, normalized size = 1.3 \begin{align*} -{\frac{2\,x}{3}\sqrt{b{x}^{3}+a}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}{\frac{1}{\sqrt{a}}}} \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^{5} + a x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.875145, size = 180, normalized size = 5.62 \begin{align*} \left [\frac{\log \left (\frac{b x^{4} + 2 \, a x - 2 \, \sqrt{b x^{5} + a x^{2}} \sqrt{a}}{x^{4}}\right )}{3 \, \sqrt{a}}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{5} + a x^{2}} \sqrt{-a}}{a x}\right )}{3 \, 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^{5}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16395, size = 63, normalized size = 1.97 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) \mathrm{sgn}\left (x\right )}{3 \, \sqrt{-a}} + \frac{2 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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