Optimal. Leaf size=28 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0128981, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1979, 620, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 1979
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\left (\frac{a}{x^2}+\frac{b}{x}\right ) x^3}} \, dx &=\int \frac{1}{\sqrt{a x+b x^2}} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [B] time = 0.0038001, size = 57, normalized size = 2.04 \[ \frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{x (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 1. \begin{align*}{\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48585, size = 154, normalized size = 5.5 \begin{align*} \left [\frac{\log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right )}{\sqrt{b}}, -\frac{2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right )}{b}\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{x^{3} \left (\frac{a}{x^{2}} + \frac{b}{x}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21856, size = 47, normalized size = 1.68 \begin{align*} -\frac{\log \left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{\sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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