Optimal. Leaf size=39 \[ -\frac{\sqrt{a+b x}}{x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0106959, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {47, 63, 208} \[ -\frac{\sqrt{a+b x}}{x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x}}{x^2} \, dx &=-\frac{\sqrt{a+b x}}{x}+\frac{1}{2} b \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=-\frac{\sqrt{a+b x}}{x}+\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )\\ &=-\frac{\sqrt{a+b x}}{x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.026963, size = 47, normalized size = 1.21 \[ -\frac{b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )+a+b x}{x \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 37, normalized size = 1. \begin{align*} 2\,b \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \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 = 1.60688, size = 225, normalized size = 5.77 \begin{align*} \left [\frac{\sqrt{a} b x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \, \sqrt{b x + a} a}{2 \, a x}, \frac{\sqrt{-a} b x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - \sqrt{b x + a} a}{a x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.90541, size = 44, normalized size = 1.13 \begin{align*} - \frac{\sqrt{b} \sqrt{\frac{a}{b x} + 1}}{\sqrt{x}} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07493, size = 55, normalized size = 1.41 \begin{align*} \frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x + a} b}{x}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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