Optimal. Leaf size=41 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{\sqrt{a+b x}}{a x} \]
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Rubi [A] time = 0.0117272, antiderivative size = 41, 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 = {51, 63, 208} \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{\sqrt{a+b x}}{a x} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{a+b x}} \, dx &=-\frac{\sqrt{a+b x}}{a x}-\frac{b \int \frac{1}{x \sqrt{a+b x}} \, dx}{2 a}\\ &=-\frac{\sqrt{a+b x}}{a x}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{a}\\ &=-\frac{\sqrt{a+b x}}{a x}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.062887, size = 47, normalized size = 1.15 \[ \frac{\sqrt{a+b x} \left (\frac{b \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )}{\sqrt{\frac{b x}{a}+1}}-\frac{a}{x}\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 40, normalized size = 1. \begin{align*} 2\,b \left ( -1/2\,{\frac{\sqrt{bx+a}}{abx}}+1/2\,{\frac{1}{{a}^{3/2}}{\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 = 1.7556, size = 232, normalized size = 5.66 \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^{2} x}, -\frac{\sqrt{-a} b x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) + \sqrt{b x + a} a}{a^{2} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.18542, size = 44, normalized size = 1.07 \begin{align*} - \frac{\sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{x}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10643, size = 63, normalized size = 1.54 \begin{align*} -\frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x + a} b}{a x}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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