Optimal. Leaf size=25 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0064869, antiderivative size = 25, 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 = {63, 205} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{-a+b x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )}{b}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.004967, size = 25, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 20, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{\sqrt{a}}\arctan \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(-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 = 1.59789, size = 140, normalized size = 5.6 \begin{align*} \left [-\frac{\sqrt{-a} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right )}{a}, \frac{2 \, \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.55866, size = 58, normalized size = 2.32 \begin{align*} \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22039, size = 26, normalized size = 1.04 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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