Optimal. Leaf size=29 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0106771, antiderivative size = 29, 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, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} (-a+b x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0050798, size = 29, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 19, normalized size = 0.7 \begin{align*} -2\,{\frac{1}{\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58672, size = 161, normalized size = 5.55 \begin{align*} \left [\frac{\sqrt{a b} \log \left (\frac{b x + a - 2 \, \sqrt{a b} \sqrt{x}}{b x - a}\right )}{a b}, \frac{2 \, \sqrt{-a b} \arctan \left (\frac{\sqrt{-a b}}{b \sqrt{x}}\right )}{a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.10954, size = 88, normalized size = 3.03 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{b \sqrt{x}} & \text{for}\: a = 0 \\- \frac{2 \sqrt{x}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{\sqrt{a} b \sqrt{\frac{1}{b}}} - \frac{\log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{\sqrt{a} b \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22255, size = 27, normalized size = 0.93 \begin{align*} \frac{2 \, \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]