Optimal. Leaf size=34 \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.028346, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {848, 50, 54, 215} \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 848
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{\sqrt{1-a^2 x^2}}{\sqrt{x} \sqrt{1-a x}} \, dx &=\int \frac{\sqrt{1+a x}}{\sqrt{x}} \, dx\\ &=\sqrt{x} \sqrt{1+a x}+\frac{1}{2} \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx\\ &=\sqrt{x} \sqrt{1+a x}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )\\ &=\sqrt{x} \sqrt{1+a x}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0128162, size = 34, normalized size = 1. \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.139, size = 86, normalized size = 2.5 \begin{align*} -{\frac{1}{2\,ax-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{x}\sqrt{-ax+1} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1 \right ){\frac{1}{\sqrt{a}}}} \right ) \right ){\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}}{\sqrt{-a x + 1} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.79546, size = 494, normalized size = 14.53 \begin{align*} \left [-\frac{4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a x + 1} a \sqrt{x} -{\left (a x - 1\right )} \sqrt{a} \log \left (-\frac{8 \, a^{3} x^{3} - 4 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x + 1\right )} \sqrt{-a x + 1} \sqrt{a} \sqrt{x} - 7 \, a x - 1}{a x - 1}\right )}{4 \,{\left (a^{2} x - a\right )}}, -\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a x + 1} a \sqrt{x} -{\left (a x - 1\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a x + 1} \sqrt{-a} \sqrt{x}}{2 \, a^{2} x^{2} - a x - 1}\right )}{2 \,{\left (a^{2} x - a\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt{x} \sqrt{- a x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]