Optimal. Leaf size=35 \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0288091, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {848, 50, 54, 216} \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 848
Rule 50
Rule 54
Rule 216
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{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0101975, size = 35, normalized size = 1. \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.156, size = 76, normalized size = 2.2 \begin{align*}{\frac{1}{2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{x} \left ( 2\,\sqrt{a}\sqrt{-x \left ( ax-1 \right ) }+\arctan \left ({\frac{2\,ax-1}{2}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) }}}} \right ) \right ){\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) }}}{\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 = 1.80723, size = 486, normalized size = 13.89 \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]