Optimal. Leaf size=62 \[ \frac{2}{3} x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{8 c x \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-a c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.153416, antiderivative size = 89, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6176, 6181, 78, 37} \[ \frac{2 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}-\frac{10 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 6176
Rule 6181
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{-\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} \sqrt{x} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{x^{5/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{\left (5 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 a \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{10 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0262458, size = 50, normalized size = 0.81 \[ \frac{2 \sqrt{\frac{1}{a x}+1} (a x-5) \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 47, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( ax-5 \right ) }{ \left ( 3\,ax-3 \right ) a}\sqrt{-acx+c}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10216, size = 73, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (a^{2} \sqrt{-c} x^{2} - 4 \, a \sqrt{-c} x - 5 \, \sqrt{-c}\right )}{\left (a x - 1\right )}}{3 \,{\left (a^{2} x - a\right )} \sqrt{a x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.53216, size = 111, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} - 4 \, a x - 5\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{2} x - a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 164.195, size = 66, normalized size = 1.06 \begin{align*} \frac{4 i c x \sqrt{\frac{1}{a c x + c}}}{3} + \frac{4 i c \sqrt{\frac{1}{a c x + c}}}{a} - \frac{2 i \left (- a c x + c\right )^{2} \sqrt{\frac{1}{a c x + c}}}{3 a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17643, size = 80, normalized size = 1.29 \begin{align*} -\frac{2 \,{\left (\frac{4 \, \sqrt{2} \sqrt{-c} c}{a} - \frac{{\left (-a c x - c\right )}^{\frac{3}{2}} + 6 \, \sqrt{-a c x - c} c}{a}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{3 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]