Optimal. Leaf size=80 \[ \frac{7 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{4 a^2}+\frac{1}{2} x^2 \sqrt{c-\frac{c}{a x}}+\frac{7 x \sqrt{c-\frac{c}{a x}}}{4 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.240854, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {6167, 6133, 25, 434, 446, 78, 51, 63, 208} \[ \frac{7 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{4 a^2}+\frac{1}{2} x^2 \sqrt{c-\frac{c}{a x}}+\frac{7 x \sqrt{c-\frac{c}{a x}}}{4 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6133
Rule 25
Rule 434
Rule 446
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx\\ &=-\int \frac{\sqrt{c-\frac{c}{a x}} x (1+a x)}{1-a x} \, dx\\ &=\frac{c \int \frac{1+a x}{\sqrt{c-\frac{c}{a x}}} \, dx}{a}\\ &=\frac{c \int \frac{\left (a+\frac{1}{x}\right ) x}{\sqrt{c-\frac{c}{a x}}} \, dx}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{a+x}{x^3 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{1}{2} \sqrt{c-\frac{c}{a x}} x^2-\frac{(7 c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{4 a}\\ &=\frac{7 \sqrt{c-\frac{c}{a x}} x}{4 a}+\frac{1}{2} \sqrt{c-\frac{c}{a x}} x^2-\frac{(7 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a^2}\\ &=\frac{7 \sqrt{c-\frac{c}{a x}} x}{4 a}+\frac{1}{2} \sqrt{c-\frac{c}{a x}} x^2+\frac{7 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{4 a}\\ &=\frac{7 \sqrt{c-\frac{c}{a x}} x}{4 a}+\frac{1}{2} \sqrt{c-\frac{c}{a x}} x^2+\frac{7 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.0713571, size = 77, normalized size = 0.96 \[ \frac{\sqrt{c-\frac{c}{a x}} \left (a x \sqrt{1-\frac{1}{a x}} (2 a x+7)+7 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}}\right )\right )}{4 a^2 \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.161, size = 139, normalized size = 1.7 \begin{align*}{\frac{x}{8}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 4\,\sqrt{a{x}^{2}-x}{a}^{5/2}x-2\,\sqrt{a{x}^{2}-x}{a}^{3/2}+16\,{a}^{3/2}\sqrt{ \left ( ax-1 \right ) x}+8\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) -\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1 \right ){\frac{1}{\sqrt{a}}}} \right ) a \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{a}^{-{\frac{5}{2}}}} \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{{\left (a x + 1\right )} \sqrt{c - \frac{c}{a x}} x}{a x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54527, size = 327, normalized size = 4.09 \begin{align*} \left [\frac{2 \,{\left (2 \, a^{2} x^{2} + 7 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} + 7 \, \sqrt{c} \log \left (-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right )}{8 \, a^{2}}, \frac{{\left (2 \, a^{2} x^{2} + 7 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} - 7 \, \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right )}{4 \, a^{2}}\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{x \sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )}{a x - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.24579, size = 153, normalized size = 1.91 \begin{align*} \frac{1}{4} \, \sqrt{a^{2} c x^{2} - a c x}{\left (\frac{2 \, x{\left | a \right |}}{a^{2} \mathrm{sgn}\left (x\right )} + \frac{7 \,{\left | a \right |}}{a^{3} \mathrm{sgn}\left (x\right )}\right )} + \frac{7 \, \sqrt{c} \log \left ({\left | a \right |} \sqrt{{\left | c \right |}}\right ) \mathrm{sgn}\left (x\right )}{8 \, a^{2}} - \frac{7 \, \sqrt{c} \log \left ({\left | -2 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}{\left | a \right |} + a \sqrt{c} \right |}\right )}{8 \, a^{2} \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]