Optimal. Leaf size=251 \[ \frac{119 x \sqrt{c-\frac{c}{a x}}}{24 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{119 \sqrt{c-\frac{c}{a x}}}{8 a^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{119 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{8 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{x^3 \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{19 x^2 \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.292824, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6182, 6180, 89, 78, 51, 63, 208} \[ \frac{119 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{119 x \sqrt{c-\frac{c}{a x}}}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{119 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{8 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{x^3 \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{19 x^2 \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6182
Rule 6180
Rule 89
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^2 \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^4 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{-\frac{19}{2 a}+\frac{3 x}{a^2}}{x^3 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (119 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{24 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{119 \sqrt{c-\frac{c}{a x}} x}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (119 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{119 \sqrt{c-\frac{c}{a x}} x}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{119 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\left (119 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a^3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{119 \sqrt{c-\frac{c}{a x}} x}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{119 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\left (119 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{8 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{119 \sqrt{c-\frac{c}{a x}} x}{12 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{119 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{19 \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{119 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{8 a^3 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.490683, size = 159, normalized size = 0.63 \[ \frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^3 x^3-38 a^2 x^2+119 a x+357\right ) \sqrt{c-\frac{c}{a x}}}{a^2 x^2-1}-357 \sqrt{c} \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )+357 \sqrt{c} \log (1-a x)}{48 a^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.187, size = 180, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax+1 \right ) x}{48\, \left ( ax-1 \right ) ^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 16\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}-76\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+238\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-357\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) xa+714\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}-357\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \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 \sqrt{c - \frac{c}{a x}} x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.94247, size = 737, normalized size = 2.94 \begin{align*} \left [\frac{357 \,{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{96 \,{\left (a^{4} x - a^{3}\right )}}, \frac{357 \,{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{48 \,{\left (a^{4} x - a^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c - \frac{c}{a x}} x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]