Optimal. Leaf size=161 \[ \frac{x \left (a-\frac{1}{x}\right )^2 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}{a^2 \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{\left (16 a+\frac{1}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{7 \left (c-\frac{c}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.124572, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6182, 6179, 98, 147, 63, 208} \[ \frac{x \left (a-\frac{1}{x}\right )^2 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}{a^2 \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{\left (16 a+\frac{1}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{7 \left (c-\frac{c}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6182
Rule 6179
Rule 98
Rule 147
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{5/2} \, dx &=\frac{\left (c-\frac{c}{a x}\right )^{5/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{5/2} \, dx}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{\left (c-\frac{c}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3}{x^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{\left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2} x}{a^2 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (c-\frac{c}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{\left (\frac{7}{2 a}+\frac{x}{2 a^2}\right ) \left (1-\frac{x}{a}\right )}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{\left (16 a+\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2} x}{a^2 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (7 \left (c-\frac{c}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{\left (16 a+\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2} x}{a^2 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (7 \left (c-\frac{c}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{\left (16 a+\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2} x}{a^2 \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{7 \left (c-\frac{c}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (1-\frac{1}{a x}\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0643764, size = 89, normalized size = 0.55 \[ \frac{c^2 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (3 a^2 x^2-22 a x+2\right )-21 a x \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{3 a^2 x \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.179, size = 144, normalized size = 0.9 \begin{align*}{\frac{ \left ( ax+1 \right ){c}^{2}}{6\, \left ( ax-1 \right ) x}\sqrt{{\frac{ax-1}{ax+1}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 6\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}-44\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-21\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{2}{a}^{2}+4\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \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{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.52613, size = 795, normalized size = 4.94 \begin{align*} \left [\frac{21 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\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 (3 \, a^{3} c^{2} x^{3} - 19 \, a^{2} c^{2} x^{2} - 20 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{12 \,{\left (a^{3} x^{2} - a^{2} x\right )}}, \frac{21 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\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 (3 \, a^{3} c^{2} x^{3} - 19 \, a^{2} c^{2} x^{2} - 20 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{6 \,{\left (a^{3} x^{2} - a^{2} x\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{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]