Optimal. Leaf size=193 \[ -\frac{a^3 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{3/2}}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{5/2}}+\frac{a^2 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}{8 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}+\frac{a^{3/2} \left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{8 \sqrt{2} \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.201913, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6176, 6181, 94, 93, 206} \[ -\frac{a^3 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{3/2}}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{5/2}}+\frac{a^2 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}{8 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}+\frac{a^{3/2} \left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{8 \sqrt{2} \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6176
Rule 6181
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{5/2} x^{5/2}} \, dx}{(c-a c x)^{5/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{\sqrt{x} \sqrt{1+\frac{x}{a}}}{\left (1-\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2} x^2}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{5/2}}+\frac{\left (a \left (1-\frac{1}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{\sqrt{x} \left (1-\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{8 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac{a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}{8 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}-\frac{a^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2} x^2}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{5/2}}+\frac{\left (a \left (1-\frac{1}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac{a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}{8 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}-\frac{a^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2} x^2}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{5/2}}+\frac{\left (a \left (1-\frac{1}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{8 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac{a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}{8 \left (a-\frac{1}{x}\right ) (c-a c x)^{5/2}}-\frac{a^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2} x^2}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{5/2}}+\frac{a^{3/2} \left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{8 \sqrt{2} \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.149885, size = 123, normalized size = 0.64 \[ \frac{x \sqrt{1-\frac{1}{a x}} \left (\sqrt{2} \sqrt{\frac{1}{x}} (a x-1)^2 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )-2 \sqrt{a} \sqrt{\frac{1}{a x}+1} (a x+3)\right )}{16 \sqrt{a} c^2 (a x-1)^2 \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.141, size = 165, normalized size = 0.9 \begin{align*} -{\frac{1}{16\, \left ( ax-1 \right ) ^{2}a}\sqrt{-c \left ( ax-1 \right ) } \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ){x}^{2}{a}^{2}c-2\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xac-2\,xa\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}+\sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) c-6\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{c}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \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{1}{{\left (-a c x + c\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 = 1.69129, size = 775, normalized size = 4.02 \begin{align*} \left [-\frac{\sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \,{\left (a^{2} x^{2} + 4 \, a x + 3\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{32 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, -\frac{\sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \,{\left (a^{2} x^{2} + 4 \, a x + 3\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{16 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{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 [A] time = 1.19808, size = 120, normalized size = 0.62 \begin{align*} -\frac{\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{c^{\frac{3}{2}}} + \frac{2 \,{\left ({\left (-a c x - c\right )}^{\frac{3}{2}} - 2 \, \sqrt{-a c x - c} c\right )}}{{\left (a c x - c\right )}^{2} c}}{16 \, a c \mathrm{sgn}\left (-a c x - c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]