Optimal. Leaf size=89 \[ \frac{2 x \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{5/2}}{7 \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{18 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{5/2}}{35 a \left (1-\frac{1}{a x}\right )^{5/2}} \]
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Rubi [A] time = 0.158009, antiderivative size = 89, normalized size of antiderivative = 1., 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} \[ \frac{2 x \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{5/2}}{7 \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{18 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{5/2}}{35 a \left (1-\frac{1}{a x}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=\frac{(c-a c x)^{5/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{5/2} x^{5/2} \, dx}{\left (1-\frac{1}{a x}\right )^{5/2} x^{5/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right ) \left (1+\frac{x}{a}\right )^{3/2}}{x^{9/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x (c-a c x)^{5/2}}{7 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (9 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 a \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{18 \left (1+\frac{1}{a x}\right )^{5/2} (c-a c x)^{5/2}}{35 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{2 \left (1+\frac{1}{a x}\right )^{5/2} x (c-a c x)^{5/2}}{7 \left (1-\frac{1}{a x}\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0388098, size = 59, normalized size = 0.66 \[ \frac{2 \sqrt{\frac{1}{a x}+1} (5 a x-9) \sqrt{c-a c x} (a c x+c)^2}{35 a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.117, size = 48, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 5\,ax-9 \right ) }{35\, \left ( ax-1 \right ) a} \left ( -acx+c \right ) ^{{\frac{5}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08365, size = 100, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (5 \, a^{3} \sqrt{-c} c^{2} x^{3} - 9 \, a^{2} \sqrt{-c} c^{2} x^{2} - 5 \, a \sqrt{-c} c^{2} x + 9 \, \sqrt{-c} c^{2}\right )}{\left (a x + 1\right )}^{\frac{3}{2}}}{35 \,{\left (a x - 1\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50827, size = 177, normalized size = 1.99 \begin{align*} \frac{2 \,{\left (5 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 12 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x - 9 \, c^{2}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\left (a^{2} x - a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.18758, size = 188, normalized size = 2.11 \begin{align*} -\frac{2 \,{\left (\frac{48 \, \sqrt{2} \sqrt{-c} c^{2}}{\mathrm{sgn}\left (c\right )} - \frac{15 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 84 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c - 140 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{2} + 14 \,{\left (3 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 10 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c\right )} c}{c \mathrm{sgn}\left (-a c x - c\right )}\right )}}{105 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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