Optimal. Leaf size=115 \[ \frac{64 a^2 c^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 (c-a c x)^{3/2}}+\frac{16 a^2 c^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{35 \sqrt{c-a c x}}+\frac{2}{7} a^2 c^2 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \sqrt{c-a c x} \]
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Rubi [A] time = 0.171192, antiderivative size = 137, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6176, 6181, 89, 78, 37} \[ \frac{142 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{5/2}}{105 a^2 x \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{36 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{5/2}}{35 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{5/2}}{7 \left (1-\frac{1}{a x}\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Rule 6176
Rule 6181
Rule 89
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=\frac{(c-a c x)^{5/2} \int e^{\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 )^2 \sqrt{1+\frac{x}{a}}}{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 )^{3/2} x (c-a c x)^{5/2}}{7 \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{\left (2 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{9}{a}+\frac{7 x}{2 a^2}\right ) \sqrt{1+\frac{x}{a}}}{x^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{36 \left (1+\frac{1}{a x}\right )^{3/2} (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 )^{3/2} x (c-a c x)^{5/2}}{7 \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{\left (71 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{5/2}} \, dx,x,\frac{1}{x}\right )}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{36 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{5/2}}{35 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{142 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{5/2}}{105 a^2 \left (1-\frac{1}{a x}\right )^{5/2} x}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{5/2}}{7 \left (1-\frac{1}{a x}\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0369356, size = 67, normalized size = 0.58 \[ \frac{2 c^2 \sqrt{\frac{1}{a x}+1} (a x+1) \left (15 a^2 x^2-54 a x+71\right ) \sqrt{c-a c x}}{105 a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 56, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 15\,{a}^{2}{x}^{2}-54\,ax+71 \right ) }{105\, \left ( ax-1 \right ) ^{2}a} \left ( -acx+c \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10341, size = 90, normalized size = 0.78 \begin{align*} \frac{2 \,{\left (15 \, a^{3} \sqrt{-c} c^{2} x^{3} - 39 \, a^{2} \sqrt{-c} c^{2} x^{2} + 17 \, a \sqrt{-c} c^{2} x + 71 \, \sqrt{-c} c^{2}\right )} \sqrt{a x + 1}}{105 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60649, size = 182, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (15 \, a^{4} c^{2} x^{4} - 24 \, a^{3} c^{2} x^{3} - 22 \, a^{2} c^{2} x^{2} + 88 \, a c^{2} x + 71 \, c^{2}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{105 \,{\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 = 1.18411, size = 134, normalized size = 1.17 \begin{align*} \frac{2 \,{\left (\frac{64 \, \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}}{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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