Optimal. Leaf size=77 \[ \frac{2 a^2 c^2 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{5 \sqrt{c-a c x}}+\frac{8 a^2 c^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{15 (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.15801, antiderivative size = 89, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6176, 6181, 78, 37} \[ \frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{3/2}}{5 \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{14 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{3/2}}{15 a \left (1-\frac{1}{a x}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Rule 6176
Rule 6181
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\frac{(c-a c x)^{3/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}}{x^{7/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{3/2}}{5 \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{\left (7 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 a \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{14 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{3/2}}{15 a \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{3/2}}{5 \left (1-\frac{1}{a x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0297813, size = 57, normalized size = 0.74 \[ -\frac{2 c \sqrt{\frac{1}{a x}+1} (a x+1) (3 a x-7) \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.039, size = 48, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 3\,ax-7 \right ) }{15\, \left ( ax-1 \right ) a} \left ( -acx+c \right ) ^{{\frac{3}{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 = 1.11792, size = 61, normalized size = 0.79 \begin{align*} -\frac{2 \,{\left (3 \, a^{2} \sqrt{-c} c x^{2} - 4 \, a \sqrt{-c} c x - 7 \, \sqrt{-c} c\right )} \sqrt{a x + 1}}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6449, size = 142, normalized size = 1.84 \begin{align*} -\frac{2 \,{\left (3 \, a^{3} c x^{3} - a^{2} c x^{2} - 11 \, a c x - 7 \, c\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\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 = 1.13195, size = 100, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{8 \, \sqrt{2} \sqrt{-c} c}{\mathrm{sgn}\left (c\right )} - \frac{3 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 10 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c}{c \mathrm{sgn}\left (-a c x - c\right )}\right )}}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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