Optimal. Leaf size=197 \[ -\frac{856 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 x^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{2 x \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{11 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{16 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 x \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{8 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac{1}{a x}\right )^{9/2}} \]
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Rubi [A] time = 0.192679, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac{856 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 x^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{2 x \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{11 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{16 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 x \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{8 \left (\frac{1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac{1}{a x}\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 89
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=\frac{(c-a c x)^{9/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{9/2} x^{9/2} \, dx}{\left (1-\frac{1}{a x}\right )^{9/2} x^{9/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3 \left (1+\frac{x}{a}\right )^{3/2}}{x^{13/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (12 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2 \left (1+\frac{x}{a}\right )^{3/2}}{x^{11/2}} \, dx,x,\frac{1}{x}\right )}{11 a \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{8 \left (1+\frac{1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{2 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (8 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{11}{a}+\frac{9 x}{2 a^2}\right ) \left (1+\frac{x}{a}\right )^{3/2}}{x^{9/2}} \, dx,x,\frac{1}{x}\right )}{33 a \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{8 \left (1+\frac{1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{16 \left (1+\frac{1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 \left (1-\frac{1}{a x}\right )^{9/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (428 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^{7/2}} \, dx,x,\frac{1}{x}\right )}{231 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{8 \left (1+\frac{1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{856 \left (1+\frac{1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 \left (1-\frac{1}{a x}\right )^{9/2} x^2}+\frac{16 \left (1+\frac{1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 \left (1-\frac{1}{a x}\right )^{9/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0457692, size = 77, normalized size = 0.39 \[ \frac{2 c^4 \sqrt{\frac{1}{a x}+1} (a x+1)^2 \left (105 a^3 x^3-455 a^2 x^2+755 a x-533\right ) \sqrt{c-a c x}}{1155 a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.118, size = 64, normalized size = 0.3 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 105\,{x}^{3}{a}^{3}-455\,{a}^{2}{x}^{2}+755\,ax-533 \right ) }{1155\, \left ( ax-1 \right ) ^{3}a} \left ( -acx+c \right ) ^{{\frac{9}{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 = 1.0703, size = 143, normalized size = 0.73 \begin{align*} \frac{2 \,{\left (105 \, a^{5} \sqrt{-c} c^{4} x^{5} - 455 \, a^{4} \sqrt{-c} c^{4} x^{4} + 650 \, a^{3} \sqrt{-c} c^{4} x^{3} - 78 \, a^{2} \sqrt{-c} c^{4} x^{2} - 755 \, a \sqrt{-c} c^{4} x + 533 \, \sqrt{-c} c^{4}\right )}{\left (a x + 1\right )}^{\frac{3}{2}}}{1155 \,{\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 = 1.6092, size = 239, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (105 \, a^{6} c^{4} x^{6} - 140 \, a^{5} c^{4} x^{5} - 295 \, a^{4} c^{4} x^{4} + 472 \, a^{3} c^{4} x^{3} + 211 \, a^{2} c^{4} x^{2} - 844 \, a c^{4} x - 533 \, c^{4}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{1155 \,{\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 = 1.29634, size = 317, normalized size = 1.61 \begin{align*} -\frac{2 \,{\left (\frac{1536 \, \sqrt{2} \sqrt{-c} c^{4}}{\mathrm{sgn}\left (c\right )} - \frac{315 \,{\left (a c x + c\right )}^{5} \sqrt{-a c x - c} - 3080 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c} c + 11880 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c^{2} - 22176 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c^{3} - 18480 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{4} + 22 \,{\left (35 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c} - 270 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c + 756 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c^{2} + 840 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{3}\right )} c}{c \mathrm{sgn}\left (-a c x - c\right )}\right )}}{3465 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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