Optimal. Leaf size=254 \[ -\frac{768 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 x^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9088 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 x^3 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{32 \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{2 x \left (a-\frac{1}{x}\right )^4 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{11 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{128 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 x \left (1-\frac{1}{a x}\right )^{9/2}} \]
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Rubi [A] time = 0.220808, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac{768 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 x^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9088 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 x^3 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{32 \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{2 x \left (a-\frac{1}{x}\right )^4 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{11 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{128 \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 x \left (1-\frac{1}{a x}\right )^{9/2}} \]
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^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=\frac{(c-a c x)^{9/2} \int e^{\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 )^4 \sqrt{1+\frac{x}{a}}}{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 )^4 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (16 \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 \sqrt{1+\frac{x}{a}}}{x^{11/2}} \, dx,x,\frac{1}{x}\right )}{11 a \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{2 \left (a-\frac{1}{x}\right )^4 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{\left (64 \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 \sqrt{1+\frac{x}{a}}}{x^{9/2}} \, dx,x,\frac{1}{x}\right )}{33 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{128 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac{1}{a x}\right )^{9/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{\left (128 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\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 )}{231 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{768 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 \left (1-\frac{1}{a x}\right )^{9/2} x^2}+\frac{128 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac{1}{a x}\right )^{9/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{\left (4544 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{5/2}} \, dx,x,\frac{1}{x}\right )}{1155 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9088 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 \left (1-\frac{1}{a x}\right )^{9/2} x^3}-\frac{768 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 \left (1-\frac{1}{a x}\right )^{9/2} x^2}+\frac{128 \left (1+\frac{1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac{1}{a x}\right )^{9/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 \left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0551238, size = 83, normalized size = 0.33 \[ \frac{2 c^4 \sqrt{\frac{1}{a x}+1} (a x+1) \left (315 a^4 x^4-1820 a^3 x^3+4530 a^2 x^2-6396 a x+5419\right ) \sqrt{c-a c x}}{3465 a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 72, normalized size = 0.3 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 315\,{x}^{4}{a}^{4}-1820\,{x}^{3}{a}^{3}+4530\,{a}^{2}{x}^{2}-6396\,ax+5419 \right ) }{3465\, \left ( ax-1 \right ) ^{4}a} \left ( -acx+c \right ) ^{{\frac{9}{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 = 1.07954, size = 134, normalized size = 0.53 \begin{align*} \frac{2 \,{\left (315 \, a^{5} \sqrt{-c} c^{4} x^{5} - 1505 \, a^{4} \sqrt{-c} c^{4} x^{4} + 2710 \, a^{3} \sqrt{-c} c^{4} x^{3} - 1866 \, a^{2} \sqrt{-c} c^{4} x^{2} - 977 \, a \sqrt{-c} c^{4} x + 5419 \, \sqrt{-c} c^{4}\right )} \sqrt{a x + 1}}{3465 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54384, size = 246, normalized size = 0.97 \begin{align*} \frac{2 \,{\left (315 \, a^{6} c^{4} x^{6} - 1190 \, a^{5} c^{4} x^{5} + 1205 \, a^{4} c^{4} x^{4} + 844 \, a^{3} c^{4} x^{3} - 2843 \, a^{2} c^{4} x^{2} + 4442 \, a c^{4} x + 5419 \, c^{4}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{3465 \,{\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 = 1.27681, size = 198, normalized size = 0.78 \begin{align*} \frac{2 \,{\left (\frac{4096 \, \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}}{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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