Optimal. Leaf size=254 \[ -\frac{256 (c-a c x)^{5/2}}{7 a^3 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{2944 (c-a c x)^{5/2}}{35 a^4 x^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^4 (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{128 (c-a c x)^{5/2}}{35 a^2 x \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.213266, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{256 (c-a c x)^{5/2}}{7 a^3 x^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{2944 (c-a c x)^{5/2}}{35 a^4 x^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^4 (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{128 (c-a c x)^{5/2}}{35 a^2 x \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}} \]
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)^{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 )^4}{x^{9/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (16 \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}{x^{7/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{7 a \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}+\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (192 \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}{x^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}+\frac{128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (128 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{-\frac{5}{a}+\frac{3 x}{2 a^2}}{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (1472 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{2944 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^3}-\frac{256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0381353, size = 68, normalized size = 0.27 \[ \frac{2 c^2 \left (5 a^4 x^4-36 a^3 x^3+142 a^2 x^2-708 a x-1451\right ) \sqrt{c-a c x}}{35 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 72, normalized size = 0.3 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 5\,{x}^{4}{a}^{4}-36\,{x}^{3}{a}^{3}+142\,{a}^{2}{x}^{2}-708\,ax-1451 \right ) }{35\,a \left ( ax-1 \right ) ^{4}} \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 = 1.11333, size = 162, normalized size = 0.64 \begin{align*} \frac{2 \,{\left (5 \, a^{5} \sqrt{-c} c^{2} x^{5} - 31 \, a^{4} \sqrt{-c} c^{2} x^{4} + 106 \, a^{3} \sqrt{-c} c^{2} x^{3} - 566 \, a^{2} \sqrt{-c} c^{2} x^{2} - 2159 \, a \sqrt{-c} c^{2} x - 1451 \, \sqrt{-c} c^{2}\right )}{\left (a x - 1\right )}^{2}}{35 \,{\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )}{\left (a x + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50044, size = 185, normalized size = 0.73 \begin{align*} \frac{2 \,{\left (5 \, a^{4} c^{2} x^{4} - 36 \, a^{3} c^{2} x^{3} + 142 \, a^{2} c^{2} x^{2} - 708 \, a c^{2} x - 1451 \, 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 = 1.29361, size = 138, normalized size = 0.54 \begin{align*} -\frac{2 \,{\left (5 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 56 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c - 280 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{2} - 1120 \, \sqrt{-a c x - c} c^{3} + \frac{560 \, c^{4}}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{35 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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