Optimal. Leaf size=128 \[ \frac{256 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}}{35 \sqrt{c-a c x}}+\frac{64}{35} c^2 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{24}{35} c x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}+\frac{2}{7} x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{5/2} \]
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Rubi [A] time = 0.202914, antiderivative size = 197, normalized size of antiderivative = 1.54, 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} \[ -\frac{344 \sqrt{\frac{1}{a x}+1} (c-a c x)^{5/2}}{35 a^3 x^2 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{2 x \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^3 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{16 \sqrt{\frac{1}{a x}+1} (c-a c x)^{5/2}}{5 a^2 x \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{24 \sqrt{\frac{1}{a x}+1} (c-a c x)^{5/2}}{35 a \left (1-\frac{1}{a x}\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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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)^{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 )^3}{x^{9/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{5/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (12 \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^{7/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{7 a \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{24 \sqrt{1+\frac{1}{a x}} (c-a c x)^{5/2}}{35 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{2 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{5/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (24 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{-\frac{7}{a}+\frac{5 x}{2 a^2}}{x^{5/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{35 a \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{24 \sqrt{1+\frac{1}{a x}} (c-a c x)^{5/2}}{35 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{16 \sqrt{1+\frac{1}{a x}} (c-a c x)^{5/2}}{5 a^2 \left (1-\frac{1}{a x}\right )^{5/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{5/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (172 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{35 a^3 \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{24 \sqrt{1+\frac{1}{a x}} (c-a c x)^{5/2}}{35 a \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{344 \sqrt{1+\frac{1}{a x}} (c-a c x)^{5/2}}{35 a^3 \left (1-\frac{1}{a x}\right )^{5/2} x^2}+\frac{16 \sqrt{1+\frac{1}{a x}} (c-a c x)^{5/2}}{5 a^2 \left (1-\frac{1}{a x}\right )^{5/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{5/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0416979, size = 70, normalized size = 0.55 \[ \frac{2 c^2 \sqrt{\frac{1}{a x}+1} \left (5 a^3 x^3-27 a^2 x^2+71 a x-177\right ) \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 64, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 5\,{x}^{3}{a}^{3}-27\,{a}^{2}{x}^{2}+71\,ax-177 \right ) }{35\,a \left ( ax-1 \right ) ^{3}} \left ( -acx+c \right ) ^{{\frac{5}{2}}}\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 = 1.11759, size = 130, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (5 \, a^{4} \sqrt{-c} c^{2} x^{4} - 22 \, a^{3} \sqrt{-c} c^{2} x^{3} + 44 \, a^{2} \sqrt{-c} c^{2} x^{2} - 106 \, a \sqrt{-c} c^{2} x - 177 \, \sqrt{-c} c^{2}\right )}{\left (a x - 1\right )}}{35 \,{\left (a^{2} x - a\right )} \sqrt{a x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88813, size = 182, normalized size = 1.42 \begin{align*} \frac{2 \,{\left (5 \, a^{4} c^{2} x^{4} - 22 \, a^{3} c^{2} x^{3} + 44 \, a^{2} c^{2} x^{2} - 106 \, a c^{2} x - 177 \, 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 = 1.19111, size = 116, normalized size = 0.91 \begin{align*} -\frac{2 \,{\left (5 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 42 \,{\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} - 280 \, \sqrt{-a c x - c} c^{3}\right )}{\left | c \right |}}{35 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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