Optimal. Leaf size=311 \[ -\frac{512 (c-a c x)^{7/2}}{63 a^3 x^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{5120 (c-a c x)^{7/2}}{63 a^4 x^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{11776 (c-a c x)^{7/2}}{63 a^5 x^4 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^5 (c-a c x)^{7/2}}{9 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}-\frac{40 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{128 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 x \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.236138, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{512 (c-a c x)^{7/2}}{63 a^3 x^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{5120 (c-a c x)^{7/2}}{63 a^4 x^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{11776 (c-a c x)^{7/2}}{63 a^5 x^4 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^5 (c-a c x)^{7/2}}{9 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}-\frac{40 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{128 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 x \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}} \]
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)^{7/2} \, dx &=\frac{(c-a c x)^{7/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{7/2} x^{7/2} \, dx}{\left (1-\frac{1}{a x}\right )^{7/2} x^{7/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^5}{x^{11/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{7/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (20 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\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 )}{9 a \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{2 \left (a-\frac{1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (320 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\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 )}{63 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{128 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (256 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\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 )}{21 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{512 (c-a c x)^{7/2}}{63 a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{128 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (512 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\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 )}{63 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{5120 (c-a c x)^{7/2}}{63 a^4 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^3}-\frac{512 (c-a c x)^{7/2}}{63 a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{128 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (5888 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{11776 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^4}+\frac{5120 (c-a c x)^{7/2}}{63 a^4 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^3}-\frac{512 (c-a c x)^{7/2}}{63 a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{128 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0448429, size = 76, normalized size = 0.24 \[ -\frac{2 c^3 \left (7 a^5 x^5-55 a^4 x^4+214 a^3 x^3-638 a^2 x^2+2867 a x+5797\right ) \sqrt{c-a c x}}{63 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 80, normalized size = 0.3 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 7\,{x}^{5}{a}^{5}-55\,{x}^{4}{a}^{4}+214\,{x}^{3}{a}^{3}-638\,{a}^{2}{x}^{2}+2867\,ax+5797 \right ) }{63\,a \left ( ax-1 \right ) ^{5}} \left ( -acx+c \right ) ^{{\frac{7}{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 = 1.11219, size = 184, normalized size = 0.59 \begin{align*} -\frac{2 \,{\left (7 \, a^{6} \sqrt{-c} c^{3} x^{6} - 48 \, a^{5} \sqrt{-c} c^{3} x^{5} + 159 \, a^{4} \sqrt{-c} c^{3} x^{4} - 424 \, a^{3} \sqrt{-c} c^{3} x^{3} + 2229 \, a^{2} \sqrt{-c} c^{3} x^{2} + 8664 \, a \sqrt{-c} c^{3} x + 5797 \, \sqrt{-c} c^{3}\right )}{\left (a x - 1\right )}^{2}}{63 \,{\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 = 1.60913, size = 212, normalized size = 0.68 \begin{align*} -\frac{2 \,{\left (7 \, a^{5} c^{3} x^{5} - 55 \, a^{4} c^{3} x^{4} + 214 \, a^{3} c^{3} x^{3} - 638 \, a^{2} c^{3} x^{2} + 2867 \, a c^{3} x + 5797 \, c^{3}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{63 \,{\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 = 1.30885, size = 170, normalized size = 0.55 \begin{align*} \frac{2 \,{\left (7 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c} - 90 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c + 504 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c^{2} + 1680 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{3} + 5040 \, \sqrt{-a c x - c} c^{4} - \frac{2016 \, c^{5}}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{63 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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