Optimal. Leaf size=161 \[ \frac{4096 c^4 x \sqrt{1-\frac{1}{a^2 x^2}}}{315 \sqrt{c-a c x}}+\frac{1024}{315} c^3 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{128}{105} c^2 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}+\frac{32}{63} c x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{5/2}+\frac{2}{9} x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{7/2} \]
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Rubi [A] time = 0.222049, antiderivative size = 254, normalized size of antiderivative = 1.58, 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} \[ -\frac{256 \sqrt{\frac{1}{a x}+1} (c-a c x)^{7/2}}{45 a^3 x^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{5504 \sqrt{\frac{1}{a x}+1} (c-a c x)^{7/2}}{315 a^4 x^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{2 x \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}{9 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{32 \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{128 \sqrt{\frac{1}{a x}+1} (c-a c x)^{7/2}}{105 a^2 x \left (1-\frac{1}{a x}\right )^{7/2}} \]
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)^{7/2} \, dx &=\frac{(c-a c x)^{7/2} \int e^{-\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 )^4}{x^{11/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{7/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (16 \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^{9/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{9 a \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{63 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{2 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{\left (64 \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^{7/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{21 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{63 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{128 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{105 a^2 \left (1-\frac{1}{a x}\right )^{7/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{\left (128 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\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 )}{105 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{63 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{256 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{45 a^3 \left (1-\frac{1}{a x}\right )^{7/2} x^2}+\frac{128 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{105 a^2 \left (1-\frac{1}{a x}\right )^{7/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{\left (2752 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{315 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{32 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{63 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{5504 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{315 a^4 \left (1-\frac{1}{a x}\right )^{7/2} x^3}-\frac{256 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{45 a^3 \left (1-\frac{1}{a x}\right )^{7/2} x^2}+\frac{128 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}{105 a^2 \left (1-\frac{1}{a x}\right )^{7/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac{1}{a x}\right )^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.045066, size = 78, normalized size = 0.48 \[ -\frac{2 c^3 \sqrt{\frac{1}{a x}+1} \left (35 a^4 x^4-220 a^3 x^3+642 a^2 x^2-1276 a x+2867\right ) \sqrt{c-a c x}}{315 a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.044, size = 72, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 35\,{x}^{4}{a}^{4}-220\,{x}^{3}{a}^{3}+642\,{a}^{2}{x}^{2}-1276\,ax+2867 \right ) }{315\,a \left ( ax-1 \right ) ^{4}} \left ( -acx+c \right ) ^{{\frac{7}{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 = 1.10925, size = 151, normalized size = 0.94 \begin{align*} -\frac{2 \,{\left (35 \, a^{5} \sqrt{-c} c^{3} x^{5} - 185 \, a^{4} \sqrt{-c} c^{3} x^{4} + 422 \, a^{3} \sqrt{-c} c^{3} x^{3} - 634 \, a^{2} \sqrt{-c} c^{3} x^{2} + 1591 \, a \sqrt{-c} c^{3} x + 2867 \, \sqrt{-c} c^{3}\right )}{\left (a x - 1\right )}}{315 \,{\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 = 1.87376, size = 216, normalized size = 1.34 \begin{align*} -\frac{2 \,{\left (35 \, a^{5} c^{3} x^{5} - 185 \, a^{4} c^{3} x^{4} + 422 \, a^{3} c^{3} x^{3} - 634 \, a^{2} c^{3} x^{2} + 1591 \, a c^{3} x + 2867 \, c^{3}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{315 \,{\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 = 1.23105, size = 149, normalized size = 0.93 \begin{align*} \frac{2 \,{\left (35 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c} - 360 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c + 1512 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c^{2} + 3360 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{3} + 5040 \, \sqrt{-a c x - c} c^{4}\right )}{\left | c \right |}}{315 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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