Optimal. Leaf size=307 \[ \frac{a^5 x^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{5/2}}{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}-\frac{a^5 x^2 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{5/2}}{8 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}-\frac{a^4 x^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{3/2}}{128 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}-\frac{3 a^3 x^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}{256 \left (a-\frac{1}{x}\right ) (c-a c x)^{7/2}}-\frac{3 a^{5/2} \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{256 \sqrt{2} \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}} \]
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Rubi [A] time = 0.238829, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6176, 6181, 94, 93, 206} \[ \frac{a^5 x^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{5/2}}{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}-\frac{a^5 x^2 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{5/2}}{8 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}-\frac{a^4 x^3 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{3/2}}{128 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}-\frac{3 a^3 x^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}{256 \left (a-\frac{1}{x}\right ) (c-a c x)^{7/2}}-\frac{3 a^{5/2} \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{256 \sqrt{2} \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{7/2} x^{7/2}} \, dx}{(c-a c x)^{7/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{7/2} \operatorname{Subst}\left (\int \frac{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}}{\left (1-\frac{x}{a}\right )^5} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^2}{8 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}+\frac{\left (3 a \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}}{\left (1-\frac{x}{a}\right )^4} \, dx,x,\frac{1}{x}\right )}{16 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^2}{8 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}+\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^3}{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}-\frac{\left (a^2 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{\sqrt{x} \left (1-\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{64 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^2}{8 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}-\frac{a^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2} x^3}{128 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}+\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^3}{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}-\frac{\left (3 a^2 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{\sqrt{x} \left (1-\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{256 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^2}{8 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}-\frac{3 a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^3}{256 \left (a-\frac{1}{x}\right ) (c-a c x)^{7/2}}-\frac{a^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2} x^3}{128 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}+\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^3}{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}-\frac{\left (3 a^2 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{512 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^2}{8 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}-\frac{3 a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^3}{256 \left (a-\frac{1}{x}\right ) (c-a c x)^{7/2}}-\frac{a^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2} x^3}{128 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}+\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^3}{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}-\frac{\left (3 a^2 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{256 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^2}{8 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{7/2}}-\frac{3 a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^3}{256 \left (a-\frac{1}{x}\right ) (c-a c x)^{7/2}}-\frac{a^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2} x^3}{128 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}+\frac{a^5 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2} x^3}{32 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{7/2}}-\frac{3 a^{5/2} \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{256 \sqrt{2} \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.188333, size = 147, normalized size = 0.48 \[ \frac{\sqrt{1-\frac{1}{a x}} \left (\frac{2 \sqrt{a} \sqrt{\frac{1}{a x}+1} \left (-3 a^3 x^3+13 a^2 x^2+79 a x+39\right )}{\sqrt{\frac{1}{x}}}+3 \sqrt{2} (a x-1)^4 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{512 \sqrt{a} c^3 \sqrt{\frac{1}{x}} (a x-1)^4 \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.25, size = 278, normalized size = 0.9 \begin{align*} -{\frac{1}{512\, \left ( ax-1 \right ) ^{3} \left ( ax+1 \right ) a}\sqrt{-c \left ( ax-1 \right ) } \left ( 3\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{4}{a}^{4}c-12\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{3}{a}^{3}c-6\,{x}^{3}{a}^{3}\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}+18\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{2}{a}^{2}c+26\,{x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}-12\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xac+158\,xa\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}+3\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) c+78\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a c x + c\right )}^{\frac{7}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68711, size = 1030, normalized size = 3.36 \begin{align*} \left [-\frac{3 \, \sqrt{2}{\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \,{\left (3 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 92 \, a^{2} x^{2} - 118 \, a x - 39\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{1024 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}}, -\frac{3 \, \sqrt{2}{\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \,{\left (3 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 92 \, a^{2} x^{2} - 118 \, a x - 39\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{512 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}}\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 = 1.23487, size = 189, normalized size = 0.62 \begin{align*} -\frac{\frac{3 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{c^{\frac{5}{2}}} - \frac{2 \,{\left (3 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 22 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c + 44 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{2} + 24 \, \sqrt{-a c x - c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{2}}}{512 \, a c \mathrm{sgn}\left (-a c x - c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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