Integrand size = 22, antiderivative size = 277 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=-\frac {3 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {23 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{8 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {7 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {79 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{8 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}} \]
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Time = 0.15 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6317, 6314, 101, 156, 162, 65, 214, 212} \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {a^2 x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{5/2}}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {7 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {79 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{8 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {23 \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{5/2}}{8 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {3 a \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{5/2}}{2 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}} \]
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Rule 65
Rule 101
Rule 156
Rule 162
Rule 212
Rule 214
Rule 6314
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-\frac {1}{a x}\right )^{5/2} \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{5/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}} \\ & = \frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {Subst}\left (\int \frac {\frac {7}{2 a}+\frac {5 x}{2 a^2}}{x \left (1-\frac {x}{a}\right )^3 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {3 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {-\frac {14}{a^2}-\frac {9 x}{a^3}}{x \left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{4 \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {3 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {23 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{8 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (a^2 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {28}{a^3}+\frac {23 x}{2 a^4}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {3 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {23 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{8 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (79 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (7 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {3 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {23 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{8 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (7 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (79 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{8 a \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {3 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {23 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{8 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {7 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {79 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{8 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.49 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a \sqrt {1+\frac {1}{a x}} x \left (23-35 a x+8 a^2 x^2\right )+112 (-1+a x)^2 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-79 \sqrt {2} (-1+a x)^2 \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{16 a c^2 \sqrt {c-\frac {c}{a x}} (-1+a x)^2} \]
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Time = 0.30 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.18
method | result | size |
risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {7 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{2 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {19 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{8 a^{5} c \left (x -\frac {1}{a}\right )}-\frac {79 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{32 a^{4} \sqrt {c}}\right ) a^{2} \sqrt {\left (a x +1\right ) a c x}\, \left (a x -1\right )}{c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(328\) |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (32 \sqrt {\left (a x +1\right ) x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}-79 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x^{2}-140 \sqrt {\left (a x +1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x +112 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {1}{a}}\, x^{2}+158 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x +92 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-224 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x +112 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-79 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{32 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right )^{2} a^{\frac {3}{2}} c^{3} \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}}\) | \(366\) |
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Time = 0.33 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.41 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {79 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 112 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (8 \, a^{4} x^{4} - 27 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 23 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{64 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, \frac {79 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 112 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (8 \, a^{4} x^{4} - 27 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 23 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{32 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {1}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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