Integrand size = 20, antiderivative size = 250 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {a^{3/2} \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{16 \sqrt {2} \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \]
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Time = 0.16 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 96, 95, 212} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {a^{3/2} \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{16 \sqrt {2} \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}-\frac {a^4 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {a^3 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}+\frac {a^2 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}} \]
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Rule 95
Rule 96
Rule 212
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}} \, dx}{(c-a c x)^{5/2}} \\ & = -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {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 )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ & = -\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {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 )}{12 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ & = \frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{16 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ & = \frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{32 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ & = \frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {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 )}{16 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ & = \frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {a^{3/2} \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{16 \sqrt {2} \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.57 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=-\frac {\sqrt {1-\frac {1}{a x}} \left (2 \sqrt {a} \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}} \left (7+22 a x+3 a^2 x^2\right )-\frac {3 \sqrt {2} (-1+a x)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{x}\right )}{96 \sqrt {a} c^2 \left (\frac {1}{x}\right )^{3/2} (-1+a x)^3 \sqrt {c-a c x}} \]
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Time = 0.43 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\sqrt {-c \left (a x -1\right )}\, \left (-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{3} c \,x^{3}+9 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{2} c \,x^{2}+6 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}-9 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x +44 a x \sqrt {c}\, \sqrt {-c \left (a x +1\right )}+3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +14 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{96 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x -1\right )^{2} \left (a x +1\right ) c^{\frac {7}{2}} \sqrt {-c \left (a x +1\right )}\, a}\) | \(226\) |
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Time = 0.25 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.57 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\left [-\frac {3 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (3 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 29 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{192 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \, {\left (3 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 29 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{96 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int { \frac {1}{{\left (-a c x + c\right )}^{\frac {5}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.42 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} - 16 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c - 12 \, \sqrt {-a c x - c} c^{2}\right )}}{{\left (a c x - c\right )}^{3} c}}{96 \, a {\left | c \right |}} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int \frac {1}{{\left (c-a\,c\,x\right )}^{5/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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