Integrand size = 27, antiderivative size = 121 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=4 a^4 \sqrt {c-\frac {c}{a x}}-\frac {14 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {18 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {10 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4} \]
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Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6302, 6268, 25, 528, 457, 78} \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}-\frac {10 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {18 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {14 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+4 a^4 \sqrt {c-\frac {c}{a x}} \]
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Rule 25
Rule 78
Rule 457
Rule 528
Rule 6268
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx \\ & = -\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{x^5 (1-a x)} \, dx \\ & = \frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x^6} \, dx}{a} \\ & = \frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x^5} \, dx}{a} \\ & = -\frac {c \text {Subst}\left (\int \frac {x^3 (a+x)}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = -\frac {c \text {Subst}\left (\int \left (\frac {2 a^4}{\sqrt {c-\frac {c x}{a}}}-\frac {7 a^4 \sqrt {c-\frac {c x}{a}}}{c}+\frac {9 a^4 \left (c-\frac {c x}{a}\right )^{3/2}}{c^2}-\frac {5 a^4 \left (c-\frac {c x}{a}\right )^{5/2}}{c^3}+\frac {a^4 \left (c-\frac {c x}{a}\right )^{7/2}}{c^4}\right ) \, dx,x,\frac {1}{x}\right )}{a} \\ & = 4 a^4 \sqrt {c-\frac {c}{a x}}-\frac {14 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {18 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {10 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.43 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (35+85 a x+102 a^2 x^2+136 a^3 x^3+272 a^4 x^4\right )}{315 x^4} \]
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Time = 0.48 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.42
method | result | size |
gosper | \(\frac {2 \left (272 a^{4} x^{4}+136 a^{3} x^{3}+102 a^{2} x^{2}+85 a x +35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4}}\) | \(51\) |
trager | \(\frac {2 \left (272 a^{4} x^{4}+136 a^{3} x^{3}+102 a^{2} x^{2}+85 a x +35\right ) \sqrt {-\frac {-a c x +c}{a x}}}{315 x^{4}}\) | \(53\) |
risch | \(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (272 a^{5} x^{5}-136 a^{4} x^{4}-34 a^{3} x^{3}-17 a^{2} x^{2}-50 a x -35\right )}{315 \left (a x -1\right ) x^{4}}\) | \(66\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-630 \sqrt {a \,x^{2}-x}\, a^{\frac {11}{2}} x^{6}-630 a^{\frac {11}{2}} \sqrt {\left (a x -1\right ) x}\, x^{6}+1260 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{4}+315 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{5} x^{6}-315 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{5} x^{6}+716 a^{\frac {7}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{3}+444 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}+240 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x +70 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{315 x^{5} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}}\) | \(230\) |
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Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.43 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \, {\left (272 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 85 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}}{315 \, x^{4}} \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{5} \left (a x - 1\right )}\, dx \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x - 1\right )} x^{5}} \,d x } \]
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Exception generated. \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {544\,a^4\,\sqrt {c-\frac {c}{a\,x}}}{315}+\frac {2\,\sqrt {c-\frac {c}{a\,x}}}{9\,x^4}+\frac {34\,a\,\sqrt {c-\frac {c}{a\,x}}}{63\,x^3}+\frac {68\,a^2\,\sqrt {c-\frac {c}{a\,x}}}{105\,x^2}+\frac {272\,a^3\,\sqrt {c-\frac {c}{a\,x}}}{315\,x} \]
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