Integrand size = 23, antiderivative size = 286 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^4}+\frac {223 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^3}-\frac {1115 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{96 \sqrt {1-\frac {1}{a x}} x^2}+\frac {1115 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}-\frac {1115 a^{7/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{64 \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.20 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6311, 6316, 91, 81, 52, 56, 221} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {1115 a^{7/2} \sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}}}+\frac {1115 a^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{64 x \sqrt {1-\frac {1}{a x}}}-\frac {1115 a^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{96 x^2 \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{4 x^4 \sqrt {1-\frac {1}{a x}}}-\frac {8 \sqrt {c-a c x}}{x^4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {223 a \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{24 x^3 \sqrt {1-\frac {1}{a x}}} \]
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Rule 52
Rule 56
Rule 81
Rule 91
Rule 221
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{x^{9/2}} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \\ & = -\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {x^{5/2} \left (1-\frac {x}{a}\right )^2}{\left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}+\frac {\left (2 a^2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {x^{5/2} \left (\frac {27}{2 a^2}-\frac {x}{2 a^3}\right )}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^4}+\frac {\left (223 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {x^{5/2}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^4}+\frac {223 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^3}-\frac {\left (1115 a \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{48 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^4}+\frac {223 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^3}-\frac {1115 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{96 \sqrt {1-\frac {1}{a x}} x^2}+\frac {\left (1115 a^2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{64 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^4}+\frac {223 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^3}-\frac {1115 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{96 \sqrt {1-\frac {1}{a x}} x^2}+\frac {1115 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}-\frac {\left (1115 a^3 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{128 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^4}+\frac {223 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^3}-\frac {1115 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{96 \sqrt {1-\frac {1}{a x}} x^2}+\frac {1115 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}-\frac {\left (1115 a^3 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,\sqrt {\frac {1}{x}}\right )}{64 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^4}+\frac {223 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^3}-\frac {1115 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{96 \sqrt {1-\frac {1}{a x}} x^2}+\frac {1115 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}-\frac {1115 a^{7/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{64 \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.37 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {\sqrt {c-a c x} \left (48-200 a x+446 a^2 x^2-1115 a^3 x^3-3345 a^4 x^4+\frac {3345 a^{9/2} \sqrt {1+\frac {1}{a x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{9/2}}\right )}{192 a \sqrt {1-\frac {1}{a^2 x^2}} x^5} \]
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Time = 0.49 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.44
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (3345 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{4} x^{4} \sqrt {-c \left (a x +1\right )}+3345 a^{4} x^{4} \sqrt {c}+1115 a^{3} x^{3} \sqrt {c}-446 \sqrt {c}\, a^{2} x^{2}+200 \sqrt {c}\, a x -48 \sqrt {c}\right )}{192 \left (a x -1\right )^{2} \sqrt {c}\, x^{4}}\) | \(125\) |
risch | \(-\frac {\left (1809 a^{4} x^{4}+1115 a^{3} x^{3}-446 a^{2} x^{2}+200 a x -48\right ) c \sqrt {\frac {a x -1}{a x +1}}}{192 x^{4} \sqrt {-c \left (a x -1\right )}}-\frac {\left (\frac {8 a^{4}}{\sqrt {-a c x -c}}+\frac {1115 a^{4} \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{64 \sqrt {c}}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}{\sqrt {-c \left (a x -1\right )}}\) | \(142\) |
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Time = 0.27 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\left [\frac {3345 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{384 \, {\left (a x^{5} - x^{4}\right )}}, -\frac {3345 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{192 \, {\left (a x^{5} - x^{4}\right )}}\right ] \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int { \frac {\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^5} \,d x \]
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