Integrand size = 23, antiderivative size = 322 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^3}+\frac {11 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^2}+\frac {107 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}+\frac {21 a^3 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{32 \sqrt {1-\frac {1}{a x}} x}+\frac {363 a^{7/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{64 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} a^{7/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {1-\frac {1}{a x}}} \]
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Time = 0.23 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6311, 6316, 103, 159, 163, 56, 221, 95, 212} \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {363 a^{7/2} \sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} a^{7/2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {21 a^3 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{32 x \sqrt {1-\frac {1}{a x}}}+\frac {107 a^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{64 x \sqrt {1-\frac {1}{a x}}}+\frac {11 a^2 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{24 x^2 \sqrt {1-\frac {1}{a x}}}+\frac {a \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{4 x^3 \sqrt {1-\frac {1}{a x}}} \]
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Rule 56
Rule 95
Rule 103
Rule 159
Rule 163
Rule 212
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 )^{3/2}}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^3}-\frac {\left (a \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {x^{3/2} \sqrt {1+\frac {x}{a}} \left (\frac {5}{2}+\frac {11 x}{2 a}\right )}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{4 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^3}+\frac {11 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^2}+\frac {\left (a^3 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {x} \left (-\frac {33}{4 a}-\frac {63 x}{4 a^2}\right ) \sqrt {1+\frac {x}{a}}}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{12 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^3}+\frac {11 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^2}+\frac {21 a^3 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{32 \sqrt {1-\frac {1}{a x}} x}-\frac {\left (a^5 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\left (\frac {63}{8 a^2}+\frac {321 x}{8 a^3}\right ) \sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{24 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^3}+\frac {11 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^2}+\frac {107 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}+\frac {21 a^3 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{32 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (a^6 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {-\frac {447}{16 a^3}-\frac {1089 x}{16 a^4}}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{24 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^3}+\frac {11 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^2}+\frac {107 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}+\frac {21 a^3 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{32 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (363 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 {\left (4 a^3 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\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 )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^3}+\frac {11 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^2}+\frac {107 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}+\frac {21 a^3 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{32 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (363 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 {\left (8 a^3 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\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 )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^3}+\frac {11 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^2}+\frac {107 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}+\frac {21 a^3 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{32 \sqrt {1-\frac {1}{a x}} x}+\frac {363 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}}}-\frac {4 \sqrt {2} a^{7/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.46 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c-a c x} \left (\sqrt {1+\frac {1}{a x}} \left (48+136 a x+214 a^2 x^2+447 a^3 x^3\right )+\frac {1089 a^{7/2} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{7/2}}-\frac {768 \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\left (\frac {1}{x}\right )^{7/2}}\right )}{192 \sqrt {1-\frac {1}{a x}} x^4} \]
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Time = 0.49 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.56
method | result | size |
risch | \(-\frac {\left (447 a^{4} x^{4}+661 a^{3} x^{3}+350 a^{2} x^{2}+184 a x +48\right ) c \left (a x -1\right )}{192 x^{4} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}-\frac {\left (\frac {4 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {363 a^{4} \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{64 \sqrt {c}}\right ) c \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}\) | \(181\) |
default | \(\frac {\left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-768 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{4} c \,x^{4}+1089 c \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{4} x^{4}+447 a^{3} x^{3} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+214 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+136 a x \sqrt {c}\, \sqrt {-c \left (a x +1\right )}+48 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{192 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {c}\, \sqrt {-c \left (a x +1\right )}\, x^{4}}\) | \(186\) |
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Time = 0.28 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.43 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\left [\frac {768 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 1089 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (447 \, a^{4} x^{4} + 661 \, a^{3} x^{3} + 350 \, a^{2} x^{2} + 184 \, a x + 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{384 \, {\left (a x^{5} - x^{4}\right )}}, -\frac {768 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 1089 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (447 \, a^{4} x^{4} + 661 \, a^{3} x^{3} + 350 \, a^{2} x^{2} + 184 \, a x + 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{192 \, {\left (a x^{5} - x^{4}\right )}}\right ] \]
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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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\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int { \frac {\sqrt {-a c x + c}}{x^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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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. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int \frac {\sqrt {c-a\,c\,x}}{x^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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