Integrand size = 23, antiderivative size = 274 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^2}+\frac {13 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {3 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x}+\frac {45 a^{5/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{8 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} a^{5/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {1-\frac {1}{a x}}} \]
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Time = 0.21 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {45 a^{5/2} \sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} a^{5/2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {3 a^2 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{4 x \sqrt {1-\frac {1}{a x}}}+\frac {13 a^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{8 x \sqrt {1-\frac {1}{a x}}}+\frac {a \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{3 x^2 \sqrt {1-\frac {1}{a x}}} \]
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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^{7/2}} \, 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^{3/2} \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}}{3 \sqrt {1-\frac {1}{a x}} x^2}-\frac {\left (a \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {x} \sqrt {1+\frac {x}{a}} \left (\frac {3}{2}+\frac {9 x}{2 a}\right )}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{3 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^2}+\frac {3 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (a^3 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\left (-\frac {9}{4 a}-\frac {39 x}{4 a^2}\right ) \sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{6 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^2}+\frac {13 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {3 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x}-\frac {\left (a^4 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\frac {57}{8 a^2}+\frac {135 x}{8 a^3}}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^2}+\frac {13 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {3 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (45 a^2 \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 )}{16 \sqrt {1-\frac {1}{a x}}}-\frac {\left (4 a^2 \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}}{3 \sqrt {1-\frac {1}{a x}} x^2}+\frac {13 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {3 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (45 a^2 \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 )}{8 \sqrt {1-\frac {1}{a x}}}-\frac {\left (8 a^2 \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}}{3 \sqrt {1-\frac {1}{a x}} x^2}+\frac {13 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {3 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x}+\frac {45 a^{5/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{8 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} a^{5/2} \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 = 0.23 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.51 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c-a c x} \left (\sqrt {1+\frac {1}{a x}} \left (8+26 a x+57 a^2 x^2\right )+\frac {135 a^{5/2} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{5/2}}-\frac {96 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\left (\frac {1}{x}\right )^{5/2}}\right )}{24 \sqrt {1-\frac {1}{a x}} x^3} \]
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Time = 0.49 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {\left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-96 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{3} c \,x^{3}+135 c \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{3} x^{3}+57 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+26 a x \sqrt {c}\, \sqrt {-c \left (a x +1\right )}+8 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{24 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {c}\, \sqrt {-c \left (a x +1\right )}\, x^{3}}\) | \(165\) |
risch | \(-\frac {\left (57 a^{3} x^{3}+83 a^{2} x^{2}+34 a x +8\right ) c \left (a x -1\right )}{24 x^{3} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}-\frac {\left (\frac {4 a^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {45 a^{3} \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{8 \sqrt {c}}\right ) c \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}\) | \(173\) |
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Time = 0.27 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.62 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\left [\frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (57 \, a^{3} x^{3} + 83 \, a^{2} x^{2} + 34 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{48 \, {\left (a x^{4} - x^{3}\right )}}, -\frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (57 \, a^{3} x^{3} + 83 \, a^{2} x^{2} + 34 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int { \frac {\sqrt {-a c x + c}}{x^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {c-a\,c\,x}}{x^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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