Integrand size = 23, antiderivative size = 110 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {75}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 110, 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 = {6302, 6265, 21, 79, 44, 65, 214} \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {75}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3} \]
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Rule 21
Rule 44
Rule 65
Rule 79
Rule 214
Rule 6265
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx \\ & = -\int \frac {(1+a x) \sqrt {c-a c x}}{x^5 (1-a x)} \, dx \\ & = -\left (c \int \frac {1+a x}{x^5 \sqrt {c-a c x}} \, dx\right ) \\ & = \frac {\sqrt {c-a c x}}{4 x^4}-\frac {1}{8} (15 a c) \int \frac {1}{x^4 \sqrt {c-a c x}} \, dx \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}-\frac {1}{16} \left (25 a^2 c\right ) \int \frac {1}{x^3 \sqrt {c-a c x}} \, dx \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}-\frac {1}{64} \left (75 a^3 c\right ) \int \frac {1}{x^2 \sqrt {c-a c x}} \, dx \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}-\frac {1}{128} \left (75 a^4 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {1}{64} \left (75 a^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right ) \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {75}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.65 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c-a c x} \left (16+40 a x+50 a^2 x^2+75 a^3 x^3\right )}{64 x^4}+\frac {75}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]
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Time = 0.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {\left (75 a^{4} x^{4}-25 a^{3} x^{3}-10 a^{2} x^{2}-24 a x -16\right ) c}{64 x^{4} \sqrt {-c \left (a x -1\right )}}+\frac {75 a^{4} \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right ) \sqrt {c}}{64}\) | \(70\) |
pseudoelliptic | \(\frac {\frac {\sqrt {-c \left (a x -1\right )}\, \left (75 a^{3} x^{3}+50 a^{2} x^{2}+40 a x +16\right ) \sqrt {c}}{64}+\frac {75 \,\operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right ) c \,a^{4} x^{4}}{64}}{\sqrt {c}\, x^{4}}\) | \(70\) |
derivativedivides | \(2 c^{4} a^{4} \left (\frac {-\frac {75 \left (-a c x +c \right )^{\frac {7}{2}}}{128 c^{3}}+\frac {275 \left (-a c x +c \right )^{\frac {5}{2}}}{128 c^{2}}-\frac {365 \left (-a c x +c \right )^{\frac {3}{2}}}{128 c}+\frac {181 \sqrt {-a c x +c}}{128}}{a^{4} c^{4} x^{4}}+\frac {75 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 c^{\frac {7}{2}}}\right )\) | \(93\) |
default | \(2 c^{4} a^{4} \left (\frac {-\frac {75 \left (-a c x +c \right )^{\frac {7}{2}}}{128 c^{3}}+\frac {275 \left (-a c x +c \right )^{\frac {5}{2}}}{128 c^{2}}-\frac {365 \left (-a c x +c \right )^{\frac {3}{2}}}{128 c}+\frac {181 \sqrt {-a c x +c}}{128}}{a^{4} c^{4} x^{4}}+\frac {75 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 c^{\frac {7}{2}}}\right )\) | \(93\) |
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Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.35 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\left [\frac {75 \, a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, {\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt {-a c x + c}}{128 \, x^{4}}, -\frac {75 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt {-a c x + c}}{64 \, x^{4}}\right ] \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )}{x^{5} \left (a x - 1\right )}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.48 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {1}{128} \, a^{4} c^{4} {\left (\frac {2 \, {\left (75 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 275 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 365 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 181 \, \sqrt {-a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{3} + 4 \, {\left (a c x - c\right )}^{3} c^{4} + 6 \, {\left (a c x - c\right )}^{2} c^{5} + 4 \, {\left (a c x - c\right )} c^{6} + c^{7}} + \frac {75 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.19 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {\frac {75 \, a^{5} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {75 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{5} c + 275 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{5} c^{2} - 365 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{5} c^{3} + 181 \, \sqrt {-a c x + c} a^{5} c^{4}}{a^{4} c^{4} x^{4}}}{64 \, a} \]
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Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {181\,\sqrt {c-a\,c\,x}}{64\,x^4}-\frac {365\,{\left (c-a\,c\,x\right )}^{3/2}}{64\,c\,x^4}+\frac {275\,{\left (c-a\,c\,x\right )}^{5/2}}{64\,c^2\,x^4}-\frac {75\,{\left (c-a\,c\,x\right )}^{7/2}}{64\,c^3\,x^4}-\frac {a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,75{}\mathrm {i}}{64} \]
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