Integrand size = 20, antiderivative size = 163 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{a^{3/2} \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 96, 95, 212} \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-a c x}}{a^{3/2} \sqrt {1-\frac {1}{a x}}}+\frac {2 x \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {4 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}}} \]
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Rule 95
Rule 96
Rule 212
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} \sqrt {x} \, 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 {\left (1+\frac {x}{a}\right )^{3/2}}{x^{5/2} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{3/2} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{a \sqrt {1-\frac {1}{a x}}} \\ & = \frac {4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {\left (4 \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 )}{a^2 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {\left (8 \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 )}{a^2 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {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 )}{a^{3/2} \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.64 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {1+\frac {1}{a x}} (7+a x)-6 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{3 a^{3/2} \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.45 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.66
method | result | size |
default | \(-\frac {2 \left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (6 \sqrt {c}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-a x \sqrt {-c \left (a x +1\right )}-7 \sqrt {-c \left (a x +1\right )}\right )}{3 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x +1\right )}\, a}\) | \(107\) |
risch | \(-\frac {2 \left (a x +7\right ) c \left (a x -1\right )}{3 a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{a \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}\) | \(121\) |
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Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.53 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (a^{2} x^{2} + 8 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{3 \, {\left (a^{2} x - a\right )}}, -\frac {2 \, {\left (6 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (a^{2} x^{2} + 8 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{3 \, {\left (a^{2} x - a\right )}}\right ] \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a c x + c}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\int \frac {\sqrt {c-a\,c\,x}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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