Integrand size = 20, antiderivative size = 95 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\frac {64 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x}{15 \sqrt {c-a c x}}+\frac {16}{15} c \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}+\frac {2}{5} \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2} \]
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Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.44, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 91, 79, 37} \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\frac {86 \sqrt {\frac {1}{a x}+1} (c-a c x)^{3/2}}{15 a^2 x \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {2 x \sqrt {\frac {1}{a x}+1} (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {28 \sqrt {\frac {1}{a x}+1} (c-a c x)^{3/2}}{15 a \left (1-\frac {1}{a x}\right )^{3/2}} \]
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Rule 37
Rule 79
Rule 91
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {(c-a c x)^{3/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}} \\ & = \frac {2 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {\left (2 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {-\frac {7}{a}+\frac {5 x}{2 a^2}}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 \left (1-\frac {1}{a x}\right )^{3/2}} \\ & = -\frac {28 \sqrt {1+\frac {1}{a x}} (c-a c x)^{3/2}}{15 a \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {2 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {\left (43 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{15 a^2 \left (1-\frac {1}{a x}\right )^{3/2}} \\ & = -\frac {28 \sqrt {1+\frac {1}{a x}} (c-a c x)^{3/2}}{15 a \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {86 \sqrt {1+\frac {1}{a x}} (c-a c x)^{3/2}}{15 a^2 \left (1-\frac {1}{a x}\right )^{3/2} x}+\frac {2 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.63 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 c \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (43-14 a x+3 a^2 x^2\right )}{15 a \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.41 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.56
method | result | size |
risch | \(\frac {2 c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (3 a^{2} x^{2}-14 a x +43\right ) \left (a x +1\right )}{15 \sqrt {-c \left (a x -1\right )}\, a}\) | \(53\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (3 a^{2} x^{2}-14 a x +43\right ) \left (-a c x +c \right )^{\frac {3}{2}} \sqrt {\frac {a x -1}{a x +1}}}{15 a \left (a x -1\right )^{2}}\) | \(56\) |
default | \(-\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c \left (3 a^{2} x^{2}-14 a x +43\right )}{15 \left (a x -1\right ) a}\) | \(58\) |
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Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 \, {\left (3 \, a^{3} c x^{3} - 11 \, a^{2} c x^{2} + 29 \, a c x + 43 \, c\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{2} x - a\right )}} \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.76 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 \, {\left (3 \, a^{3} \sqrt {-c} c x^{3} - 11 \, a^{2} \sqrt {-c} c x^{2} + 29 \, a \sqrt {-c} c x + 43 \, \sqrt {-c} c\right )} {\left (a x - 1\right )}}{15 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \]
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Exception generated. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.40 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (3\,a^2\,x^2-8\,a\,x+21\right )}{15\,a}-\frac {128\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{15\,a\,\left (a\,x-1\right )} \]
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