Integrand size = 21, antiderivative size = 104 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=-\frac {8 c \sqrt {1-\frac {1}{a^2 x^2}} x}{5 a \sqrt {c-a c x}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}}{5 a}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2}}{5 a c} \]
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Time = 0.14 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6311, 6316, 79, 47, 37} \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {12 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{5 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {6 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{5 a \sqrt {1-\frac {1}{a x}}} \]
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Rule 37
Rule 47
Rule 79
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{3/2} \, 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 {1-\frac {x}{a}}{x^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {\left (9 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 a \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {6 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{5 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {\left (6 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 a^2 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {12 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{5 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {6 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{5 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.56 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (6-3 a x+a^2 x^2\right )}{5 a^2 \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.44 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.48
method | result | size |
risch | \(-\frac {2 c \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-3 a x +6\right ) \left (a x +1\right )}{5 \sqrt {-c \left (a x -1\right )}\, a^{2}}\) | \(50\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (a^{2} x^{2}-3 a x +6\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{5 a^{2} \left (a x -1\right )}\) | \(55\) |
default | \(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (a^{2} x^{2}-3 a x +6\right )}{5 \left (a x -1\right ) a^{2}}\) | \(56\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.58 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \, a x + 6\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{3} x - a^{2}\right )}} \]
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\[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\int x \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{3} \sqrt {-c} x^{3} - 2 \, a^{2} \sqrt {-c} x^{2} + 3 \, a \sqrt {-c} x + 6 \, \sqrt {-c}\right )} {\left (a x - 1\right )}}{5 \, {\left (a^{3} x - a^{2}\right )} \sqrt {a x + 1}} \]
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=-\frac {4 \, \sqrt {-a c x - c} {\left | c \right |}}{a^{2} c} - \frac {2 \, {\left ({\left (a c x + c\right )}^{2} \sqrt {-a c x - c} {\left | c \right |} + 5 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c {\left | c \right |}\right )}}{5 \, a^{2} c^{3}} \]
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Time = 4.51 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.55 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (a^3\,x^3-2\,a^2\,x^2+3\,a\,x+6\right )}{5\,a^2\,\left (a\,x-1\right )} \]
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