Integrand size = 23, antiderivative size = 142 \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {152 c \sqrt {1-\frac {1}{a^2 x^2}} x}{105 a^2 \sqrt {c-a c x}}+\frac {38 \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}}{105 a^2}+\frac {6 \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2}}{35 a^2 c}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 (c-a c x)^{3/2}}{7 a c} \]
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Time = 0.17 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.30, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6311, 6316, 79, 47, 37} \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {208 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{105 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {104 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}-\frac {26 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}} \]
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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^{5/2} \, 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^{9/2} \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^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {\left (13 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{x^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{7 a \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {26 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}-\frac {\left (52 \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 )}{35 a^2 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {104 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {26 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {\left (104 \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 )}{105 a^3 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {208 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{105 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {104 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {26 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.47 \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (-104+52 a x-39 a^2 x^2+15 a^3 x^3\right )}{105 a^3 \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.46 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.42
method | result | size |
risch | \(-\frac {2 c \sqrt {\frac {a x -1}{a x +1}}\, \left (15 a^{3} x^{3}-39 a^{2} x^{2}+52 a x -104\right ) \left (a x +1\right )}{105 \sqrt {-c \left (a x -1\right )}\, a^{3}}\) | \(59\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (15 a^{3} x^{3}-39 a^{2} x^{2}+52 a x -104\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{105 a^{3} \left (a x -1\right )}\) | \(64\) |
default | \(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (15 a^{3} x^{3}-39 a^{2} x^{2}+52 a x -104\right )}{105 \left (a x -1\right ) a^{3}}\) | \(65\) |
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Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.49 \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 13 \, a^{2} x^{2} - 52 \, a x - 104\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{4} x - a^{3}\right )}} \]
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\[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\int x^{2} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58 \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (15 \, a^{4} \sqrt {-c} x^{4} - 24 \, a^{3} \sqrt {-c} x^{3} + 13 \, a^{2} \sqrt {-c} x^{2} - 52 \, a \sqrt {-c} x - 104 \, \sqrt {-c}\right )} {\left (a x - 1\right )}}{105 \, {\left (a^{4} x - a^{3}\right )} \sqrt {a x + 1}} \]
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Exception generated. \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.54 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.62 \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (15\,a^3\,x^3-9\,a^2\,x^2+4\,a\,x-48\right )}{105\,a^3}-\frac {304\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{105\,a^3\,\left (a\,x-1\right )} \]
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