Integrand size = 21, antiderivative size = 57 \[ \int e^{2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {4 \sqrt {c-a c x}}{a^2}-\frac {2 (c-a c x)^{3/2}}{a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2} \]
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Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6302, 6265, 21, 78} \[ \int e^{2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2}-\frac {2 (c-a c x)^{3/2}}{a^2 c}+\frac {4 \sqrt {c-a c x}}{a^2} \]
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Rule 21
Rule 78
Rule 6265
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} x \sqrt {c-a c x} \, dx \\ & = -\int \frac {x (1+a x) \sqrt {c-a c x}}{1-a x} \, dx \\ & = -\left (c \int \frac {x (1+a x)}{\sqrt {c-a c x}} \, dx\right ) \\ & = -\left (c \int \left (\frac {2}{a \sqrt {c-a c x}}-\frac {3 \sqrt {c-a c x}}{a c}+\frac {(c-a c x)^{3/2}}{a c^2}\right ) \, dx\right ) \\ & = \frac {4 \sqrt {c-a c x}}{a^2}-\frac {2 (c-a c x)^{3/2}}{a^2 c}+\frac {2 (c-a c x)^{5/2}}{5 a^2 c^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.54 \[ \int e^{2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (6+3 a x+a^2 x^2\right )}{5 a^2} \]
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Time = 0.55 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.49
method | result | size |
gosper | \(\frac {2 \sqrt {-a c x +c}\, \left (a^{2} x^{2}+3 a x +6\right )}{5 a^{2}}\) | \(28\) |
trager | \(\frac {2 \sqrt {-a c x +c}\, \left (a^{2} x^{2}+3 a x +6\right )}{5 a^{2}}\) | \(28\) |
pseudoelliptic | \(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (a^{2} x^{2}+3 a x +6\right )}{5 a^{2}}\) | \(29\) |
risch | \(-\frac {2 c \left (a^{2} x^{2}+3 a x +6\right ) \left (a x -1\right )}{5 a^{2} \sqrt {-c \left (a x -1\right )}}\) | \(35\) |
derivativedivides | \(\frac {\frac {2 \left (-a c x +c \right )^{\frac {5}{2}}}{5}-2 c \left (-a c x +c \right )^{\frac {3}{2}}+4 c^{2} \sqrt {-a c x +c}}{a^{2} c^{2}}\) | \(47\) |
default | \(\frac {\frac {2 \left (-a c x +c \right )^{\frac {5}{2}}}{5}-2 c \left (-a c x +c \right )^{\frac {3}{2}}+4 c^{2} \sqrt {-a c x +c}}{a^{2} c^{2}}\) | \(47\) |
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Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.47 \[ \int e^{2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{2} x^{2} + 3 \, a x + 6\right )} \sqrt {-a c x + c}}{5 \, a^{2}} \]
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Time = 2.53 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.40 \[ \int e^{2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\begin {cases} \frac {2 \cdot \left (2 c^{2} \sqrt {- a c x + c} - c \left (- a c x + c\right )^{\frac {3}{2}} + \frac {\left (- a c x + c\right )^{\frac {5}{2}}}{5}\right )}{a^{2} c^{2}} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\frac {x^{2}}{2} + \frac {2 x}{a} + \frac {2 \left (\begin {cases} - x & \text {for}\: a = 0 \\\frac {\log {\left (a x - 1 \right )}}{a} & \text {otherwise} \end {cases}\right )}{a}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.77 \[ \int e^{2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {5}{2}} - 5 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 10 \, \sqrt {-a c x + c} c^{2}\right )}}{5 \, a^{2} c^{2}} \]
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Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.61 \[ \int e^{2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=-\frac {2 \, {\left (\frac {5 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-a c x + c} c\right )}}{a c} - \frac {3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} - 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {-a c x + c} c^{2}}{a c^{2}}\right )}}{15 \, a} \]
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Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int e^{2 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2\,{\left (c-a\,c\,x\right )}^{5/2}-10\,c\,{\left (c-a\,c\,x\right )}^{3/2}+20\,c^2\,\sqrt {c-a\,c\,x}}{5\,a^2\,c^2} \]
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