Integrand size = 23, antiderivative size = 80 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {4 \sqrt {c-a c x}}{a^3}-\frac {10 (c-a c x)^{3/2}}{3 a^3 c}+\frac {8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3} \]
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Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6302, 6265, 21, 78} \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac {10 (c-a c x)^{3/2}}{3 a^3 c}+\frac {4 \sqrt {c-a c x}}{a^3} \]
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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^2 \sqrt {c-a c x} \, dx \\ & = -\int \frac {x^2 (1+a x) \sqrt {c-a c x}}{1-a x} \, dx \\ & = -\left (c \int \frac {x^2 (1+a x)}{\sqrt {c-a c x}} \, dx\right ) \\ & = -\left (c \int \left (\frac {2}{a^2 \sqrt {c-a c x}}-\frac {5 \sqrt {c-a c x}}{a^2 c}+\frac {4 (c-a c x)^{3/2}}{a^2 c^2}-\frac {(c-a c x)^{5/2}}{a^2 c^3}\right ) \, dx\right ) \\ & = \frac {4 \sqrt {c-a c x}}{a^3}-\frac {10 (c-a c x)^{3/2}}{3 a^3 c}+\frac {8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.50 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (104+52 a x+39 a^2 x^2+15 a^3 x^3\right )}{105 a^3} \]
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Time = 0.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(\frac {2 \sqrt {-a c x +c}\, \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right )}{105 a^{3}}\) | \(37\) |
trager | \(\frac {2 \sqrt {-a c x +c}\, \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right )}{105 a^{3}}\) | \(37\) |
pseudoelliptic | \(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right )}{105 a^{3}}\) | \(38\) |
risch | \(-\frac {2 c \left (15 a^{3} x^{3}+39 a^{2} x^{2}+52 a x +104\right ) \left (a x -1\right )}{105 a^{3} \sqrt {-c \left (a x -1\right )}}\) | \(44\) |
derivativedivides | \(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {7}{2}}}{7}-\frac {4 c \left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {5 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}-2 c^{3} \sqrt {-a c x +c}\right )}{c^{3} a^{3}}\) | \(61\) |
default | \(\frac {-\frac {2 \left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {8 c \left (-a c x +c \right )^{\frac {5}{2}}}{5}-\frac {10 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 c^{3} \sqrt {-a c x +c}}{a^{3} c^{3}}\) | \(61\) |
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Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.45 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (15 \, a^{3} x^{3} + 39 \, a^{2} x^{2} + 52 \, a x + 104\right )} \sqrt {-a c x + c}}{105 \, a^{3}} \]
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Time = 2.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\begin {cases} - \frac {2 \left (- 2 c^{3} \sqrt {- a c x + c} + \frac {5 c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {4 c \left (- a c x + c\right )^{\frac {5}{2}}}{5} + \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a^{3} c^{3}} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\frac {x^{3}}{3} + \frac {x^{2}}{a} + \frac {2 x}{a^{2}} + \frac {2 \left (\begin {cases} - x & \text {for}\: a = 0 \\\frac {\log {\left (a x - 1 \right )}}{a} & \text {otherwise} \end {cases}\right )}{a^{2}}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {2 \, {\left (15 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 84 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 175 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 210 \, \sqrt {-a c x + c} c^{3}\right )}}{105 \, a^{3} c^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (66) = 132\).
Time = 0.26 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.78 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (\frac {7 \, {\left (3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} - 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {-a c x + c} c^{2}\right )}}{a^{2} c^{2}} + \frac {3 \, {\left (5 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} + 21 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} c - 35 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, \sqrt {-a c x + c} c^{3}\right )}}{a^{2} c^{3}}\right )}}{105 \, a} \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {c-a\,c\,x}}{a^3}-\frac {10\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^3\,c}+\frac {8\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a^3\,c^2}-\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a^3\,c^3} \]
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