Integrand size = 24, antiderivative size = 93 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=-\frac {2 (1+a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}+\frac {(1+a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \]
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Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6327, 6328, 45} \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {(a x+1)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {2 (a x+1)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]
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Rule 45
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-a^2 c x^2\right )^{5/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \\ & = \frac {\left (c-a^2 c x^2\right )^{5/2} \int (-1+a x) (1+a x)^4 \, dx}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \\ & = \frac {\left (c-a^2 c x^2\right )^{5/2} \int \left (-2 (1+a x)^4+(1+a x)^5\right ) \, dx}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \\ & = -\frac {2 (1+a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}+\frac {(1+a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.59 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {c^2 (1+a x)^5 (-7+5 a x) \sqrt {c-a^2 c x^2}}{30 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.51 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(\frac {x \left (5 a^{5} x^{5}+18 a^{4} x^{4}+15 a^{3} x^{3}-20 a^{2} x^{2}-45 a x -30\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{30 \left (a x +1\right )^{4} \left (a x -1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(84\) |
default | \(\frac {\left (5 a^{5} x^{5}+18 a^{4} x^{4}+15 a^{3} x^{3}-20 a^{2} x^{2}-45 a x -30\right ) x \,c^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x -1\right )}{30 \left (a x +1\right )^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(86\) |
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {{\left (5 \, a^{5} c^{2} x^{6} + 18 \, a^{4} c^{2} x^{5} + 15 \, a^{3} c^{2} x^{4} - 20 \, a^{2} c^{2} x^{3} - 45 \, a c^{2} x^{2} - 30 \, c^{2} x\right )} \sqrt {-a^{2} c}}{30 \, a} \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.51 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {{\left (5 \, a^{7} \sqrt {-c} c^{2} x^{7} + 13 \, a^{6} \sqrt {-c} c^{2} x^{6} - 3 \, a^{5} \sqrt {-c} c^{2} x^{5} - 35 \, a^{4} \sqrt {-c} c^{2} x^{4} - 25 \, a^{3} \sqrt {-c} c^{2} x^{3} + 15 \, a^{2} \sqrt {-c} c^{2} x^{2} + 30 \, \sqrt {-c} c^{2}\right )} {\left (a x + 1\right )}^{2}}{30 \, {\left (a^{3} x^{2} + 2 \, a^{2} x + a\right )} {\left (a x - 1\right )}} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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