Integrand size = 20, antiderivative size = 137 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {44 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{7/2}}{63 a \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {214 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{7/2}}{315 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{7/2}}{9 \left (1-\frac {1}{a x}\right )^{7/2}} \]
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Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 91, 79, 37} \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\frac {214 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{7/2}}{315 a^2 x \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {44 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{7/2}}{63 a \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {2 x \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{7/2}}{9 \left (1-\frac {1}{a x}\right )^{7/2}} \]
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Rule 37
Rule 79
Rule 91
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {(c-a c x)^{7/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{7/2} x^{7/2} \, dx}{\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2 \left (1+\frac {x}{a}\right )^{3/2}}{x^{11/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}} \\ & = \frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{7/2}}{9 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {\left (2 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {\left (-\frac {11}{a}+\frac {9 x}{2 a^2}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x^{9/2}} \, dx,x,\frac {1}{x}\right )}{9 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {44 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{7/2}}{63 a \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{7/2}}{9 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {\left (107 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{63 a^2 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {44 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{7/2}}{63 a \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {214 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{7/2}}{315 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{7/2}}{9 \left (1-\frac {1}{a x}\right )^{7/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.50 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 c^3 \sqrt {1+\frac {1}{a x}} (1+a x)^2 \sqrt {c-a c x} \left (107-110 a x+35 a^2 x^2\right )}{315 a \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.43 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.41
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (35 a^{2} x^{2}-110 a x +107\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(56\) |
default | \(-\frac {2 \left (a x -1\right ) \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{3} \left (35 a^{2} x^{2}-110 a x +107\right )}{315 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(58\) |
risch | \(\frac {2 c^{4} \left (a x -1\right ) \left (35 a^{4} x^{4}-40 a^{3} x^{3}-78 a^{2} x^{2}+104 a x +107\right )}{315 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\) | \(69\) |
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Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.69 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{5} c^{3} x^{5} - 5 \, a^{4} c^{3} x^{4} - 118 \, a^{3} c^{3} x^{3} + 26 \, a^{2} c^{3} x^{2} + 211 \, a c^{3} x + 107 \, c^{3}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{2} x - a\right )}} \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.66 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{4} \sqrt {-c} c^{3} x^{4} - 110 \, a^{3} \sqrt {-c} c^{3} x^{3} + 72 \, a^{2} \sqrt {-c} c^{3} x^{2} + 110 \, a \sqrt {-c} c^{3} x - 107 \, \sqrt {-c} c^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}}{315 \, {\left (a x - 1\right )} a} \]
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Exception generated. \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.74 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (35\,a^4\,x^4+30\,a^3\,x^3-88\,a^2\,x^2-62\,a\,x+149\right )}{315\,a}-\frac {512\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{315\,a\,\left (a\,x-1\right )} \]
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