Integrand size = 20, antiderivative size = 254 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=-\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {2944 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}} \]
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Time = 0.16 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6311, 6316, 96, 91, 79, 37} \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=-\frac {2944 (c-a c x)^{5/2}}{35 a^4 x^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \left (a-\frac {1}{x}\right )^4 (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}-\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}-\frac {256 (c-a c x)^{5/2}}{7 a^3 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}+\frac {128 (c-a c x)^{5/2}}{35 a^2 x \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}} \]
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Rule 37
Rule 79
Rule 91
Rule 96
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {(c-a c x)^{5/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{5/2} x^{5/2} \, dx}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^4}{x^{9/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{5/2}} \\ & = \frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (16 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3}{x^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{7 a \left (1-\frac {1}{a x}\right )^{5/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (192 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (128 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {-\frac {5}{a}+\frac {3 x}{2 a^2}}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (1472 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {2944 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.27 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 c^2 \sqrt {c-a c x} \left (-1451-708 a x+142 a^2 x^2-36 a^3 x^3+5 a^4 x^4\right )}{35 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.47 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.28
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (5 a^{4} x^{4}-36 a^{3} x^{3}+142 a^{2} x^{2}-708 a x -1451\right ) \left (-a c x +c \right )^{\frac {5}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{35 a \left (a x -1\right )^{4}}\) | \(72\) |
default | \(\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{2} \left (5 a^{4} x^{4}-36 a^{3} x^{3}+142 a^{2} x^{2}-708 a x -1451\right )}{35 \left (a x -1\right )^{2} a}\) | \(76\) |
risch | \(-\frac {2 \left (5 a^{3} x^{3}-41 a^{2} x^{2}+183 a x -891\right ) \left (a x +1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{35 a \sqrt {-c \left (a x -1\right )}}+\frac {32 c^{3} \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-c \left (a x -1\right )}}\) | \(95\) |
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Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.33 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \, {\left (5 \, a^{4} c^{2} x^{4} - 36 \, a^{3} c^{2} x^{3} + 142 \, a^{2} c^{2} x^{2} - 708 \, a c^{2} x - 1451 \, c^{2}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{2} x - a\right )}} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.47 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \, {\left (5 \, a^{5} \sqrt {-c} c^{2} x^{5} - 31 \, a^{4} \sqrt {-c} c^{2} x^{4} + 106 \, a^{3} \sqrt {-c} c^{2} x^{3} - 566 \, a^{2} \sqrt {-c} c^{2} x^{2} - 2159 \, a \sqrt {-c} c^{2} x - 1451 \, \sqrt {-c} c^{2}\right )} {\left (a x - 1\right )}^{2}}{35 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
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Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.37 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (5\,a^3\,x^3-31\,a^2\,x^2+111\,a\,x-597\right )}{35\,a}-\frac {4096\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{35\,a\,\left (a\,x-1\right )} \]
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