Integrand size = 20, antiderivative size = 197 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=-\frac {8 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {856 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {16 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6311, 6316, 96, 91, 79, 37} \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=-\frac {856 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 x^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 x \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {16 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 x \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {8 \left (\frac {1}{a x}+1\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}} \]
[In]
[Out]
Rule 37
Rule 79
Rule 91
Rule 96
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {(c-a c x)^{9/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{9/2} x^{9/2} \, dx}{\left (1-\frac {1}{a x}\right )^{9/2} x^{9/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3 \left (1+\frac {x}{a}\right )^{3/2}}{x^{13/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}} \\ & = \frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (12 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\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 )}{11 a \left (1-\frac {1}{a x}\right )^{9/2}} \\ & = -\frac {8 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (8 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\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 )}{33 a \left (1-\frac {1}{a x}\right )^{9/2}} \\ & = -\frac {8 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {16 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (428 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{231 a^3 \left (1-\frac {1}{a x}\right )^{9/2}} \\ & = -\frac {8 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {856 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {16 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.39 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 c^4 \sqrt {1+\frac {1}{a x}} (1+a x)^2 \sqrt {c-a c x} \left (-533+755 a x-455 a^2 x^2+105 a^3 x^3\right )}{1155 a \sqrt {1-\frac {1}{a x}}} \]
[In]
[Out]
Time = 0.42 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.32
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (105 a^{3} x^{3}-455 a^{2} x^{2}+755 a x -533\right ) \left (-a c x +c \right )^{\frac {9}{2}}}{1155 a \left (a x -1\right )^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(64\) |
default | \(\frac {2 \left (a x -1\right ) \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{4} \left (105 a^{3} x^{3}-455 a^{2} x^{2}+755 a x -533\right )}{1155 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(66\) |
risch | \(-\frac {2 c^{5} \left (a x -1\right ) \left (105 a^{5} x^{5}-245 a^{4} x^{4}-50 a^{3} x^{3}+522 a^{2} x^{2}-311 a x -533\right )}{1155 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\) | \(77\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.53 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (105 \, a^{6} c^{4} x^{6} - 140 \, a^{5} c^{4} x^{5} - 295 \, a^{4} c^{4} x^{4} + 472 \, a^{3} c^{4} x^{3} + 211 \, a^{2} c^{4} x^{2} - 844 \, a c^{4} x - 533 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{1155 \, {\left (a^{2} x - a\right )}} \]
[In]
[Out]
Timed out. \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.54 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (105 \, a^{5} \sqrt {-c} c^{4} x^{5} - 455 \, a^{4} \sqrt {-c} c^{4} x^{4} + 650 \, a^{3} \sqrt {-c} c^{4} x^{3} - 78 \, a^{2} \sqrt {-c} c^{4} x^{2} - 755 \, a \sqrt {-c} c^{4} x + 533 \, \sqrt {-c} c^{4}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}}{1155 \, {\left (a x - 1\right )} a} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.66 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=-\frac {2 \, {\left (512 \, \sqrt {2} \sqrt {-c} c^{3} + \frac {105 \, {\left (a c x + c\right )}^{5} \sqrt {-a c x - c} - 770 \, {\left (a c x + c\right )}^{4} \sqrt {-a c x - c} c + 1980 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} c^{2} - 1848 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c^{3}}{c^{2}}\right )} c^{2}}{1155 \, a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \]
[In]
[Out]
Time = 4.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.56 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (105\,a^5\,x^5-35\,a^4\,x^4-330\,a^3\,x^3+142\,a^2\,x^2+353\,a\,x-491\right )}{1155\,a}-\frac {2048\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{1155\,a\,\left (a\,x-1\right )} \]
[In]
[Out]