Integrand size = 18, antiderivative size = 197 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {568 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 \left (1-\frac {1}{a x}\right )^{7/2} x^2}+\frac {48 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6311, 6316, 96, 91, 79, 37} \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {568 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 x^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {2 x \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {48 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 x \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {8 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}} \]
[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)^{7/2} \int e^{\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 )^3 \sqrt {1+\frac {x}{a}}}{x^{11/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}} \\ & = \frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (4 \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 \sqrt {1+\frac {x}{a}}}{x^{9/2}} \, dx,x,\frac {1}{x}\right )}{3 a \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (8 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {\left (-\frac {9}{a}+\frac {7 x}{2 a^2}\right ) \sqrt {1+\frac {x}{a}}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{21 a \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {48 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (284 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{105 a^3 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {568 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 \left (1-\frac {1}{a x}\right )^{7/2} x^2}+\frac {48 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.40 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 c^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (-319+2 a x+156 a^2 x^2-130 a^3 x^3+35 a^4 x^4\right )}{315 a \sqrt {1-\frac {1}{a x}}} \]
[In]
[Out]
Time = 0.42 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.31
method | result | size |
default | \(-\frac {2 \sqrt {-c \left (a x -1\right )}\, c^{3} \left (a x +1\right ) \left (35 a^{3} x^{3}-165 a^{2} x^{2}+321 a x -319\right )}{315 \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(61\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (35 a^{3} x^{3}-165 a^{2} x^{2}+321 a x -319\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{3} \sqrt {\frac {a x -1}{a x +1}}}\) | \(64\) |
risch | \(\frac {2 c^{4} \left (a x -1\right ) \left (35 a^{4} x^{4}-130 a^{3} x^{3}+156 a^{2} x^{2}+2 a x -319\right )}{315 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\) | \(69\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.48 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{5} c^{3} x^{5} - 95 \, a^{4} c^{3} x^{4} + 26 \, a^{3} c^{3} x^{3} + 158 \, a^{2} c^{3} x^{2} - 317 \, a c^{3} x - 319 \, c^{3}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{2} x - a\right )}} \]
[In]
[Out]
Timed out. \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.42 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{4} \sqrt {-c} c^{3} x^{4} - 130 \, a^{3} \sqrt {-c} c^{3} x^{3} + 156 \, a^{2} \sqrt {-c} c^{3} x^{2} + 2 \, a \sqrt {-c} c^{3} x - 319 \, \sqrt {-c} c^{3}\right )} \sqrt {a x + 1}}{315 \, a} \]
[In]
[Out]
Exception generated. \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 4.45 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.52 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\frac {2\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (-35\,a^4\,x^4+60\,a^3\,x^3+34\,a^2\,x^2-124\,a\,x+193\right )}{315\,a}+\frac {1024\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{315\,a\,\left (a\,x-1\right )} \]
[In]
[Out]