Integrand size = 82, antiderivative size = 27 \[ \int \frac {e^{-\frac {2 \left (-5+x^2\right )}{x}} \left (5 e^{\frac {2 \left (-5+x^2\right )}{x}} x^2+e^{32 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (320+64 x^2\right )+e^{16 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (960+192 x^2\right )\right )}{x^2} \, dx=-2-\left (3+e^{16 e^{-\frac {2 \left (-5+x^2\right )}{x}}}\right )^2+5 x \] Output:
-2+5*x-(exp(16/exp((x^2-5)/x)^2)+3)^2
Time = 0.33 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-\frac {2 \left (-5+x^2\right )}{x}} \left (5 e^{\frac {2 \left (-5+x^2\right )}{x}} x^2+e^{32 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (320+64 x^2\right )+e^{16 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (960+192 x^2\right )\right )}{x^2} \, dx=-6 e^{16 e^{\frac {10}{x}-2 x}}-e^{32 e^{\frac {10}{x}-2 x}}+5 x \] Input:
Integrate[(5*E^((2*(-5 + x^2))/x)*x^2 + E^(32/E^((2*(-5 + x^2))/x))*(320 + 64*x^2) + E^(16/E^((2*(-5 + x^2))/x))*(960 + 192*x^2))/(E^((2*(-5 + x^2)) /x)*x^2),x]
Output:
-6*E^(16*E^(10/x - 2*x)) - E^(32*E^(10/x - 2*x)) + 5*x
Time = 1.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{-\frac {2 \left (x^2-5\right )}{x}} \left (5 e^{\frac {2 \left (x^2-5\right )}{x}} x^2+e^{32 e^{-\frac {2 \left (x^2-5\right )}{x}}} \left (64 x^2+320\right )+e^{16 e^{-\frac {2 \left (x^2-5\right )}{x}}} \left (192 x^2+960\right )\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {64 e^{16 e^{-\frac {2 \left (x^2-5\right )}{x}}-\frac {2 \left (x^2-5\right )}{x}} \left (e^{16 e^{\frac {10}{x}-2 x}}+3\right ) \left (x^2+5\right )}{x^2}+5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 5 x-\left (e^{16 e^{\frac {10}{x}-2 x}}+3\right )^2\) |
Input:
Int[(5*E^((2*(-5 + x^2))/x)*x^2 + E^(32/E^((2*(-5 + x^2))/x))*(320 + 64*x^ 2) + E^(16/E^((2*(-5 + x^2))/x))*(960 + 192*x^2))/(E^((2*(-5 + x^2))/x)*x^ 2),x]
Output:
-(3 + E^(16*E^(10/x - 2*x)))^2 + 5*x
Time = 0.38 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37
method | result | size |
risch | \(-{\mathrm e}^{32 \,{\mathrm e}^{-\frac {2 \left (x^{2}-5\right )}{x}}}+5 x -6 \,{\mathrm e}^{16 \,{\mathrm e}^{-\frac {2 \left (x^{2}-5\right )}{x}}}\) | \(37\) |
parts | \(5 x +\frac {\left (-6 \,{\mathrm e}^{\frac {2 x^{2}-10}{x}} {\mathrm e}^{16 \,{\mathrm e}^{-\frac {2 \left (x^{2}-5\right )}{x}}} x -x \,{\mathrm e}^{\frac {2 x^{2}-10}{x}} {\mathrm e}^{32 \,{\mathrm e}^{-\frac {2 \left (x^{2}-5\right )}{x}}}\right ) {\mathrm e}^{-\frac {2 \left (x^{2}-5\right )}{x}}}{x}\) | \(84\) |
norman | \(\frac {\left (5 x^{2} {\mathrm e}^{\frac {2 x^{2}-10}{x}}-x \,{\mathrm e}^{\frac {2 x^{2}-10}{x}} {\mathrm e}^{32 \,{\mathrm e}^{-\frac {2 \left (x^{2}-5\right )}{x}}}-6 \,{\mathrm e}^{\frac {2 x^{2}-10}{x}} {\mathrm e}^{16 \,{\mathrm e}^{-\frac {2 \left (x^{2}-5\right )}{x}}} x \right ) {\mathrm e}^{-\frac {2 \left (x^{2}-5\right )}{x}}}{x}\) | \(97\) |
Input:
int(((64*x^2+320)*exp(16/exp((x^2-5)/x)^2)^2+(192*x^2+960)*exp(16/exp((x^2 -5)/x)^2)+5*x^2*exp((x^2-5)/x)^2)/x^2/exp((x^2-5)/x)^2,x,method=_RETURNVER BOSE)
Output:
-exp(32*exp(-2*(x^2-5)/x))+5*x-6*exp(16*exp(-2*(x^2-5)/x))
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-\frac {2 \left (-5+x^2\right )}{x}} \left (5 e^{\frac {2 \left (-5+x^2\right )}{x}} x^2+e^{32 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (320+64 x^2\right )+e^{16 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (960+192 x^2\right )\right )}{x^2} \, dx=5 \, x - e^{\left (32 \, e^{\left (-\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )}\right )} - 6 \, e^{\left (16 \, e^{\left (-\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )}\right )} \] Input:
integrate(((64*x^2+320)*exp(16/exp((x^2-5)/x)^2)^2+(192*x^2+960)*exp(16/ex p((x^2-5)/x)^2)+5*x^2*exp((x^2-5)/x)^2)/x^2/exp((x^2-5)/x)^2,x, algorithm= "fricas")
Output:
5*x - e^(32*e^(-2*(x^2 - 5)/x)) - 6*e^(16*e^(-2*(x^2 - 5)/x))
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-\frac {2 \left (-5+x^2\right )}{x}} \left (5 e^{\frac {2 \left (-5+x^2\right )}{x}} x^2+e^{32 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (320+64 x^2\right )+e^{16 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (960+192 x^2\right )\right )}{x^2} \, dx=5 x - e^{32 e^{- \frac {2 \left (x^{2} - 5\right )}{x}}} - 6 e^{16 e^{- \frac {2 \left (x^{2} - 5\right )}{x}}} \] Input:
integrate(((64*x**2+320)*exp(16/exp((x**2-5)/x)**2)**2+(192*x**2+960)*exp( 16/exp((x**2-5)/x)**2)+5*x**2*exp((x**2-5)/x)**2)/x**2/exp((x**2-5)/x)**2, x)
Output:
5*x - exp(32*exp(-2*(x**2 - 5)/x)) - 6*exp(16*exp(-2*(x**2 - 5)/x))
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-\frac {2 \left (-5+x^2\right )}{x}} \left (5 e^{\frac {2 \left (-5+x^2\right )}{x}} x^2+e^{32 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (320+64 x^2\right )+e^{16 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (960+192 x^2\right )\right )}{x^2} \, dx=5 \, x - e^{\left (32 \, e^{\left (-2 \, x + \frac {10}{x}\right )}\right )} - 6 \, e^{\left (16 \, e^{\left (-2 \, x + \frac {10}{x}\right )}\right )} \] Input:
integrate(((64*x^2+320)*exp(16/exp((x^2-5)/x)^2)^2+(192*x^2+960)*exp(16/ex p((x^2-5)/x)^2)+5*x^2*exp((x^2-5)/x)^2)/x^2/exp((x^2-5)/x)^2,x, algorithm= "maxima")
Output:
5*x - e^(32*e^(-2*x + 10/x)) - 6*e^(16*e^(-2*x + 10/x))
\[ \int \frac {e^{-\frac {2 \left (-5+x^2\right )}{x}} \left (5 e^{\frac {2 \left (-5+x^2\right )}{x}} x^2+e^{32 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (320+64 x^2\right )+e^{16 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (960+192 x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (5 \, x^{2} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} + 64 \, {\left (x^{2} + 5\right )} e^{\left (32 \, e^{\left (-\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )}\right )} + 192 \, {\left (x^{2} + 5\right )} e^{\left (16 \, e^{\left (-\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )}\right )}\right )} e^{\left (-\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )}}{x^{2}} \,d x } \] Input:
integrate(((64*x^2+320)*exp(16/exp((x^2-5)/x)^2)^2+(192*x^2+960)*exp(16/ex p((x^2-5)/x)^2)+5*x^2*exp((x^2-5)/x)^2)/x^2/exp((x^2-5)/x)^2,x, algorithm= "giac")
Output:
integrate((5*x^2*e^(2*(x^2 - 5)/x) + 64*(x^2 + 5)*e^(32*e^(-2*(x^2 - 5)/x) ) + 192*(x^2 + 5)*e^(16*e^(-2*(x^2 - 5)/x)))*e^(-2*(x^2 - 5)/x)/x^2, x)
Time = 2.80 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-\frac {2 \left (-5+x^2\right )}{x}} \left (5 e^{\frac {2 \left (-5+x^2\right )}{x}} x^2+e^{32 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (320+64 x^2\right )+e^{16 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (960+192 x^2\right )\right )}{x^2} \, dx=5\,x-6\,{\mathrm {e}}^{16\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{10/x}}-{\mathrm {e}}^{32\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{10/x}} \] Input:
int((exp(-(2*(x^2 - 5))/x)*(5*x^2*exp((2*(x^2 - 5))/x) + exp(32*exp(-(2*(x ^2 - 5))/x))*(64*x^2 + 320) + exp(16*exp(-(2*(x^2 - 5))/x))*(192*x^2 + 960 )))/x^2,x)
Output:
5*x - 6*exp(16*exp(-2*x)*exp(10/x)) - exp(32*exp(-2*x)*exp(10/x))
\[ \int \frac {e^{-\frac {2 \left (-5+x^2\right )}{x}} \left (5 e^{\frac {2 \left (-5+x^2\right )}{x}} x^2+e^{32 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (320+64 x^2\right )+e^{16 e^{-\frac {2 \left (-5+x^2\right )}{x}}} \left (960+192 x^2\right )\right )}{x^2} \, dx=\int \frac {\left (64 x^{2}+320\right ) \left ({\mathrm e}^{\frac {16}{\left ({\mathrm e}^{\frac {x^{2}-5}{x}}\right )^{2}}}\right )^{2}+\left (192 x^{2}+960\right ) {\mathrm e}^{\frac {16}{\left ({\mathrm e}^{\frac {x^{2}-5}{x}}\right )^{2}}}+5 x^{2} \left ({\mathrm e}^{\frac {x^{2}-5}{x}}\right )^{2}}{x^{2} \left ({\mathrm e}^{\frac {x^{2}-5}{x}}\right )^{2}}d x \] Input:
int(((64*x^2+320)*exp(16/exp((x^2-5)/x)^2)^2+(192*x^2+960)*exp(16/exp((x^2 -5)/x)^2)+5*x^2*exp((x^2-5)/x)^2)/x^2/exp((x^2-5)/x)^2,x)
Output:
int(((64*x^2+320)*exp(16/exp((x^2-5)/x)^2)^2+(192*x^2+960)*exp(16/exp((x^2 -5)/x)^2)+5*x^2*exp((x^2-5)/x)^2)/x^2/exp((x^2-5)/x)^2,x)