Integrand size = 36, antiderivative size = 29 \[ \int \frac {e^{-\frac {-3+x}{x}} \left (-30-19 x-8 e^{\frac {-3+x}{x}} x^6\right )}{2 x^3} \, dx=\frac {3 e^{-1+\frac {3}{x}} \left (\frac {5}{3}+\frac {x}{2}\right )}{x}-x^4 \] Output:
3*(5/3+1/2*x)/x/exp(1-3/x)-x^4
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-\frac {-3+x}{x}} \left (-30-19 x-8 e^{\frac {-3+x}{x}} x^6\right )}{2 x^3} \, dx=\frac {3}{2} e^{-1+\frac {3}{x}}+\frac {5 e^{-1+\frac {3}{x}}}{x}-x^4 \] Input:
Integrate[(-30 - 19*x - 8*E^((-3 + x)/x)*x^6)/(2*E^((-3 + x)/x)*x^3),x]
Output:
(3*E^(-1 + 3/x))/2 + (5*E^(-1 + 3/x))/x - x^4
Time = 0.71 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {27, 25, 7292, 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.
\(\displaystyle \int \frac {e^{-\frac {x-3}{x}} \left (-8 e^{\frac {x-3}{x}} x^6-19 x-30\right )}{2 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int -\frac {e^{\frac {3-x}{x}} \left (8 e^{-\frac {3-x}{x}} x^6+19 x+30\right )}{x^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {e^{\frac {3-x}{x}} \left (8 e^{-\frac {3-x}{x}} x^6+19 x+30\right )}{x^3}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\frac {1}{2} \int \frac {e^{\frac {3}{x}-1} \left (8 e^{-\frac {3-x}{x}} x^6+19 x+30\right )}{x^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{2} \int \left (8 x^3+\frac {e^{\frac {3}{x}-1} (19 x+30)}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-2 x^4+3 e^{\frac {3}{x}-1}+\frac {10 e^{\frac {3}{x}-1}}{x}\right )\) |
Input:
Int[(-30 - 19*x - 8*E^((-3 + x)/x)*x^6)/(2*E^((-3 + x)/x)*x^3),x]
Output:
(3*E^(-1 + 3/x) + (10*E^(-1 + 3/x))/x - 2*x^4)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-x^{4}+\frac {\left (3 x +10\right ) {\mathrm e}^{-\frac {-3+x}{x}}}{2 x}\) | \(26\) |
parallelrisch | \(\frac {\left (-2 x^{5} {\mathrm e}^{\frac {-3+x}{x}}+10+3 x \right ) {\mathrm e}^{-\frac {-3+x}{x}}}{2 x}\) | \(34\) |
derivativedivides | \(-x^{4}-\frac {5 \,{\mathrm e}^{\frac {3}{x}-1} \left (1-\frac {3}{x}\right )}{3}+\frac {19 \,{\mathrm e}^{\frac {3}{x}-1}}{6}\) | \(36\) |
default | \(-x^{4}-\frac {5 \,{\mathrm e}^{\frac {3}{x}-1} \left (1-\frac {3}{x}\right )}{3}+\frac {19 \,{\mathrm e}^{\frac {3}{x}-1}}{6}\) | \(36\) |
parts | \(-x^{4}-\frac {5 \,{\mathrm e}^{\frac {3}{x}-1} \left (1-\frac {3}{x}\right )}{3}+\frac {19 \,{\mathrm e}^{\frac {3}{x}-1}}{6}\) | \(36\) |
norman | \(\frac {\left (5 x +\frac {3 x^{2}}{2}-x^{6} {\mathrm e}^{\frac {-3+x}{x}}\right ) {\mathrm e}^{-\frac {-3+x}{x}}}{x^{2}}\) | \(37\) |
orering | \(-\frac {\left (36 x^{3}+82 x^{2}-147 x -90\right ) \left (-8 x^{6} {\mathrm e}^{\frac {-3+x}{x}}-19 x -30\right ) {\mathrm e}^{-\frac {-3+x}{x}}}{8 x^{2} \left (95 x^{2}+237 x +90\right )}+\frac {x^{3} \left (12 x^{2}+59 x +30\right ) \left (\frac {\left (-48 x^{5} {\mathrm e}^{\frac {-3+x}{x}}-8 x^{6} \left (\frac {1}{x}-\frac {-3+x}{x^{2}}\right ) {\mathrm e}^{\frac {-3+x}{x}}-19\right ) {\mathrm e}^{-\frac {-3+x}{x}}}{2 x^{3}}-\frac {3 \left (-8 x^{6} {\mathrm e}^{\frac {-3+x}{x}}-19 x -30\right ) {\mathrm e}^{-\frac {-3+x}{x}}}{2 x^{4}}-\frac {\left (-8 x^{6} {\mathrm e}^{\frac {-3+x}{x}}-19 x -30\right ) {\mathrm e}^{-\frac {-3+x}{x}} \left (\frac {1}{x}-\frac {-3+x}{x^{2}}\right )}{2 x^{3}}\right )}{380 x^{2}+948 x +360}\) | \(223\) |
meijerg | \(\frac {5 \,{\mathrm e}^{1+\frac {3}{x}-\frac {3 \,{\mathrm e}^{-1}}{x}} \left (1-\frac {\left (2-\frac {6 \,{\mathrm e}^{-1}}{x}\right ) {\mathrm e}^{\frac {3 \,{\mathrm e}^{-1}}{x}}}{2}\right )}{3}+324 \,{\mathrm e}^{\frac {3}{x}-4-\frac {3 \,{\mathrm e}^{-1}}{x}} \left (1-{\mathrm e}\right )^{4} \left (-\frac {x^{4} {\mathrm e}^{4}}{324 \left (1-{\mathrm e}\right )^{4}}-\frac {x^{3} {\mathrm e}^{3}}{81 \left (1-{\mathrm e}\right )^{3}}-\frac {x^{2} {\mathrm e}^{2}}{36 \left (1-{\mathrm e}\right )^{2}}-\frac {x \,{\mathrm e}}{18 \left (1-{\mathrm e}\right )}-\frac {37}{288}-\frac {\ln \left (x \right )}{24}+\frac {\ln \left (3\right )}{24}+\frac {i \pi }{24}+\frac {\ln \left (1-{\mathrm e}\right )}{24}+\frac {x^{4} {\mathrm e}^{4} \left (\frac {10125 \,{\mathrm e}^{-4} \left (1-{\mathrm e}\right )^{4}}{x^{4}}+\frac {6480 \,{\mathrm e}^{-3} \left (1-{\mathrm e}\right )^{3}}{x^{3}}+\frac {3240 \,{\mathrm e}^{-2} \left (1-{\mathrm e}\right )^{2}}{x^{2}}+\frac {1440 \,{\mathrm e}^{-1} \left (1-{\mathrm e}\right )}{x}+360\right )}{116640 \left (1-{\mathrm e}\right )^{4}}-\frac {x^{4} {\mathrm e}^{4+\frac {3 \,{\mathrm e}^{-1} \left (1-{\mathrm e}\right )}{x}} \left (\frac {135 \,{\mathrm e}^{-3} \left (1-{\mathrm e}\right )^{3}}{x^{3}}+\frac {45 \,{\mathrm e}^{-2} \left (1-{\mathrm e}\right )^{2}}{x^{2}}+\frac {30 \,{\mathrm e}^{-1} \left (1-{\mathrm e}\right )}{x}+30\right )}{9720 \left (1-{\mathrm e}\right )^{4}}-\frac {\ln \left (-\frac {3 \,{\mathrm e}^{-1} \left (1-{\mathrm e}\right )}{x}\right )}{24}-\frac {\operatorname {expIntegral}_{1}\left (-\frac {3 \,{\mathrm e}^{-1} \left (1-{\mathrm e}\right )}{x}\right )}{24}\right )-\frac {19 \,{\mathrm e}^{\frac {3}{x}-\frac {3 \,{\mathrm e}^{-1}}{x}} \left (1-{\mathrm e}^{\frac {3 \,{\mathrm e}^{-1}}{x}}\right )}{6}\) | \(356\) |
Input:
int(1/2*(-8*x^6*exp((-3+x)/x)-19*x-30)/x^3/exp((-3+x)/x),x,method=_RETURNV ERBOSE)
Output:
-x^4+1/2*(3*x+10)/x*exp(-(-3+x)/x)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-\frac {-3+x}{x}} \left (-30-19 x-8 e^{\frac {-3+x}{x}} x^6\right )}{2 x^3} \, dx=-\frac {{\left (2 \, x^{5} e^{\left (\frac {x - 3}{x}\right )} - 3 \, x - 10\right )} e^{\left (-\frac {x - 3}{x}\right )}}{2 \, x} \] Input:
integrate(1/2*(-8*x^6*exp((-3+x)/x)-19*x-30)/x^3/exp((-3+x)/x),x, algorith m="fricas")
Output:
-1/2*(2*x^5*e^((x - 3)/x) - 3*x - 10)*e^(-(x - 3)/x)/x
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59 \[ \int \frac {e^{-\frac {-3+x}{x}} \left (-30-19 x-8 e^{\frac {-3+x}{x}} x^6\right )}{2 x^3} \, dx=- x^{4} + \frac {\left (3 x + 10\right ) e^{- \frac {x - 3}{x}}}{2 x} \] Input:
integrate(1/2*(-8*x**6*exp((-3+x)/x)-19*x-30)/x**3/exp((-3+x)/x),x)
Output:
-x**4 + (3*x + 10)*exp(-(x - 3)/x)/(2*x)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-\frac {-3+x}{x}} \left (-30-19 x-8 e^{\frac {-3+x}{x}} x^6\right )}{2 x^3} \, dx=-x^{4} - \frac {5}{3} \, e^{\left (-1\right )} \Gamma \left (2, -\frac {3}{x}\right ) + \frac {19}{6} \, e^{\left (\frac {3}{x} - 1\right )} \] Input:
integrate(1/2*(-8*x^6*exp((-3+x)/x)-19*x-30)/x^3/exp((-3+x)/x),x, algorith m="maxima")
Output:
-x^4 - 5/3*e^(-1)*gamma(2, -3/x) + 19/6*e^(3/x - 1)
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-\frac {-3+x}{x}} \left (-30-19 x-8 e^{\frac {-3+x}{x}} x^6\right )}{2 x^3} \, dx=\frac {1}{2} \, x^{4} {\left (\frac {3 \, e^{\frac {3}{x}}}{x^{4}} + \frac {10 \, e^{\frac {3}{x}}}{x^{5}} - 2 \, e\right )} e^{\left (-1\right )} \] Input:
integrate(1/2*(-8*x^6*exp((-3+x)/x)-19*x-30)/x^3/exp((-3+x)/x),x, algorith m="giac")
Output:
1/2*x^4*(3*e^(3/x)/x^4 + 10*e^(3/x)/x^5 - 2*e)*e^(-1)
Time = 3.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {-3+x}{x}} \left (-30-19 x-8 e^{\frac {-3+x}{x}} x^6\right )}{2 x^3} \, dx=\frac {3\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{3/x}}{2}-x^4+\frac {5\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{3/x}}{x} \] Input:
int(-(exp(-(x - 3)/x)*((19*x)/2 + 4*x^6*exp((x - 3)/x) + 15))/x^3,x)
Output:
(3*exp(-1)*exp(3/x))/2 - x^4 + (5*exp(-1)*exp(3/x))/x
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-\frac {-3+x}{x}} \left (-30-19 x-8 e^{\frac {-3+x}{x}} x^6\right )}{2 x^3} \, dx=\frac {3 e^{\frac {3}{x}} x +10 e^{\frac {3}{x}}-2 e \,x^{5}}{2 e x} \] Input:
int(1/2*(-8*x^6*exp((-3+x)/x)-19*x-30)/x^3/exp((-3+x)/x),x)
Output:
(3*e**(3/x)*x + 10*e**(3/x) - 2*e*x**5)/(2*e*x)