Integrand size = 46, antiderivative size = 19 \[ \int \frac {e^{\frac {8 x+5 x^2+e^5 (8+4 x)}{x^2}} \left (e^5 (-16-4 x)-8 x\right )+x^3}{x^3} \, dx=e^{1+\frac {4 (2+x) \left (e^5+x\right )}{x^2}}+x \] Output:
exp(1+4*(2+x)/x^2*(exp(5)+x))+x
Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {8 x+5 x^2+e^5 (8+4 x)}{x^2}} \left (e^5 (-16-4 x)-8 x\right )+x^3}{x^3} \, dx=e^{5+\frac {8 e^5}{x^2}+\frac {4 \left (2+e^5\right )}{x}}+x \] Input:
Integrate[(E^((8*x + 5*x^2 + E^5*(8 + 4*x))/x^2)*(E^5*(-16 - 4*x) - 8*x) + x^3)/x^3,x]
Output:
E^(5 + (8*E^5)/x^2 + (4*(2 + E^5))/x) + x
Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2010, 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.
\(\displaystyle \int \frac {x^3+e^{\frac {5 x^2+8 x+e^5 (4 x+8)}{x^2}} \left (e^5 (-4 x-16)-8 x\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {4 e^{\frac {4 e^5 (x+2)}{x^2}+\frac {8}{x}+5} \left (-\left (\left (2+e^5\right ) x\right )-4 e^5\right )}{x^3}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^{\frac {4 e^5 (x+2)}{x^2}+\frac {8}{x}+5}+x\) |
Input:
Int[(E^((8*x + 5*x^2 + E^5*(8 + 4*x))/x^2)*(E^5*(-16 - 4*x) - 8*x) + x^3)/ x^3,x]
Output:
E^(5 + 8/x + (4*E^5*(2 + x))/x^2) + x
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 0.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32
method | result | size |
parallelrisch | \(x +{\mathrm e}^{\frac {\left (4 x +8\right ) {\mathrm e}^{5}+5 x^{2}+8 x}{x^{2}}}\) | \(25\) |
parts | \(x +{\mathrm e}^{\frac {\left (4 x +8\right ) {\mathrm e}^{5}+5 x^{2}+8 x}{x^{2}}}\) | \(25\) |
risch | \(x +{\mathrm e}^{\frac {4 x \,{\mathrm e}^{5}+5 x^{2}+8 \,{\mathrm e}^{5}+8 x}{x^{2}}}\) | \(26\) |
norman | \(\frac {x^{3}+x^{2} {\mathrm e}^{\frac {\left (4 x +8\right ) {\mathrm e}^{5}+5 x^{2}+8 x}{x^{2}}}}{x^{2}}\) | \(35\) |
derivativedivides | \(x -i {\mathrm e}^{5} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (4 \,{\mathrm e}^{5}+8\right )^{2} {\mathrm e}^{-5}}{32}} \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\frac {2 i \sqrt {2}\, {\mathrm e}^{\frac {5}{2}}}{x}+\frac {i \left (4 \,{\mathrm e}^{5}+8\right ) \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}}}{8}\right )-\frac {i {\mathrm e}^{10} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (4 \,{\mathrm e}^{5}+8\right )^{2} {\mathrm e}^{-5}}{32}} \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\frac {2 i \sqrt {2}\, {\mathrm e}^{\frac {5}{2}}}{x}+\frac {i \left (4 \,{\mathrm e}^{5}+8\right ) \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}}}{8}\right )}{2}+16 \,{\mathrm e}^{10} \left (\frac {{\mathrm e}^{-5} {\mathrm e}^{\frac {4 \,{\mathrm e}^{5}+8}{x}+\frac {8 \,{\mathrm e}^{5}}{x^{2}}}}{16}+\frac {i \left (4 \,{\mathrm e}^{5}+8\right ) {\mathrm e}^{-5} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (4 \,{\mathrm e}^{5}+8\right )^{2} {\mathrm e}^{-5}}{32}} \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\frac {2 i \sqrt {2}\, {\mathrm e}^{\frac {5}{2}}}{x}+\frac {i \left (4 \,{\mathrm e}^{5}+8\right ) \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}}}{8}\right )}{128}\right )\) | \(222\) |
default | \(x -i {\mathrm e}^{5} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (4 \,{\mathrm e}^{5}+8\right )^{2} {\mathrm e}^{-5}}{32}} \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\frac {2 i \sqrt {2}\, {\mathrm e}^{\frac {5}{2}}}{x}+\frac {i \left (4 \,{\mathrm e}^{5}+8\right ) \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}}}{8}\right )-\frac {i {\mathrm e}^{10} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (4 \,{\mathrm e}^{5}+8\right )^{2} {\mathrm e}^{-5}}{32}} \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\frac {2 i \sqrt {2}\, {\mathrm e}^{\frac {5}{2}}}{x}+\frac {i \left (4 \,{\mathrm e}^{5}+8\right ) \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}}}{8}\right )}{2}+16 \,{\mathrm e}^{10} \left (\frac {{\mathrm e}^{-5} {\mathrm e}^{\frac {4 \,{\mathrm e}^{5}+8}{x}+\frac {8 \,{\mathrm e}^{5}}{x^{2}}}}{16}+\frac {i \left (4 \,{\mathrm e}^{5}+8\right ) {\mathrm e}^{-5} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (4 \,{\mathrm e}^{5}+8\right )^{2} {\mathrm e}^{-5}}{32}} \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\frac {2 i \sqrt {2}\, {\mathrm e}^{\frac {5}{2}}}{x}+\frac {i \left (4 \,{\mathrm e}^{5}+8\right ) \sqrt {2}\, {\mathrm e}^{-\frac {5}{2}}}{8}\right )}{128}\right )\) | \(222\) |
Input:
int((((-16-4*x)*exp(5)-8*x)*exp(((4*x+8)*exp(5)+5*x^2+8*x)/x^2)+x^3)/x^3,x ,method=_RETURNVERBOSE)
Output:
x+exp(((4*x+8)*exp(5)+5*x^2+8*x)/x^2)
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {8 x+5 x^2+e^5 (8+4 x)}{x^2}} \left (e^5 (-16-4 x)-8 x\right )+x^3}{x^3} \, dx=x + e^{\left (\frac {5 \, x^{2} + 4 \, {\left (x + 2\right )} e^{5} + 8 \, x}{x^{2}}\right )} \] Input:
integrate((((-16-4*x)*exp(5)-8*x)*exp(((4*x+8)*exp(5)+5*x^2+8*x)/x^2)+x^3) /x^3,x, algorithm="fricas")
Output:
x + e^((5*x^2 + 4*(x + 2)*e^5 + 8*x)/x^2)
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {8 x+5 x^2+e^5 (8+4 x)}{x^2}} \left (e^5 (-16-4 x)-8 x\right )+x^3}{x^3} \, dx=x + e^{\frac {5 x^{2} + 8 x + \left (4 x + 8\right ) e^{5}}{x^{2}}} \] Input:
integrate((((-16-4*x)*exp(5)-8*x)*exp(((4*x+8)*exp(5)+5*x**2+8*x)/x**2)+x* *3)/x**3,x)
Output:
x + exp((5*x**2 + 8*x + (4*x + 8)*exp(5))/x**2)
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {8 x+5 x^2+e^5 (8+4 x)}{x^2}} \left (e^5 (-16-4 x)-8 x\right )+x^3}{x^3} \, dx=x + e^{\left (\frac {4 \, e^{5}}{x} + \frac {8}{x} + \frac {8 \, e^{5}}{x^{2}} + 5\right )} \] Input:
integrate((((-16-4*x)*exp(5)-8*x)*exp(((4*x+8)*exp(5)+5*x^2+8*x)/x^2)+x^3) /x^3,x, algorithm="maxima")
Output:
x + e^(4*e^5/x + 8/x + 8*e^5/x^2 + 5)
\[ \int \frac {e^{\frac {8 x+5 x^2+e^5 (8+4 x)}{x^2}} \left (e^5 (-16-4 x)-8 x\right )+x^3}{x^3} \, dx=\int { \frac {x^{3} - 4 \, {\left ({\left (x + 4\right )} e^{5} + 2 \, x\right )} e^{\left (\frac {5 \, x^{2} + 4 \, {\left (x + 2\right )} e^{5} + 8 \, x}{x^{2}}\right )}}{x^{3}} \,d x } \] Input:
integrate((((-16-4*x)*exp(5)-8*x)*exp(((4*x+8)*exp(5)+5*x^2+8*x)/x^2)+x^3) /x^3,x, algorithm="giac")
Output:
integrate((x^3 - 4*((x + 4)*e^5 + 2*x)*e^((5*x^2 + 4*(x + 2)*e^5 + 8*x)/x^ 2))/x^3, x)
Time = 4.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\frac {8 x+5 x^2+e^5 (8+4 x)}{x^2}} \left (e^5 (-16-4 x)-8 x\right )+x^3}{x^3} \, dx=x+{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^5}{x}}\,{\mathrm {e}}^{\frac {8\,{\mathrm {e}}^5}{x^2}}\,{\mathrm {e}}^5\,{\mathrm {e}}^{8/x} \] Input:
int(-(exp((8*x + 5*x^2 + exp(5)*(4*x + 8))/x^2)*(8*x + exp(5)*(4*x + 16)) - x^3)/x^3,x)
Output:
x + exp((4*exp(5))/x)*exp((8*exp(5))/x^2)*exp(5)*exp(8/x)
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\frac {8 x+5 x^2+e^5 (8+4 x)}{x^2}} \left (e^5 (-16-4 x)-8 x\right )+x^3}{x^3} \, dx=e^{\frac {4 e^{5} x +8 e^{5}+8 x}{x^{2}}} e^{5}+x \] Input:
int((((-16-4*x)*exp(5)-8*x)*exp(((4*x+8)*exp(5)+5*x^2+8*x)/x^2)+x^3)/x^3,x )
Output:
e**((4*e**5*x + 8*e**5 + 8*x)/x**2)*e**5 + x