Integrand size = 42, antiderivative size = 27 \begin {dmath*} \int \frac {e^{4 x} \left (16120 x^4+e \left (-1690 x-3380 x^2\right )\right )}{169 e^2-1612 e x^2+3844 x^4} \, dx=\frac {e^{4 x} x}{x-\frac {e+\frac {3 x^2}{13}}{5 x}} \end {dmath*}
Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \begin {dmath*} \int \frac {e^{4 x} \left (16120 x^4+e \left (-1690 x-3380 x^2\right )\right )}{169 e^2-1612 e x^2+3844 x^4} \, dx=\frac {130 e^{4 x} x^2}{-26 e+124 x^2} \end {dmath*}
Integrate[(E^(4*x)*(16120*x^4 + E*(-1690*x - 3380*x^2)))/(169*E^2 - 1612*E *x^2 + 3844*x^4),x]
Time = 0.42 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1380, 27, 7292, 2726}
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.
\(\displaystyle \int \frac {e^{4 x} \left (16120 x^4+e \left (-3380 x^2-1690 x\right )\right )}{3844 x^4-1612 e x^2+169 e^2} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle 3844 \int \frac {65 e^{4 x} \left (124 x^4-13 e \left (2 x^2+x\right )\right )}{1922 \left (13 e-62 x^2\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 130 \int \frac {e^{4 x} \left (124 x^4-13 e \left (2 x^2+x\right )\right )}{\left (13 e-62 x^2\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle 130 \int \frac {e^{4 x} x \left (124 x^3-26 e x-13 e\right )}{\left (13 e-62 x^2\right )^2}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle -\frac {65 e^{4 x} x \left (13 e x-62 x^3\right )}{\left (13 e-62 x^2\right )^2}\) |
3.6.61.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(-\frac {65 \,{\mathrm e}^{4 x} x^{2}}{-62 x^{2}+13 \,{\mathrm e}}\) | \(22\) |
norman | \(-\frac {65 \,{\mathrm e}^{4 x} x^{2}}{-62 x^{2}+13 \,{\mathrm e}}\) | \(22\) |
parallelrisch | \(-\frac {65 \,{\mathrm e}^{4 x} x^{2}}{-62 x^{2}+13 \,{\mathrm e}}\) | \(22\) |
derivativedivides | \(\frac {65 \,{\mathrm e}^{4 x}}{62}+\frac {6760 \,{\mathrm e}^{4 x} {\mathrm e} x}{31 \left (-496 x^{2}+104 \,{\mathrm e}\right )}-\frac {65 \,{\mathrm e} \sqrt {806}\, \left (2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-93 \,{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+93 \,{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{119164}-3380 \,{\mathrm e} \left (\frac {2 \,{\mathrm e}^{4 x} x}{31 \left (-496 x^{2}+104 \,{\mathrm e}\right )}-\frac {\sqrt {806}\, \left (2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-31 \,{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+31 \,{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{6196528}\right )-6760 \,{\mathrm e} \left (\frac {{\mathrm e}^{4 x}}{-30752 x^{2}+6448 \,{\mathrm e}}+\frac {\sqrt {806}\, \left ({\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{199888}\right )\) | \(357\) |
default | \(\frac {65 \,{\mathrm e}^{4 x}}{62}+\frac {6760 \,{\mathrm e}^{4 x} {\mathrm e} x}{31 \left (-496 x^{2}+104 \,{\mathrm e}\right )}-\frac {65 \,{\mathrm e} \sqrt {806}\, \left (2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-93 \,{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+93 \,{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{119164}-3380 \,{\mathrm e} \left (\frac {2 \,{\mathrm e}^{4 x} x}{31 \left (-496 x^{2}+104 \,{\mathrm e}\right )}-\frac {\sqrt {806}\, \left (2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-31 \,{\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )+31 \,{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{6196528}\right )-6760 \,{\mathrm e} \left (\frac {{\mathrm e}^{4 x}}{-30752 x^{2}+6448 \,{\mathrm e}}+\frac {\sqrt {806}\, \left ({\mathrm e}^{-\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x -\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )-{\mathrm e}^{\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}} \operatorname {Ei}_{1}\left (-4 x +\frac {2 \sqrt {806}\, {\mathrm e}^{\frac {1}{2}}}{31}\right )\right ) {\mathrm e}^{-\frac {1}{2}}}{199888}\right )\) | \(357\) |
int(((-3380*x^2-1690*x)*exp(1)+16120*x^4)*exp(4*x)/(169*exp(1)^2-1612*x^2* exp(1)+3844*x^4),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \begin {dmath*} \int \frac {e^{4 x} \left (16120 x^4+e \left (-1690 x-3380 x^2\right )\right )}{169 e^2-1612 e x^2+3844 x^4} \, dx=\frac {65 \, x^{2} e^{\left (4 \, x\right )}}{62 \, x^{2} - 13 \, e} \end {dmath*}
integrate(((-3380*x^2-1690*x)*exp(1)+16120*x^4)*exp(4*x)/(169*exp(1)^2-161 2*x^2*exp(1)+3844*x^4),x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \begin {dmath*} \int \frac {e^{4 x} \left (16120 x^4+e \left (-1690 x-3380 x^2\right )\right )}{169 e^2-1612 e x^2+3844 x^4} \, dx=\frac {65 x^{2} e^{4 x}}{62 x^{2} - 13 e} \end {dmath*}
integrate(((-3380*x**2-1690*x)*exp(1)+16120*x**4)*exp(4*x)/(169*exp(1)**2- 1612*x**2*exp(1)+3844*x**4),x)
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \begin {dmath*} \int \frac {e^{4 x} \left (16120 x^4+e \left (-1690 x-3380 x^2\right )\right )}{169 e^2-1612 e x^2+3844 x^4} \, dx=\frac {65 \, x^{2} e^{\left (4 \, x\right )}}{62 \, x^{2} - 13 \, e} \end {dmath*}
integrate(((-3380*x^2-1690*x)*exp(1)+16120*x^4)*exp(4*x)/(169*exp(1)^2-161 2*x^2*exp(1)+3844*x^4),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \begin {dmath*} \int \frac {e^{4 x} \left (16120 x^4+e \left (-1690 x-3380 x^2\right )\right )}{169 e^2-1612 e x^2+3844 x^4} \, dx=\frac {65 \, {\left (62 \, x^{2} e^{\left (4 \, x\right )} + 13 \, e^{\left (4 \, x + 1\right )}\right )}}{62 \, {\left (62 \, x^{2} - 13 \, e\right )}} \end {dmath*}
integrate(((-3380*x^2-1690*x)*exp(1)+16120*x^4)*exp(4*x)/(169*exp(1)^2-161 2*x^2*exp(1)+3844*x^4),x, algorithm=\
Time = 14.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \begin {dmath*} \int \frac {e^{4 x} \left (16120 x^4+e \left (-1690 x-3380 x^2\right )\right )}{169 e^2-1612 e x^2+3844 x^4} \, dx=-\frac {65\,x^2\,{\mathrm {e}}^{4\,x}}{62\,\left (\frac {13\,\mathrm {e}}{62}-x^2\right )} \end {dmath*}