\(\int \frac {e^{2 x}}{2-3 x+x^2} \, dx\) [174]

Optimal result

Integrand size = 16, antiderivative size = 22 \[ \int \frac {e^{2 x}}{2-3 x+x^2} \, dx=e^4 \operatorname {ExpIntegralEi}(-4+2 x)-e^2 \operatorname {ExpIntegralEi}(-2+2 x) \]

[Out]

exp(4)*Ei(-4+2*x)-exp(2)*Ei(-2+2*x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2300, 2209} \[ \int \frac {e^{2 x}}{2-3 x+x^2} \, dx=e^4 \operatorname {ExpIntegralEi}(2 x-4)-e^2 \operatorname {ExpIntegralEi}(2 x-2) \]

[In]

Int[E^(2*x)/(2 - 3*x + x^2),x]

[Out]

E^4*ExpIntegralEi[-4 + 2*x] - E^2*ExpIntegralEi[-2 + 2*x]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2300

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 e^{2 x}}{4-2 x}-\frac {2 e^{2 x}}{-2+2 x}\right ) \, dx \\ & = -\left (2 \int \frac {e^{2 x}}{4-2 x} \, dx\right )-2 \int \frac {e^{2 x}}{-2+2 x} \, dx \\ & = e^4 \operatorname {ExpIntegralEi}(-4+2 x)-e^2 \operatorname {ExpIntegralEi}(-2+2 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x}}{2-3 x+x^2} \, dx=e^4 \operatorname {ExpIntegralEi}(-4+2 x)-e^2 \operatorname {ExpIntegralEi}(-2+2 x) \]

[In]

Integrate[E^(2*x)/(2 - 3*x + x^2),x]

[Out]

E^4*ExpIntegralEi[-4 + 2*x] - E^2*ExpIntegralEi[-2 + 2*x]

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05

method	result	size
derivativedivides	\(-{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-2 x +4\right )+{\mathrm e}^{2} \operatorname {Ei}_{1}\left (-2 x +2\right )\)	\(23\)
default	\(-{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-2 x +4\right )+{\mathrm e}^{2} \operatorname {Ei}_{1}\left (-2 x +2\right )\)	\(23\)
risch	\(-{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-2 x +4\right )+{\mathrm e}^{2} \operatorname {Ei}_{1}\left (-2 x +2\right )\)	\(23\)

[In]

int(exp(2*x)/(x^2-3*x+2),x,method=_RETURNVERBOSE)

[Out]

-exp(4)*Ei(1,-2*x+4)+exp(2)*Ei(1,-2*x+2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 x}}{2-3 x+x^2} \, dx={\rm Ei}\left (2 \, x - 4\right ) e^{4} - {\rm Ei}\left (2 \, x - 2\right ) e^{2} \]

[In]

integrate(exp(2*x)/(x^2-3*x+2),x, algorithm="fricas")

[Out]

Ei(2*x - 4)*e^4 - Ei(2*x - 2)*e^2

Sympy [F]

\[ \int \frac {e^{2 x}}{2-3 x+x^2} \, dx=\int \frac {e^{2 x}}{\left (x - 2\right ) \left (x - 1\right )}\, dx \]

[In]

integrate(exp(2*x)/(x**2-3*x+2),x)

[Out]

Integral(exp(2*x)/((x - 2)*(x - 1)), x)

Maxima [F]

\[ \int \frac {e^{2 x}}{2-3 x+x^2} \, dx=\int { \frac {e^{\left (2 \, x\right )}}{x^{2} - 3 \, x + 2} \,d x } \]

[In]

integrate(exp(2*x)/(x^2-3*x+2),x, algorithm="maxima")

[Out]

integrate(e^(2*x)/(x^2 - 3*x + 2), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 x}}{2-3 x+x^2} \, dx={\rm Ei}\left (2 \, x - 4\right ) e^{4} - {\rm Ei}\left (2 \, x - 2\right ) e^{2} \]

[In]

integrate(exp(2*x)/(x^2-3*x+2),x, algorithm="giac")

[Out]

Ei(2*x - 4)*e^4 - Ei(2*x - 2)*e^2

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 x}}{2-3 x+x^2} \, dx=\int \frac {{\mathrm {e}}^{2\,x}}{x^2-3\,x+2} \,d x \]

[In]

int(exp(2*x)/(x^2 - 3*x + 2),x)

[Out]

int(exp(2*x)/(x^2 - 3*x + 2), x)