\(\int \frac {e^{d+e x}}{a+b x+c x^2} \, dx\) [469]

Optimal result

Integrand size = 20, antiderivative size = 138 \[ \int \frac {e^{d+e x}}{a+b x+c x^2} \, dx=\frac {e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}} \]

[Out]

exp(d-1/2*e*(b-(-4*a*c+b^2)^(1/2))/c)*Ei(1/2*e*(b+2*c*x-(-4*a*c+b^2)^(1/2))/c)/(-4*a*c+b^2)^(1/2)-exp(d-1/2*e*
(b+(-4*a*c+b^2)^(1/2))/c)*Ei(1/2*e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/c)/(-4*a*c+b^2)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2300, 2209} \[ \int \frac {e^{d+e x}}{a+b x+c x^2} \, dx=\frac {e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}} \]

[In]

Int[E^(d + e*x)/(a + b*x + c*x^2),x]

[Out]

(E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/Sqrt[b^2
- 4*a*c] - (E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)]
)/Sqrt[b^2 - 4*a*c]

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 c e^{d+e x}}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}-\frac {2 c e^{d+e x}}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \, dx \\ & = \frac {(2 c) \int \frac {e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}} \\ & = \frac {e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92 \[ \int \frac {e^{d+e x}}{a+b x+c x^2} \, dx=\frac {e^{d+\frac {\left (-b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )-e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}} \]

[In]

Integrate[E^(d + e*x)/(a + b*x + c*x^2),x]

[Out]

(E^(d + ((-b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)] - E^(d -
((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/Sqrt[b^2 - 4*a*c]

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(-\frac {e \left ({\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )-{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\)	\(169\)
default	\(-\frac {e \left ({\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )-{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\)	\(169\)
risch	\(-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\)	\(186\)

[In]

int(exp(e*x+d)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

-e*(exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-b*e+2*c*d-2*c*(e*x+d)+(-4*a*c*e^2+b^2*e^2)^(
1/2))/c)-exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(b*e-2*c*d+2*c*(e*x+d)+(-4*a*c*e^2+b^2*e
^2)^(1/2))/c))/(-4*a*c*e^2+b^2*e^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.39 \[ \int \frac {e^{d+e x}}{a+b x+c x^2} \, dx=\frac {c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{{\left (b^{2} - 4 \, a c\right )} e} \]

[In]

integrate(exp(e*x+d)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

(c*sqrt((b^2 - 4*a*c)*e^2/c^2)*Ei(1/2*(2*c*e*x + b*e - c*sqrt((b^2 - 4*a*c)*e^2/c^2))/c)*e^(1/2*(2*c*d - b*e +
c*sqrt((b^2 - 4*a*c)*e^2/c^2))/c) - c*sqrt((b^2 - 4*a*c)*e^2/c^2)*Ei(1/2*(2*c*e*x + b*e + c*sqrt((b^2 - 4*a*c
)*e^2/c^2))/c)*e^(1/2*(2*c*d - b*e - c*sqrt((b^2 - 4*a*c)*e^2/c^2))/c))/((b^2 - 4*a*c)*e)

Sympy [F]

\[ \int \frac {e^{d+e x}}{a+b x+c x^2} \, dx=e^{d} \int \frac {e^{e x}}{a + b x + c x^{2}}\, dx \]

[In]

integrate(exp(e*x+d)/(c*x**2+b*x+a),x)

[Out]

exp(d)*Integral(exp(e*x)/(a + b*x + c*x**2), x)

Maxima [F]

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

[In]

integrate(exp(e*x+d)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate(e^(e*x + d)/(c*x^2 + b*x + a), x)

Giac [F]

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

[In]

integrate(exp(e*x+d)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

integrate(e^(e*x + d)/(c*x^2 + b*x + a), x)

Mupad [F(-1)]

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

[In]

int(exp(d + e*x)/(a + b*x + c*x^2),x)

[Out]

int(exp(d + e*x)/(a + b*x + c*x^2), x)