\(\int \frac {e^{d+e x}}{a+b x+c x^2} \, dx\) [35]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92 \[ \int \frac {e^{d+e x}}{a+b x+c x^2} \, dx=\frac {e^{d+\frac {\left (-b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )-e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}} \] Input:

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

Output:

(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))*Exp 
IntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/Sqrt[b^2 - 4*a*c]
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2698, 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.

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

\(\Big \downarrow \) 2698

\(\displaystyle \int \left (\frac {2 c e^{d+e x}}{\sqrt {b^2-4 a c} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}-\frac {2 c e^{d+e x}}{\sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}\)

Input:

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

Output:

(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]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2698

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_) + (c_.)*(x_ 
)^2), x_Symbol] :> Int[ExpandIntegrand[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]
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.22

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

Input:

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

Output:

-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)

Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.39 \[ \int \frac {e^{d+e x}}{a+b x+c x^2} \, dx=\frac {c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{{\left (b^{2} - 4 \, a c\right )} e} \] Input:

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

Output:

(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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