\(\int \frac {e^{a+b x}}{c+d x^2} \, dx\) [29]

Optimal result

Integrand size = 17, antiderivative size = 118 \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=-\frac {i e^{a-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (b \left (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )\right )}{2 \sqrt {c} \sqrt {d}} \] Input:

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

Output:

((-1/2*I)*E^(a - (I*b*Sqrt[c])/Sqrt[d])*(E^(((2*I)*b*Sqrt[c])/Sqrt[d])*Exp 
IntegralEi[b*(((-I)*Sqrt[c])/Sqrt[d] + x)] - ExpIntegralEi[b*((I*Sqrt[c])/ 
Sqrt[d] + x)]))/(Sqrt[c]*Sqrt[d])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2699, 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.

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

\(\Big \downarrow \) 2699

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(E^(a + (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/S 
qrt[d])])/(2*Sqrt[-c]*Sqrt[d]) - (E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegral 
Ei[(b*(Sqrt[-c] + Sqrt[d]*x))/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d])
 

Defintions of rubi rules used

rule 2009

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

rule 2699

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(-\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right )-{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right )}{2 \sqrt {-c d}}\)	\(102\)
default	\(-\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right )-{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right )}{2 \sqrt {-c d}}\)	\(102\)
risch	\(-\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right )}{2 \sqrt {-c d}}+\frac {{\mathrm e}^{\frac {-b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right )}{2 \sqrt {-c d}}\)	\(106\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.83 \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=-\frac {\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )}}{2 \, b c} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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