\(\int \frac {e^{a+b x}}{c+d x^2} \, dx\) [464]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2301, 2209} \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]

[In]

Int[E^(a + b*x)/(c + d*x^2),x]

[Out]

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

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 2301

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]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=-\frac {i e^{a-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (b \left (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )\right )}{2 \sqrt {c} \sqrt {d}} \]

[In]

Integrate[E^(a + b*x)/(c + d*x^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.83 \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=-\frac {\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )}}{2 \, b c} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(e^(b*x + a)/(d*x^2 + c), x)

Giac [F]

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

[In]

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

[Out]

integrate(e^(b*x + a)/(d*x^2 + c), x)

Mupad [F(-1)]

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

[In]

int(exp(a + b*x)/(c + d*x^2),x)

[Out]

int(exp(a + b*x)/(c + d*x^2), x)