\(\int \frac {e^{a+b x}}{x (c+d x^2)} \, dx\) [463]

Optimal result

Integrand size = 20, antiderivative size = 111 \[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=\frac {e^a \operatorname {ExpIntegralEi}(b x)}{c}-\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2303, 2209} \[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=-\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{\sqrt {d}}\right )}{2 c}+\frac {e^a \operatorname {ExpIntegralEi}(b x)}{c} \]

[In]

Int[E^(a + b*x)/(x*(c + d*x^2)),x]

[Out]

(E^a*ExpIntegralEi[b*x])/c - (E^(a + (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/Sqrt[d])
])/(2*c) - (E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[(b*(Sqrt[-c] + Sqrt[d]*x))/Sqrt[d]])/(2*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 2303

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_) + (c_)*(x_)^2), x_Symbol] :> Int[ExpandIntegran
d[F^(g*(d + e*x)^n), u^m/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x] && PolynomialQ[u, x] && Intege
rQ[m]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=\frac {e^a \left (2 \operatorname {ExpIntegralEi}(b x)-e^{-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (b \left (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )+\operatorname {ExpIntegralEi}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )\right )\right )}{2 c} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (\frac {b \sqrt {-c d}+a d -d \left (b x +a \right )}{d}\right )+{\mathrm e}^{-\frac {b \sqrt {-c d}-a d}{d}} \operatorname {Ei}_{1}\left (-\frac {b \sqrt {-c d}-a d +d \left (b x +a \right )}{d}\right )}{2 c}-\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b x \right )}{c}\)	\(112\)
default	\(\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (\frac {b \sqrt {-c d}+a d -d \left (b x +a \right )}{d}\right )+{\mathrm e}^{-\frac {b \sqrt {-c d}-a d}{d}} \operatorname {Ei}_{1}\left (-\frac {b \sqrt {-c d}-a d +d \left (b x +a \right )}{d}\right )}{2 c}-\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b x \right )}{c}\)	\(112\)
risch	\(-\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b x \right )}{c}+\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (\frac {b \sqrt {-c d}+a d -d \left (b x +a \right )}{d}\right )}{2 c}+\frac {{\mathrm e}^{\frac {-b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (-\frac {b \sqrt {-c d}-a d +d \left (b x +a \right )}{d}\right )}{2 c}\)	\(113\)

[In]

int(exp(b*x+a)/x/(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-1/c*exp(a)*Ei(1,-b*x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.72 \[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=-\frac {{\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} + {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )} - 2 \, {\rm Ei}\left (b x\right ) e^{a}}{2 \, c} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

exp(a)*Integral(exp(b*x)/(c*x + d*x**3), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(exp(a + b*x)/(x*(c + d*x^2)),x)

[Out]

int(exp(a + b*x)/(x*(c + d*x^2)), x)