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

Optimal. Leaf size=118 \[ \frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}} \]

[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])

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Rubi [A]  time = 0.119093, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2269, 2178} \[ \frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}} \]

Antiderivative was successfully verified.

[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 2269

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]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{e^{a+b x}}{c+d x^2} \, dx &=\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]  time = 0.0537059, size = 94, normalized size = 0.8 \[ -\frac{i e^{a-\frac{i b \sqrt{c}}{\sqrt{d}}} \left (e^{\frac{2 i b \sqrt{c}}{\sqrt{d}}} \text{Ei}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )-\text{Ei}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )\right )}{2 \sqrt{c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a + b*x)/(c + d*x^2),x]

[Out]

((-I/2)*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])

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Maple [A]  time = 0.01, size = 102, normalized size = 0.9 \begin{align*} -{\frac{1}{2} \left ({{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) -{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) \right ){\frac{1}{\sqrt{-cd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (b x + a\right )}}{d x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(b*x + a)/(d*x^2 + c), x)

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Fricas [A]  time = 1.51285, size = 192, normalized size = 1.63 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int \frac{e^{b x}}{c + d x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (b x + a\right )}}{d x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(b*x + a)/(d*x^2 + c), x)