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]

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)

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Rubi [A]  time = 0.12, antiderivative size = 118, normalized size of antiderivative = 1.00, 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 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

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]

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*}

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Mathematica [C]  time = 0.06, size = 94, normalized size = 0.80 \[ -\frac {i e^{a-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \text {Ei}\left (b \left (x-\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )-\text {Ei}\left (b \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )\right )}{2 \sqrt {c} \sqrt {d}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a + b*x)/(c + d*x^2),x]

[Out]

((-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])

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fricas [A]  time = 0.39, size = 98, normalized size = 0.83 \[ -\frac {\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )}}{2 \, b c} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (b x + a\right )}}{d x^{2} + c}\,{d x} \]

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)

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maple [A]  time = 0.02, size = 102, normalized size = 0.86 \[ -\frac {\Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}-\Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}}{2 \sqrt {-c d}} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (b x + a\right )}}{d x^{2} + c}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{a+b\,x}}{d\,x^2+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x)/(c + d*x^2),x)

[Out]

int(exp(a + b*x)/(c + d*x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{a} \int \frac {e^{b x}}{c + d x^{2}}\, dx \]

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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