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

Optimal. Leaf size=111 \[ -\frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}-\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}+\frac{e^a \text{ExpIntegralEi}(b x)}{c} \]

[Out]

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

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Rubi [A]  time = 0.39909, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}-\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}+\frac{e^a \text{ExpIntegralEi}(b x)}{c} \]

Antiderivative was successfully verified.

[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*(S
qrt[-c] - Sqrt[d]*x))/Sqrt[d])])/(2*c) - (E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegr
alEi[(b*(Sqrt[-c] + Sqrt[d]*x))/Sqrt[d]])/(2*c)

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Rubi in Sympy [A]  time = 43.1497, size = 94, normalized size = 0.85 \[ \frac{e^{a} \operatorname{Ei}{\left (b x \right )}}{c} - \frac{e^{a - \frac{b \sqrt{- c}}{\sqrt{d}}} \operatorname{Ei}{\left (\frac{b \left (\sqrt{d} x + \sqrt{- c}\right )}{\sqrt{d}} \right )}}{2 c} - \frac{e^{a + \frac{b \sqrt{- c}}{\sqrt{d}}} \operatorname{Ei}{\left (\frac{b \left (\sqrt{d} x - \sqrt{- c}\right )}{\sqrt{d}} \right )}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(b*x+a)/x/(d*x**2+c),x)

[Out]

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

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Mathematica [C]  time = 0.0944766, size = 93, normalized size = 0.84 \[ \frac{e^a \left (2 \text{ExpIntegralEi}(b x)-e^{-\frac{i b \sqrt{c}}{\sqrt{d}}} \left (e^{\frac{2 i b \sqrt{c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+\text{ExpIntegralEi}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )\right )\right )}{2 c} \]

Antiderivative was successfully verified.

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

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Maple [A]  time = 0.026, size = 112, normalized size = 1. \[ -{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{c}}+{\frac{1}{2\,c} \left ({{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) +{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(b*x+a)/x/(d*x^2+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(b*x + a)/((d*x^2 + c)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.268467, size = 108, normalized size = 0.97 \[ -\frac{{\rm Ei}\left (b x - \sqrt{-\frac{b^{2} c}{d}}\right ) e^{\left (a + \sqrt{-\frac{b^{2} c}{d}}\right )} +{\rm Ei}\left (b x + \sqrt{-\frac{b^{2} c}{d}}\right ) e^{\left (a - \sqrt{-\frac{b^{2} c}{d}}\right )} - 2 \,{\rm Ei}\left (b x\right ) e^{a}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{a} \int \frac{e^{b x}}{c x + d x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(b*x + a)/((d*x^2 + c)*x),x, algorithm="giac")

[Out]

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