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

Optimal. Leaf size=145 \[ \frac{\sqrt{d} e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 (-c)^{3/2}}-\frac{\sqrt{d} e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 (-c)^{3/2}}+\frac{e^a b \text{ExpIntegralEi}(b x)}{c}-\frac{e^{a+b x}}{c x} \]

[Out]

-(E^(a + b*x)/(c*x)) + (b*E^a*ExpIntegralEi[b*x])/c + (Sqrt[d]*E^(a + (b*Sqrt[-c
])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/Sqrt[d])])/(2*(-c)^(3/2))
- (Sqrt[d]*E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[(b*(Sqrt[-c] + Sqrt[d]*x)
)/Sqrt[d]])/(2*(-c)^(3/2))

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Rubi [A]  time = 0.575842, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{d} e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 (-c)^{3/2}}-\frac{\sqrt{d} e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 (-c)^{3/2}}+\frac{e^a b \text{ExpIntegralEi}(b x)}{c}-\frac{e^{a+b x}}{c x} \]

Antiderivative was successfully verified.

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

[Out]

-(E^(a + b*x)/(c*x)) + (b*E^a*ExpIntegralEi[b*x])/c + (Sqrt[d]*E^(a + (b*Sqrt[-c
])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/Sqrt[d])])/(2*(-c)^(3/2))
- (Sqrt[d]*E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[(b*(Sqrt[-c] + Sqrt[d]*x)
)/Sqrt[d]])/(2*(-c)^(3/2))

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Rubi in Sympy [A]  time = 54.354, size = 126, normalized size = 0.87 \[ \frac{b e^{a} \operatorname{Ei}{\left (b x \right )}}{c} - \frac{\sqrt{d} e^{a - \frac{b \sqrt{- c}}{\sqrt{d}}} \operatorname{Ei}{\left (\frac{b \left (\sqrt{d} x + \sqrt{- c}\right )}{\sqrt{d}} \right )}}{2 \left (- c\right )^{\frac{3}{2}}} + \frac{\sqrt{d} e^{a + \frac{b \sqrt{- c}}{\sqrt{d}}} \operatorname{Ei}{\left (\frac{b \left (\sqrt{d} x - \sqrt{- c}\right )}{\sqrt{d}} \right )}}{2 \left (- c\right )^{\frac{3}{2}}} - \frac{e^{a + b x}}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*exp(a)*Ei(b*x)/c - sqrt(d)*exp(a - b*sqrt(-c)/sqrt(d))*Ei(b*(sqrt(d)*x + sqrt(
-c))/sqrt(d))/(2*(-c)**(3/2)) + sqrt(d)*exp(a + b*sqrt(-c)/sqrt(d))*Ei(b*(sqrt(d
)*x - sqrt(-c))/sqrt(d))/(2*(-c)**(3/2)) - exp(a + b*x)/(c*x)

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Mathematica [C]  time = 0.232373, size = 133, normalized size = 0.92 \[ \frac{e^a \left (i \sqrt{d} x e^{\frac{i b \sqrt{c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )-i \sqrt{d} x e^{-\frac{i b \sqrt{c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+2 b \sqrt{c} x \text{ExpIntegralEi}(b x)-2 \sqrt{c} e^{b x}\right )}{2 c^{3/2} x} \]

Antiderivative was successfully verified.

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

[Out]

(E^a*(-2*Sqrt[c]*E^(b*x) + 2*b*Sqrt[c]*x*ExpIntegralEi[b*x] + I*Sqrt[d]*E^((I*b*
Sqrt[c])/Sqrt[d])*x*ExpIntegralEi[b*(((-I)*Sqrt[c])/Sqrt[d] + x)] - (I*Sqrt[d]*x
*ExpIntegralEi[b*((I*Sqrt[c])/Sqrt[d] + x)])/E^((I*b*Sqrt[c])/Sqrt[d])))/(2*c^(3
/2)*x)

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Maple [A]  time = 0.037, size = 142, normalized size = 1. \[ b \left ( -{\frac{{{\rm e}^{bx+a}}}{bcx}}-{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{c}}+{\frac{d}{2\,cb} \left ({{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) -{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) \right ){\frac{1}{\sqrt{-cd}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(b*x+a)/x^2/(d*x^2+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.265082, size = 165, normalized size = 1.14 \[ -\frac{b x{\rm Ei}\left (b x - \sqrt{-\frac{b^{2} c}{d}}\right ) e^{\left (a + \sqrt{-\frac{b^{2} c}{d}}\right )} - b x{\rm Ei}\left (b x + \sqrt{-\frac{b^{2} c}{d}}\right ) e^{\left (a - \sqrt{-\frac{b^{2} c}{d}}\right )} - 2 \,{\left (b x{\rm Ei}\left (b x\right ) e^{a} - e^{\left (b x + a\right )}\right )} \sqrt{-\frac{b^{2} c}{d}}}{2 \, \sqrt{-\frac{b^{2} c}{d}} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{a} \int \frac{e^{b x}}{c x^{2} + d x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

exp(a)*Integral(exp(b*x)/(c*x**2 + d*x**4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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