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

Optimal. Leaf size=38 \[ \text{ExpIntegralEi}\left (a+b x+c x^2\right )-\frac{e^{a+b x+c x^2}}{a+b x+c x^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.293068, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \text{ExpIntegralEi}\left (a+b x+c x^2\right )-\frac{e^{a+b x+c x^2}}{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 82.2818, size = 32, normalized size = 0.84 \[ \operatorname{Ei}{\left (a + b x + c x^{2} \right )} - \frac{e^{a + b x + c x^{2}}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0314182, size = 35, normalized size = 0.92 \[ \text{ExpIntegralEi}(a+x (b+c x))-\frac{e^{a+x (b+c x)}}{a+x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

-(E^(a + x*(b + c*x))/(a + x*(b + c*x))) + ExpIntegralEi[a + x*(b + c*x)]

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 45, normalized size = 1.2 \[ -{\frac{{{\rm e}^{c{x}^{2}+bx+a}}}{c{x}^{2}+bx+a}}-{\it Ei} \left ( 1,-c{x}^{2}-bx-a \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^2,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.247456, size = 66, normalized size = 1.74 \[ \frac{{\left (c x^{2} + b x + a\right )}{\rm Ei}\left (c x^{2} + b x + a\right ) - e^{\left (c x^{2} + b x + a\right )}}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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