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

Optimal. Leaf size=11 \[ \text{ExpIntegralEi}\left (a+b x+c x^2\right ) \]

[Out]

ExpIntegralEi[a + b*x + c*x^2]

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Rubi [A]  time = 0.266157, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \text{ExpIntegralEi}\left (a+b x+c x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

ExpIntegralEi[a + b*x + c*x^2]

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Rubi in Sympy [A]  time = 79.4201, size = 10, normalized size = 0.91 \[ \operatorname{Ei}{\left (a + b x + c x^{2} \right )} \]

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),x)

[Out]

Ei(a + b*x + c*x**2)

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Mathematica [A]  time = 0.008948, size = 10, normalized size = 0.91 \[ \text{ExpIntegralEi}(a+x (b+c x)) \]

Antiderivative was successfully verified.

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

[Out]

ExpIntegralEi[a + x*(b + c*x)]

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Maple [A]  time = 0.008, size = 19, normalized size = 1.7 \[ -{\it Ei} \left ( 1,-c{x}^{2}-bx-a \right ) \]

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),x)

[Out]

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

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

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.232682, size = 15, normalized size = 1.36 \[{\rm Ei}\left (c x^{2} + b x + a\right ) \]

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),x, algorithm="fricas")

[Out]

Ei(c*x^2 + b*x + a)

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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),x)

[Out]

Timed out

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

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),x, algorithm="giac")

[Out]

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