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

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

[Out]

ExpIntegralEi[a + b*x + c*x^2]

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Rubi [A]  time = 0.176594, 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, Rules used = {6707, 2178} \[ \text{Ei}\left (a+b x+c x^2\right ) \]

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]

Rule 6707

Int[(F_)^(v_)*(u_)*(w_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Dist[q, Subst[Int[x^m*F^x,
x], x, v], x] /;  !FalseQ[q]] /; FreeQ[{F, m}, x] && EqQ[w, v]

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

Rubi steps

\begin{align*} \int \frac{e^{a+b x+c x^2} (b+2 c x)}{a+b x+c x^2} \, dx &=\operatorname{Subst}\left (\int \frac{e^x}{x} \, dx,x,a+b x+c x^2\right )\\ &=\text{Ei}\left (a+b x+c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0237218, size = 10, normalized size = 0.91 \[ \text{Ei}(a+x (b+c x)) \]

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 = 0.036, size = 19, normalized size = 1.7 \begin{align*} -{\it Ei} \left ( 1,-c{x}^{2}-bx-a \right ) \end{align*}

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. \begin{align*} \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} \end{align*}

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, 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 = 0.806188, size = 28, normalized size = 2.55 \begin{align*}{\rm Ei}\left (c x^{2} + b x + a\right ) \end{align*}

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, algorithm="fricas")

[Out]

Ei(c*x^2 + b*x + a)

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Sympy [A]  time = 34.1153, size = 10, normalized size = 0.91 \begin{align*} \operatorname{Ei}{\left (a + b x + c x^{2} \right )} \end{align*}

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]

Ei(a + b*x + c*x**2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, algorithm="giac")

[Out]

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