3.7.24 \(\int \frac {e^{a+b x+c x^2} (b+2 c x)}{a+b x+c x^2} \, dx\) [624]

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

[Out]

Ei(c*x^2+b*x+a)

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Rubi [A]
time = 0.12, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = {6839, 2209} \begin {gather*} \text {Ei}\left (a+b x+c x^2\right ) \end {gather*}

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 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 6839

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]

Rubi steps

\begin {align*} \int \frac {e^{a+b x+c x^2} (b+2 c x)}{a+b x+c x^2} \, dx &=\text {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*}

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Mathematica [A]
time = 0.10, size = 10, normalized size = 0.91 \begin {gather*} \text {Ei}(a+x (b+c x)) \end {gather*}

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 = 0.10, size = 19, normalized size = 1.73

method	result	size
derivativedivides	\(-\expIntegral \left (1, -c \,x^{2}-b x -a \right )\)	\(19\)
default	\(-\expIntegral \left (1, -c \,x^{2}-b x -a \right )\)	\(19\)
risch	\(-\expIntegral \left (1, -c \,x^{2}-b x -a \right )\)	\(19\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.40, size = 11, normalized size = 1.00 \begin {gather*} {\rm Ei}\left (c x^{2} + b x + a\right ) \end {gather*}

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 = 9.08, size = 10, normalized size = 0.91 \begin {gather*} \operatorname {Ei}{\left (a + b x + c x^{2} \right )} \end {gather*}

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 [A]
time = 3.84, size = 11, normalized size = 1.00 \begin {gather*} {\rm Ei}\left (c x^{2} + b x + a\right ) \end {gather*}

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]

Ei(c*x^2 + b*x + a)

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Mupad [B]
time = 3.79, size = 11, normalized size = 1.00 \begin {gather*} \mathrm {ei}\left (c\,x^2+b\,x+a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ei(a + b*x + c*x^2)

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