\(\int \frac {e^{a+b x+c x^2} (b+2 c x)}{(a+b x+c x^2)^3} \, dx\) [626]

Optimal result

Integrand size = 31, antiderivative size = 72 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )^2}-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \operatorname {ExpIntegralEi}\left (a+b x+c x^2\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6839, 2208, 2209} \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \operatorname {ExpIntegralEi}\left (c x^2+b x+a\right )-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )}-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )^2} \]

[In]

Int[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^3,x]

[Out]

-1/2*E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^2 - E^(a + b*x + c*x^2)/(2*(a + b*x + c*x^2)) + ExpIntegralEi[a + b
*x + c*x^2]/2

Rule 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

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*} \text {integral}& = \text {Subst}\left (\int \frac {e^x}{x^3} \, dx,x,a+b x+c x^2\right ) \\ & = -\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} \text {Subst}\left (\int \frac {e^x}{x^2} \, dx,x,a+b x+c x^2\right ) \\ & = -\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )^2}-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \text {Subst}\left (\int \frac {e^x}{x} \, dx,x,a+b x+c x^2\right ) \\ & = -\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )^2}-\frac {e^{a+b x+c x^2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \text {Ei}\left (a+b x+c x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (-\frac {e^{a+x (b+c x)} \left (1+a+b x+c x^2\right )}{(a+x (b+c x))^2}+\operatorname {ExpIntegralEi}(a+x (b+c x))\right ) \]

[In]

Integrate[(E^(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^3,x]

[Out]

(-((E^(a + x*(b + c*x))*(1 + a + b*x + c*x^2))/(a + x*(b + c*x))^2) + ExpIntegralEi[a + x*(b + c*x)])/2

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )^{2}}-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )}-\frac {\operatorname {Ei}_{1}\left (-c \,x^{2}-b x -a \right )}{2}\)	\(70\)
default	\(-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )^{2}}-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )}-\frac {\operatorname {Ei}_{1}\left (-c \,x^{2}-b x -a \right )}{2}\)	\(70\)
risch	\(-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )^{2}}-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{2 \left (c \,x^{2}+b x +a \right )}-\frac {\operatorname {Ei}_{1}\left (-c \,x^{2}-b x -a \right )}{2}\)	\(70\)

[In]

int(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.54 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} {\rm Ei}\left (c x^{2} + b x + a\right ) - {\left (c x^{2} + b x + a + 1\right )} e^{\left (c x^{2} + b x + a\right )}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

[In]

integrate(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/2*((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*Ei(c*x^2 + b*x + a) - (c*x^2 + b*x + a + 1)*e^(
c*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(exp(c*x**2+b*x+a)*(2*c*x+b)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \]

[In]

integrate(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^3, x)

Giac [F]

\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \]

[In]

integrate(exp(c*x^2+b*x+a)*(2*c*x+b)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

integrate((2*c*x + b)*e^(c*x^2 + b*x + a)/(c*x^2 + b*x + a)^3, x)

Mupad [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\mathrm {expint}\left (-c\,x^2-b\,x-a\right )}{2}-{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (\frac {1}{2\,\left (c\,x^2+b\,x+a\right )}+\frac {1}{2\,{\left (c\,x^2+b\,x+a\right )}^2}\right ) \]

[In]

int((exp(a + b*x + c*x^2)*(b + 2*c*x))/(a + b*x + c*x^2)^3,x)

[Out]

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