\(\int e^{a+b x+c x^2} (b+2 c x) (a+b x+c x^2)^m \, dx\) [619]

Optimal result

Integrand size = 31, antiderivative size = 49 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^m \, dx=\left (-a-b x-c x^2\right )^{-m} \left (a+b x+c x^2\right )^m \Gamma \left (1+m,-a-b x-c x^2\right ) \]

[Out]

(c*x^2+b*x+a)^m*GAMMA(1+m,-c*x^2-b*x-a)/((-c*x^2-b*x-a)^m)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6839, 2212} \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^m \, dx=\left (-a-b x-c x^2\right )^{-m} \left (a+b x+c x^2\right )^m \Gamma \left (m+1,-c x^2-b x-a\right ) \]

[In]

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

[Out]

((a + b*x + c*x^2)^m*Gamma[1 + m, -a - b*x - c*x^2])/(-a - b*x - c*x^2)^m

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

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 e^x x^m \, dx,x,a+b x+c x^2\right ) \\ & = \left (-a-b x-c x^2\right )^{-m} \left (a+b x+c x^2\right )^m \Gamma \left (1+m,-a-b x-c x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^m \, dx=(-a-x (b+c x))^{-m} (a+x (b+c x))^m \Gamma (1+m,-a-x (b+c x)) \]

[In]

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

[Out]

((a + x*(b + c*x))^m*Gamma[1 + m, -a - x*(b + c*x)])/(-a - x*(b + c*x))^m

Maple [F]

\[\int {\mathrm e}^{c \,x^{2}+b x +a} \left (2 x c +b \right ) \left (c \,x^{2}+b x +a \right )^{m}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.49 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^m \, dx=e^{\left (-i \, \pi m\right )} \Gamma \left (m + 1, -c x^{2} - b x - a\right ) \]

[In]

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

[Out]

e^(-I*pi*m)*gamma(m + 1, -c*x^2 - b*x - a)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^m \, dx=\frac {\Gamma \left (m+1,-c\,x^2-b\,x-a\right )\,{\left (c\,x^2+b\,x+a\right )}^m}{{\left (-c\,x^2-b\,x-a\right )}^m} \]

[In]

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

[Out]

(igamma(m + 1, - a - b*x - c*x^2)*(a + b*x + c*x^2)^m)/(- a - b*x - c*x^2)^m