\(\int e^{a+b x+c x^2} (b+2 c x) \, dx\) [623]

Optimal result

Integrand size = 19, antiderivative size = 12 \[ \int e^{a+b x+c x^2} (b+2 c x) \, dx=e^{a+b x+c x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2268} \[ \int e^{a+b x+c x^2} (b+2 c x) \, dx=e^{a+b x+c x^2} \]

[In]

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

[Out]

E^(a + b*x + c*x^2)

Rule 2268

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[e*(F^(a + b*x + c*x^2)/(2
*c*Log[F])), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rubi steps \begin{align*} \text {integral}& = e^{a+b x+c x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int e^{a+b x+c x^2} (b+2 c x) \, dx=e^{a+b x+c x^2} \]

[In]

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

[Out]

E^(a + b*x + c*x^2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

method	result	size
gosper	\({\mathrm e}^{c \,x^{2}+b x +a}\)	\(12\)
derivativedivides	\({\mathrm e}^{c \,x^{2}+b x +a}\)	\(12\)
default	\({\mathrm e}^{c \,x^{2}+b x +a}\)	\(12\)
norman	\({\mathrm e}^{c \,x^{2}+b x +a}\)	\(12\)
risch	\({\mathrm e}^{c \,x^{2}+b x +a}\)	\(12\)
parallelrisch	\({\mathrm e}^{c \,x^{2}+b x +a}\)	\(12\)
parts	\(-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) x c}{\sqrt {-c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) b}{2 \sqrt {-c}}+\frac {{\mathrm e}^{a -\frac {b^{2}}{4 c}} \left (2 x \,\operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) \sqrt {\pi }\, c +\operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) b \sqrt {\pi }+2 \sqrt {-c}\, {\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}}\right )}{2 \sqrt {-c}}\)	\(163\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int e^{a+b x+c x^2} (b+2 c x) \, dx=e^{\left (c x^{2} + b x + a\right )} \]

[In]

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

[Out]

e^(c*x^2 + b*x + a)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int e^{a+b x+c x^2} (b+2 c x) \, dx=e^{a + b x + c x^{2}} \]

[In]

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

[Out]

exp(a + b*x + c*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int e^{a+b x+c x^2} (b+2 c x) \, dx=e^{\left (c x^{2} + b x + a\right )} \]

[In]

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

[Out]

e^(c*x^2 + b*x + a)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int e^{a+b x+c x^2} (b+2 c x) \, dx=e^{\left (c x^{2} + b x + a\right )} \]

[In]

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

[Out]

e^(c*x^2 + b*x + a)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int e^{a+b x+c x^2} (b+2 c x) \, dx={\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2} \]

[In]

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

[Out]

exp(b*x)*exp(a)*exp(c*x^2)