3.32.11 \(\int (2+e^{2+8 x+x^2} (2+16 x+4 x^2)) \, dx\) [3111]

Optimal. Leaf size=22 \[ 2 \left (5+x+e^{2+8 x+x^2} x\right )-\log (4) \]

[Out]

2*x*exp(x^2+8*x+2)+10+2*x-2*ln(2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2326} \begin {gather*} \frac {2 e^{x^2+8 x+2} \left (x^2+4 x\right )}{x+4}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + E^(2 + 8*x + x^2)*(2 + 16*x + 4*x^2),x]

[Out]

2*x + (2*E^(2 + 8*x + x^2)*(4*x + x^2))/(4 + x)

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 x+\int e^{2+8 x+x^2} \left (2+16 x+4 x^2\right ) \, dx\\ &=2 x+\frac {2 e^{2+8 x+x^2} \left (4 x+x^2\right )}{4+x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.12, size = 15, normalized size = 0.68 \begin {gather*} 2 \left (1+e^{2+8 x+x^2}\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 + E^(2 + 8*x + x^2)*(2 + 16*x + 4*x^2),x]

[Out]

2*(1 + E^(2 + 8*x + x^2))*x

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 17, normalized size = 0.77

method	result	size
default	\(2 x +2 x \,{\mathrm e}^{x^{2}+8 x +2}\)	\(17\)
norman	\(2 x +2 x \,{\mathrm e}^{x^{2}+8 x +2}\)	\(17\)
risch	\(2 x +2 x \,{\mathrm e}^{x^{2}+8 x +2}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+16*x+2)*exp(x^2+8*x+2)+2,x,method=_RETURNVERBOSE)

[Out]

2*x+2*x*exp(x^2+8*x+2)

________________________________________________________________________________________

Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.30, size = 125, normalized size = 5.68 \begin {gather*} -i \, \sqrt {\pi } \operatorname {erf}\left (i \, x + 4 i\right ) e^{\left (-14\right )} - 2 \, {\left (\frac {{\left (x + 4\right )}^{3} \Gamma \left (\frac {3}{2}, -{\left (x + 4\right )}^{2}\right )}{\left (-{\left (x + 4\right )}^{2}\right )^{\frac {3}{2}}} - \frac {16 \, \sqrt {\pi } {\left (x + 4\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x + 4\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x + 4\right )}^{2}}} + 8 \, e^{\left ({\left (x + 4\right )}^{2}\right )}\right )} e^{\left (-14\right )} - 8 \, {\left (\frac {4 \, \sqrt {\pi } {\left (x + 4\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x + 4\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x + 4\right )}^{2}}} - e^{\left ({\left (x + 4\right )}^{2}\right )}\right )} e^{\left (-14\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+16*x+2)*exp(x^2+8*x+2)+2,x, algorithm="maxima")

[Out]

-I*sqrt(pi)*erf(I*x + 4*I)*e^(-14) - 2*((x + 4)^3*gamma(3/2, -(x + 4)^2)/(-(x + 4)^2)^(3/2) - 16*sqrt(pi)*(x +
4)*(erf(sqrt(-(x + 4)^2)) - 1)/sqrt(-(x + 4)^2) + 8*e^((x + 4)^2))*e^(-14) - 8*(4*sqrt(pi)*(x + 4)*(erf(sqrt(
-(x + 4)^2)) - 1)/sqrt(-(x + 4)^2) - e^((x + 4)^2))*e^(-14) + 2*x

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 16, normalized size = 0.73 \begin {gather*} 2 \, x e^{\left (x^{2} + 8 \, x + 2\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+16*x+2)*exp(x^2+8*x+2)+2,x, algorithm="fricas")

[Out]

2*x*e^(x^2 + 8*x + 2) + 2*x

________________________________________________________________________________________

Sympy [A]
time = 0.03, size = 15, normalized size = 0.68 \begin {gather*} 2 x e^{x^{2} + 8 x + 2} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2+16*x+2)*exp(x**2+8*x+2)+2,x)

[Out]

2*x*exp(x**2 + 8*x + 2) + 2*x

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 16, normalized size = 0.73 \begin {gather*} 2 \, x e^{\left (x^{2} + 8 \, x + 2\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+16*x+2)*exp(x^2+8*x+2)+2,x, algorithm="giac")

[Out]

2*x*e^(x^2 + 8*x + 2) + 2*x

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 16, normalized size = 0.73 \begin {gather*} 2\,x\,\left ({\mathrm {e}}^{8\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^2+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(8*x + x^2 + 2)*(16*x + 4*x^2 + 2) + 2,x)

[Out]

2*x*(exp(8*x)*exp(x^2)*exp(2) + 1)

________________________________________________________________________________________