\(\int \frac {1}{441} e^{\frac {1}{441} (441+42 x+x^2)} (42+2 x) \, dx\) [4014]

Optimal result

Integrand size = 23, antiderivative size = 11 \[ \int \frac {1}{441} e^{\frac {1}{441} \left (441+42 x+x^2\right )} (42+2 x) \, dx=e^{\left (1+\frac {x}{21}\right )^2} \]

[Out]

exp((1+1/21*x)^2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 2259, 2240} \[ \int \frac {1}{441} e^{\frac {1}{441} \left (441+42 x+x^2\right )} (42+2 x) \, dx=e^{\frac {1}{441} (x+21)^2} \]

[In]

Int[(E^((441 + 42*x + x^2)/441)*(42 + 2*x))/441,x]

[Out]

E^((21 + x)^2/441)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2259

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{441} \int e^{\frac {1}{441} \left (441+42 x+x^2\right )} (42+2 x) \, dx \\ & = \frac {1}{441} \int e^{\frac {1}{441} (21+x)^2} (42+2 x) \, dx \\ & = e^{\frac {1}{441} (21+x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{441} e^{\frac {1}{441} \left (441+42 x+x^2\right )} (42+2 x) \, dx=e^{\frac {1}{441} (21+x)^2} \]

[In]

Integrate[(E^((441 + 42*x + x^2)/441)*(42 + 2*x))/441,x]

[Out]

E^((21 + x)^2/441)

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82

method	result	size
risch	\({\mathrm e}^{\frac {\left (x +21\right )^{2}}{441}}\)	\(9\)
gosper	\({\mathrm e}^{\frac {1}{441} x^{2}+\frac {2}{21} x +1}\)	\(12\)
default	\({\mathrm e}^{\frac {1}{441} x^{2}+\frac {2}{21} x +1}\)	\(12\)
norman	\({\mathrm e}^{\frac {1}{441} x^{2}+\frac {2}{21} x +1}\)	\(12\)
parallelrisch	\({\mathrm e}^{\frac {1}{441} x^{2}+\frac {2}{21} x +1}\)	\(12\)
parts	\(-\frac {i \sqrt {\pi }\, \operatorname {erf}\left (\frac {1}{21} i x +i\right ) x}{21}-i \sqrt {\pi }\, \operatorname {erf}\left (\frac {1}{21} i x +i\right )+\sqrt {\pi }\, \left (\operatorname {erf}\left (\frac {1}{21} i x +i\right ) \left (\frac {1}{21} i x +i\right )+\frac {{\mathrm e}^{-\left (\frac {1}{21} i x +i\right )^{2}}}{\sqrt {\pi }}\right )\)	\(68\)

[In]

int(1/441*(2*x+42)*exp(1/441*x^2+2/21*x+1),x,method=_RETURNVERBOSE)

[Out]

exp(1/441*(x+21)^2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{441} e^{\frac {1}{441} \left (441+42 x+x^2\right )} (42+2 x) \, dx=e^{\left (\frac {1}{441} \, x^{2} + \frac {2}{21} \, x + 1\right )} \]

[In]

integrate(1/441*(2*x+42)*exp(1/441*x^2+2/21*x+1),x, algorithm="fricas")

[Out]

e^(1/441*x^2 + 2/21*x + 1)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {1}{441} e^{\frac {1}{441} \left (441+42 x+x^2\right )} (42+2 x) \, dx=e^{\frac {x^{2}}{441} + \frac {2 x}{21} + 1} \]

[In]

integrate(1/441*(2*x+42)*exp(1/441*x**2+2/21*x+1),x)

[Out]

exp(x**2/441 + 2*x/21 + 1)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{441} e^{\frac {1}{441} \left (441+42 x+x^2\right )} (42+2 x) \, dx=e^{\left (\frac {1}{441} \, x^{2} + \frac {2}{21} \, x + 1\right )} \]

[In]

integrate(1/441*(2*x+42)*exp(1/441*x^2+2/21*x+1),x, algorithm="maxima")

[Out]

e^(1/441*x^2 + 2/21*x + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{441} e^{\frac {1}{441} \left (441+42 x+x^2\right )} (42+2 x) \, dx=e^{\left (\frac {1}{441} \, x^{2} + \frac {2}{21} \, x + 1\right )} \]

[In]

integrate(1/441*(2*x+42)*exp(1/441*x^2+2/21*x+1),x, algorithm="giac")

[Out]

e^(1/441*x^2 + 2/21*x + 1)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {1}{441} e^{\frac {1}{441} \left (441+42 x+x^2\right )} (42+2 x) \, dx={\mathrm {e}}^{\frac {2\,x}{21}}\,\mathrm {e}\,{\mathrm {e}}^{\frac {x^2}{441}} \]

[In]

int((exp((2*x)/21 + x^2/441 + 1)*(2*x + 42))/441,x)

[Out]

exp((2*x)/21)*exp(1)*exp(x^2/441)