\(\int 4 e^{-3+3 x+25 x^2} (3+50 x) \, dx\) [1692]

Optimal result

Integrand size = 19, antiderivative size = 14 \[ \int 4 e^{-3+3 x+25 x^2} (3+50 x) \, dx=4 e^{-3+3 x+25 x^2} \]

[Out]

exp(2*ln(2)+25*x^2+3*x-3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 2268} \[ \int 4 e^{-3+3 x+25 x^2} (3+50 x) \, dx=4 e^{25 x^2+3 x-3} \]

[In]

Int[4*E^(-3 + 3*x + 25*x^2)*(3 + 50*x),x]

[Out]

4*E^(-3 + 3*x + 25*x^2)

Rule 12

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

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}& = 4 \int e^{-3+3 x+25 x^2} (3+50 x) \, dx \\ & = 4 e^{-3+3 x+25 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int 4 e^{-3+3 x+25 x^2} (3+50 x) \, dx=4 e^{-3+3 x+25 x^2} \]

[In]

Integrate[4*E^(-3 + 3*x + 25*x^2)*(3 + 50*x),x]

[Out]

4*E^(-3 + 3*x + 25*x^2)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

method	result	size
risch	\(4 \,{\mathrm e}^{25 x^{2}+3 x -3}\)	\(14\)
gosper	\({\mathrm e}^{2 \ln \left (2\right )+25 x^{2}+3 x -3}\)	\(16\)
derivativedivides	\({\mathrm e}^{2 \ln \left (2\right )+25 x^{2}+3 x -3}\)	\(16\)
default	\({\mathrm e}^{2 \ln \left (2\right )+25 x^{2}+3 x -3}\)	\(16\)
norman	\({\mathrm e}^{2 \ln \left (2\right )+25 x^{2}+3 x -3}\)	\(16\)
parallelrisch	\({\mathrm e}^{2 \ln \left (2\right )+25 x^{2}+3 x -3}\)	\(16\)
parts	\(-5 i \sqrt {\pi }\, {\mathrm e}^{2 \ln \left (2\right )-\frac {309}{100}} \operatorname {erf}\left (5 i x +\frac {3}{10} i\right ) x -\frac {3 i \sqrt {\pi }\, {\mathrm e}^{2 \ln \left (2\right )-\frac {309}{100}} \operatorname {erf}\left (5 i x +\frac {3}{10} i\right )}{10}+\frac {i {\mathrm e}^{2 \ln \left (2\right )-\frac {309}{100}} \left (50 x \,\operatorname {erf}\left (\frac {i \left (50 x +3\right )}{10}\right ) \sqrt {\pi }+3 \,\operatorname {erf}\left (\frac {i \left (50 x +3\right )}{10}\right ) \sqrt {\pi }-10 i {\mathrm e}^{\frac {\left (50 x +3\right )^{2}}{100}}\right )}{10}\)	\(98\)

[In]

int((50*x+3)*exp(2*ln(2)+25*x^2+3*x-3),x,method=_RETURNVERBOSE)

[Out]

4*exp(25*x^2+3*x-3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int 4 e^{-3+3 x+25 x^2} (3+50 x) \, dx=e^{\left (25 \, x^{2} + 3 \, x + 2 \, \log \left (2\right ) - 3\right )} \]

[In]

integrate((50*x+3)*exp(2*log(2)+25*x^2+3*x-3),x, algorithm="fricas")

[Out]

e^(25*x^2 + 3*x + 2*log(2) - 3)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int 4 e^{-3+3 x+25 x^2} (3+50 x) \, dx=4 e^{25 x^{2} + 3 x - 3} \]

[In]

integrate((50*x+3)*exp(2*ln(2)+25*x**2+3*x-3),x)

[Out]

4*exp(25*x**2 + 3*x - 3)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int 4 e^{-3+3 x+25 x^2} (3+50 x) \, dx=4 \, e^{\left (25 \, x^{2} + 3 \, x - 3\right )} \]

[In]

integrate((50*x+3)*exp(2*log(2)+25*x^2+3*x-3),x, algorithm="maxima")

[Out]

4*e^(25*x^2 + 3*x - 3)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int 4 e^{-3+3 x+25 x^2} (3+50 x) \, dx=e^{\left (25 \, x^{2} + 3 \, x + 2 \, \log \left (2\right ) - 3\right )} \]

[In]

integrate((50*x+3)*exp(2*log(2)+25*x^2+3*x-3),x, algorithm="giac")

[Out]

e^(25*x^2 + 3*x + 2*log(2) - 3)

Mupad [B] (verification not implemented)

Time = 8.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int 4 e^{-3+3 x+25 x^2} (3+50 x) \, dx=4\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{25\,x^2} \]

[In]

int(exp(3*x + 2*log(2) + 25*x^2 - 3)*(50*x + 3),x)

[Out]

4*exp(3*x)*exp(-3)*exp(25*x^2)