3.94.49 \(\int (e^{4 e^4+6 x-3 x^2} (60-60 x)+e^{8 e^4+12 x-6 x^2} (12-12 x)+e^5 (1+2 x)) \, dx\)

Optimal. Leaf size=29 \[ \left (5+e^{4 e^4+3 (2-x) x}\right )^2+e^5 x (1+x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.69, number of steps used = 3, number of rules used = 1, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2236} \begin {gather*} e^{-6 x^2+12 x+8 e^4}+10 e^{-3 x^2+6 x+4 e^4}+\frac {1}{4} e^5 (2 x+1)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4*E^4 + 6*x - 3*x^2)*(60 - 60*x) + E^(8*E^4 + 12*x - 6*x^2)*(12 - 12*x) + E^5*(1 + 2*x),x]

[Out]

E^(8*E^4 + 12*x - 6*x^2) + 10*E^(4*E^4 + 6*x - 3*x^2) + (E^5*(1 + 2*x)^2)/4

Rule 2236

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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} e^5 (1+2 x)^2+\int e^{4 e^4+6 x-3 x^2} (60-60 x) \, dx+\int e^{8 e^4+12 x-6 x^2} (12-12 x) \, dx\\ &=e^{8 e^4+12 x-6 x^2}+10 e^{4 e^4+6 x-3 x^2}+\frac {1}{4} e^5 (1+2 x)^2\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 47, normalized size = 1.62 \begin {gather*} e^{8 e^4+12 x-6 x^2}+10 e^{4 e^4+6 x-3 x^2}+e^5 x+e^5 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*E^4 + 6*x - 3*x^2)*(60 - 60*x) + E^(8*E^4 + 12*x - 6*x^2)*(12 - 12*x) + E^5*(1 + 2*x),x]

[Out]

E^(8*E^4 + 12*x - 6*x^2) + 10*E^(4*E^4 + 6*x - 3*x^2) + E^5*x + E^5*x^2

________________________________________________________________________________________

fricas [A]  time = 0.79, size = 39, normalized size = 1.34 \begin {gather*} {\left (x^{2} + x\right )} e^{5} + 10 \, e^{\left (-3 \, x^{2} + 6 \, x + 4 \, e^{4}\right )} + e^{\left (-6 \, x^{2} + 12 \, x + 8 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x+12)*exp(4*exp(4)-3*x^2+6*x)^2+(-60*x+60)*exp(4*exp(4)-3*x^2+6*x)+(2*x+1)*exp(5),x, algorithm=
"fricas")

[Out]

(x^2 + x)*e^5 + 10*e^(-3*x^2 + 6*x + 4*e^4) + e^(-6*x^2 + 12*x + 8*e^4)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 39, normalized size = 1.34 \begin {gather*} {\left (x^{2} + x\right )} e^{5} + 10 \, e^{\left (-3 \, x^{2} + 6 \, x + 4 \, e^{4}\right )} + e^{\left (-6 \, x^{2} + 12 \, x + 8 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x+12)*exp(4*exp(4)-3*x^2+6*x)^2+(-60*x+60)*exp(4*exp(4)-3*x^2+6*x)+(2*x+1)*exp(5),x, algorithm=
"giac")

[Out]

(x^2 + x)*e^5 + 10*e^(-3*x^2 + 6*x + 4*e^4) + e^(-6*x^2 + 12*x + 8*e^4)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 40, normalized size = 1.38

method	result	size
default	\(10 \,{\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}+{\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x}+{\mathrm e}^{5} \left (x^{2}+x \right )\)	\(40\)
risch	\({\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x}+x \,{\mathrm e}^{5}+x^{2} {\mathrm e}^{5}+10 \,{\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}\)	\(42\)
norman	\({\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x}+x \,{\mathrm e}^{5}+x^{2} {\mathrm e}^{5}+10 \,{\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12*x+12)*exp(4*exp(4)-3*x^2+6*x)^2+(-60*x+60)*exp(4*exp(4)-3*x^2+6*x)+(2*x+1)*exp(5),x,method=_RETURNVER
BOSE)

[Out]

10*exp(4*exp(4)-3*x^2+6*x)+exp(8*exp(4)-6*x^2+12*x)+exp(5)*(x^2+x)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 39, normalized size = 1.34 \begin {gather*} {\left (x^{2} + x\right )} e^{5} + 10 \, e^{\left (-3 \, x^{2} + 6 \, x + 4 \, e^{4}\right )} + e^{\left (-6 \, x^{2} + 12 \, x + 8 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x+12)*exp(4*exp(4)-3*x^2+6*x)^2+(-60*x+60)*exp(4*exp(4)-3*x^2+6*x)+(2*x+1)*exp(5),x, algorithm=
"maxima")

[Out]

(x^2 + x)*e^5 + 10*e^(-3*x^2 + 6*x + 4*e^4) + e^(-6*x^2 + 12*x + 8*e^4)

________________________________________________________________________________________

mupad [B]  time = 0.21, size = 41, normalized size = 1.41 \begin {gather*} 10\,{\mathrm {e}}^{-3\,x^2+6\,x+4\,{\mathrm {e}}^4}+{\mathrm {e}}^{-6\,x^2+12\,x+8\,{\mathrm {e}}^4}+x\,{\mathrm {e}}^5+x^2\,{\mathrm {e}}^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(5)*(2*x + 1) - exp(6*x + 4*exp(4) - 3*x^2)*(60*x - 60) - exp(12*x + 8*exp(4) - 6*x^2)*(12*x - 12),x)

[Out]

10*exp(6*x + 4*exp(4) - 3*x^2) + exp(12*x + 8*exp(4) - 6*x^2) + x*exp(5) + x^2*exp(5)

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 42, normalized size = 1.45 \begin {gather*} x^{2} e^{5} + x e^{5} + e^{- 6 x^{2} + 12 x + 8 e^{4}} + 10 e^{- 3 x^{2} + 6 x + 4 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x+12)*exp(4*exp(4)-3*x**2+6*x)**2+(-60*x+60)*exp(4*exp(4)-3*x**2+6*x)+(2*x+1)*exp(5),x)

[Out]

x**2*exp(5) + x*exp(5) + exp(-6*x**2 + 12*x + 8*exp(4)) + 10*exp(-3*x**2 + 6*x + 4*exp(4))

________________________________________________________________________________________