3.24.43 \(\int \frac {1}{2} e^{-23-10 e^3-e^6} (1+e^{23+10 e^3+e^6} (-5+x)) \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{4} \left (-5+e^{2-\left (5+e^3\right )^2}+x\right )^2 \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.41, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {12} \begin {gather*} \frac {1}{4} (5-x)^2+\frac {1}{2} e^{-23-10 e^3-e^6} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-23 - 10*E^3 - E^6)*(1 + E^(23 + 10*E^3 + E^6)*(-5 + x)))/2,x]

[Out]

(5 - x)^2/4 + (E^(-23 - 10*E^3 - E^6)*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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} e^{-23-10 e^3-e^6} \int \left (1+e^{23+10 e^3+e^6} (-5+x)\right ) \, dx\\ &=\frac {1}{4} (5-x)^2+\frac {1}{2} e^{-23-10 e^3-e^6} x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 31, normalized size = 1.41 \begin {gather*} \frac {1}{2} \left (-5 x+e^{-23-10 e^3-e^6} x+\frac {x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-23 - 10*E^3 - E^6)*(1 + E^(23 + 10*E^3 + E^6)*(-5 + x)))/2,x]

[Out]

(-5*x + E^(-23 - 10*E^3 - E^6)*x + x^2/2)/2

________________________________________________________________________________________

fricas [A]  time = 0.92, size = 34, normalized size = 1.55 \begin {gather*} \frac {1}{4} \, {\left ({\left (x^{2} - 10 \, x\right )} e^{\left (e^{6} + 10 \, e^{3} + 23\right )} + 2 \, x\right )} e^{\left (-e^{6} - 10 \, e^{3} - 23\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((x-5)*exp(exp(3)^2+10*exp(3)+23)+1)/exp(exp(3)^2+10*exp(3)+23),x, algorithm="fricas")

[Out]

1/4*((x^2 - 10*x)*e^(e^6 + 10*e^3 + 23) + 2*x)*e^(-e^6 - 10*e^3 - 23)

________________________________________________________________________________________

giac [A]  time = 0.24, size = 34, normalized size = 1.55 \begin {gather*} \frac {1}{4} \, {\left ({\left (x^{2} - 10 \, x\right )} e^{\left (e^{6} + 10 \, e^{3} + 23\right )} + 2 \, x\right )} e^{\left (-e^{6} - 10 \, e^{3} - 23\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((x-5)*exp(exp(3)^2+10*exp(3)+23)+1)/exp(exp(3)^2+10*exp(3)+23),x, algorithm="giac")

[Out]

1/4*((x^2 - 10*x)*e^(e^6 + 10*e^3 + 23) + 2*x)*e^(-e^6 - 10*e^3 - 23)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 24, normalized size = 1.09

method	result	size
risch	\(\frac {x^{2}}{4}-\frac {5 x}{2}+\frac {x \,{\mathrm e}^{-{\mathrm e}^{6}-10 \,{\mathrm e}^{3}-23}}{2}\)	\(24\)
default	\(\frac {{\mathrm e}^{-{\mathrm e}^{6}-10 \,{\mathrm e}^{3}-23} \left ({\mathrm e}^{{\mathrm e}^{6}+10 \,{\mathrm e}^{3}+23} \left (\frac {1}{2} x^{2}-5 x \right )+x \right )}{2}\)	\(39\)
norman	\(\frac {x^{2}}{4}-\frac {{\mathrm e}^{-{\mathrm e}^{6}} {\mathrm e}^{-10 \,{\mathrm e}^{3}} {\mathrm e}^{-23} \left (5 \,{\mathrm e}^{{\mathrm e}^{6}} {\mathrm e}^{10 \,{\mathrm e}^{3}} {\mathrm e}^{23}-1\right ) x}{2}\)	\(42\)
gosper	\(\frac {x \left ({\mathrm e}^{{\mathrm e}^{6}+10 \,{\mathrm e}^{3}+23} x -10 \,{\mathrm e}^{{\mathrm e}^{6}+10 \,{\mathrm e}^{3}+23}+2\right ) {\mathrm e}^{-{\mathrm e}^{6}-10 \,{\mathrm e}^{3}-23}}{4}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((x-5)*exp(exp(3)^2+10*exp(3)+23)+1)/exp(exp(3)^2+10*exp(3)+23),x,method=_RETURNVERBOSE)

[Out]

1/4*x^2-5/2*x+1/2*x*exp(-exp(6)-10*exp(3)-23)

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 34, normalized size = 1.55 \begin {gather*} \frac {1}{4} \, {\left ({\left (x^{2} - 10 \, x\right )} e^{\left (e^{6} + 10 \, e^{3} + 23\right )} + 2 \, x\right )} e^{\left (-e^{6} - 10 \, e^{3} - 23\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((x-5)*exp(exp(3)^2+10*exp(3)+23)+1)/exp(exp(3)^2+10*exp(3)+23),x, algorithm="maxima")

[Out]

1/4*((x^2 - 10*x)*e^(e^6 + 10*e^3 + 23) + 2*x)*e^(-e^6 - 10*e^3 - 23)

________________________________________________________________________________________

mupad [B]  time = 0.24, size = 30, normalized size = 1.36 \begin {gather*} \frac {{\mathrm {e}}^{-20\,{\mathrm {e}}^3-2\,{\mathrm {e}}^6-46}\,{\left ({\mathrm {e}}^{10\,{\mathrm {e}}^3+{\mathrm {e}}^6+23}\,\left (x-5\right )+1\right )}^2}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(- 10*exp(3) - exp(6) - 23)*((exp(10*exp(3) + exp(6) + 23)*(x - 5))/2 + 1/2),x)

[Out]

(exp(- 20*exp(3) - 2*exp(6) - 46)*(exp(10*exp(3) + exp(6) + 23)*(x - 5) + 1)^2)/4

________________________________________________________________________________________

sympy [B]  time = 0.07, size = 41, normalized size = 1.86 \begin {gather*} \frac {x^{2}}{4} + \frac {x \left (- 5 e^{23} e^{10 e^{3}} e^{e^{6}} + 1\right )}{2 e^{23} e^{10 e^{3}} e^{e^{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((x-5)*exp(exp(3)**2+10*exp(3)+23)+1)/exp(exp(3)**2+10*exp(3)+23),x)

[Out]

x**2/4 + x*(-5*exp(23)*exp(10*exp(3))*exp(exp(6)) + 1)*exp(-23)*exp(-10*exp(3))*exp(-exp(6))/2

________________________________________________________________________________________