\(\int \frac {1}{3} (-e^2-4 x) \, dx\) [6546]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{3} \left (-e^2-4 x\right ) \, dx=\frac {19}{3}-\frac {1}{3} x \left (e^2+2 x\right ) \]

[Out]

19/3-1/3*(exp(2)+2*x)*x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {9} \[ \int \frac {1}{3} \left (-e^2-4 x\right ) \, dx=-\frac {1}{24} \left (4 x+e^2\right )^2 \]

[In]

Int[(-E^2 - 4*x)/3,x]

[Out]

-1/24*(E^2 + 4*x)^2

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[a*((b + c*x)^2/(2*c)), x] /; FreeQ[{a, b, c}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{24} \left (e^2+4 x\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{3} \left (-e^2-4 x\right ) \, dx=\frac {1}{3} \left (-e^2 x-2 x^2\right ) \]

[In]

Integrate[(-E^2 - 4*x)/3,x]

[Out]

(-(E^2*x) - 2*x^2)/3

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62

method	result	size
gosper	\(-\frac {\left ({\mathrm e}^{2}+2 x \right ) x}{3}\)	\(10\)
default	\(-\frac {{\mathrm e}^{2} x}{3}-\frac {2 x^{2}}{3}\)	\(12\)
norman	\(-\frac {{\mathrm e}^{2} x}{3}-\frac {2 x^{2}}{3}\)	\(12\)
risch	\(-\frac {{\mathrm e}^{2} x}{3}-\frac {2 x^{2}}{3}\)	\(12\)
parallelrisch	\(-\frac {{\mathrm e}^{2} x}{3}-\frac {2 x^{2}}{3}\)	\(12\)
parts	\(-\frac {{\mathrm e}^{2} x}{3}-\frac {2 x^{2}}{3}\)	\(12\)

[In]

int(-1/3*exp(2)-4/3*x,x,method=_RETURNVERBOSE)

[Out]

-1/3*(exp(2)+2*x)*x

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int \frac {1}{3} \left (-e^2-4 x\right ) \, dx=-\frac {2}{3} \, x^{2} - \frac {1}{3} \, x e^{2} \]

[In]

integrate(-1/3*exp(2)-4/3*x,x, algorithm="fricas")

[Out]

-2/3*x^2 - 1/3*x*e^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{3} \left (-e^2-4 x\right ) \, dx=- \frac {2 x^{2}}{3} - \frac {x e^{2}}{3} \]

[In]

integrate(-1/3*exp(2)-4/3*x,x)

[Out]

-2*x**2/3 - x*exp(2)/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int \frac {1}{3} \left (-e^2-4 x\right ) \, dx=-\frac {2}{3} \, x^{2} - \frac {1}{3} \, x e^{2} \]

[In]

integrate(-1/3*exp(2)-4/3*x,x, algorithm="maxima")

[Out]

-2/3*x^2 - 1/3*x*e^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int \frac {1}{3} \left (-e^2-4 x\right ) \, dx=-\frac {2}{3} \, x^{2} - \frac {1}{3} \, x e^{2} \]

[In]

integrate(-1/3*exp(2)-4/3*x,x, algorithm="giac")

[Out]

-2/3*x^2 - 1/3*x*e^2

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {1}{3} \left (-e^2-4 x\right ) \, dx=-\frac {x\,\left (2\,x+{\mathrm {e}}^2\right )}{3} \]

[In]

int(- (4*x)/3 - exp(2)/3,x)

[Out]

-(x*(2*x + exp(2)))/3