\(\int \frac {1}{2} (-1+e^2-e^{25}+2 x-9 x^2) \, dx\) [3851]

Optimal result

Integrand size = 22, antiderivative size = 21 \[ \int \frac {1}{2} \left (-1+e^2-e^{25}+2 x-9 x^2\right ) \, dx=\frac {1}{2} x \left (-1+e^2-e^{25}+x-3 x^2\right ) \]

[Out]

1/2*(x-1+exp(2)-3*x^2-exp(25))*x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {12} \[ \int \frac {1}{2} \left (-1+e^2-e^{25}+2 x-9 x^2\right ) \, dx=-\frac {3 x^3}{2}+\frac {x^2}{2}-\frac {1}{2} \left (1-e^2+e^{25}\right ) x \]

[In]

Int[(-1 + E^2 - E^25 + 2*x - 9*x^2)/2,x]

[Out]

-1/2*((1 - E^2 + E^25)*x) + x^2/2 - (3*x^3)/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{align*} \text {integral}& = \frac {1}{2} \int \left (-1+e^2-e^{25}+2 x-9 x^2\right ) \, dx \\ & = -\frac {1}{2} \left (1-e^2+e^{25}\right ) x+\frac {x^2}{2}-\frac {3 x^3}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {1}{2} \left (-1+e^2-e^{25}+2 x-9 x^2\right ) \, dx=\frac {1}{2} \left (-x+e^2 x-e^{25} x+x^2-3 x^3\right ) \]

[In]

Integrate[(-1 + E^2 - E^25 + 2*x - 9*x^2)/2,x]

[Out]

(-x + E^2*x - E^25*x + x^2 - 3*x^3)/2

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {\left (x -1+{\mathrm e}^{2}-3 x^{2}-{\mathrm e}^{25}\right ) x}{2}\)	\(18\)
norman	\(\left (\frac {{\mathrm e}^{2}}{2}-\frac {{\mathrm e}^{25}}{2}-\frac {1}{2}\right ) x +\frac {x^{2}}{2}-\frac {3 x^{3}}{2}\)	\(24\)
parallelrisch	\(\left (\frac {{\mathrm e}^{2}}{2}-\frac {{\mathrm e}^{25}}{2}-\frac {1}{2}\right ) x +\frac {x^{2}}{2}-\frac {3 x^{3}}{2}\)	\(24\)
default	\(-\frac {3 x^{3}}{2}+\frac {{\mathrm e}^{2} x}{2}-\frac {x \,{\mathrm e}^{25}}{2}+\frac {x^{2}}{2}-\frac {x}{2}\)	\(25\)
risch	\(-\frac {3 x^{3}}{2}+\frac {{\mathrm e}^{2} x}{2}-\frac {x \,{\mathrm e}^{25}}{2}+\frac {x^{2}}{2}-\frac {x}{2}\)	\(25\)
parts	\(-\frac {3 x^{3}}{2}+\frac {{\mathrm e}^{2} x}{2}-\frac {x \,{\mathrm e}^{25}}{2}+\frac {x^{2}}{2}-\frac {x}{2}\)	\(25\)

[In]

int(-1/2*exp(25)+1/2*exp(2)-9/2*x^2+x-1/2,x,method=_RETURNVERBOSE)

[Out]

1/2*(x-1+exp(2)-3*x^2-exp(25))*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {1}{2} \left (-1+e^2-e^{25}+2 x-9 x^2\right ) \, dx=-\frac {3}{2} \, x^{3} + \frac {1}{2} \, x^{2} - \frac {1}{2} \, x e^{25} + \frac {1}{2} \, x e^{2} - \frac {1}{2} \, x \]

[In]

integrate(-1/2*exp(25)+1/2*exp(2)-9/2*x^2+x-1/2,x, algorithm="fricas")

[Out]

-3/2*x^3 + 1/2*x^2 - 1/2*x*e^25 + 1/2*x*e^2 - 1/2*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {1}{2} \left (-1+e^2-e^{25}+2 x-9 x^2\right ) \, dx=- \frac {3 x^{3}}{2} + \frac {x^{2}}{2} + x \left (- \frac {e^{25}}{2} - \frac {1}{2} + \frac {e^{2}}{2}\right ) \]

[In]

integrate(-1/2*exp(25)+1/2*exp(2)-9/2*x**2+x-1/2,x)

[Out]

-3*x**3/2 + x**2/2 + x*(-exp(25)/2 - 1/2 + exp(2)/2)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {1}{2} \left (-1+e^2-e^{25}+2 x-9 x^2\right ) \, dx=-\frac {3}{2} \, x^{3} + \frac {1}{2} \, x^{2} - \frac {1}{2} \, x e^{25} + \frac {1}{2} \, x e^{2} - \frac {1}{2} \, x \]

[In]

integrate(-1/2*exp(25)+1/2*exp(2)-9/2*x^2+x-1/2,x, algorithm="maxima")

[Out]

-3/2*x^3 + 1/2*x^2 - 1/2*x*e^25 + 1/2*x*e^2 - 1/2*x

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {1}{2} \left (-1+e^2-e^{25}+2 x-9 x^2\right ) \, dx=-\frac {3}{2} \, x^{3} + \frac {1}{2} \, x^{2} - \frac {1}{2} \, x e^{25} + \frac {1}{2} \, x e^{2} - \frac {1}{2} \, x \]

[In]

integrate(-1/2*exp(25)+1/2*exp(2)-9/2*x^2+x-1/2,x, algorithm="giac")

[Out]

-3/2*x^3 + 1/2*x^2 - 1/2*x*e^25 + 1/2*x*e^2 - 1/2*x

Mupad [B] (verification not implemented)

Time = 9.41 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{2} \left (-1+e^2-e^{25}+2 x-9 x^2\right ) \, dx=-\frac {x\,\left (3\,x^2-x-{\mathrm {e}}^2+{\mathrm {e}}^{25}+1\right )}{2} \]

[In]

int(x + exp(2)/2 - exp(25)/2 - (9*x^2)/2 - 1/2,x)

[Out]

-(x*(exp(25) - exp(2) - x + 3*x^2 + 1))/2