\(\int (-22+e^x-2 x+18 x^2-4 x^3) \, dx\) [7714]

Optimal result

Integrand size = 18, antiderivative size = 21 \[ \int \left (-22+e^x-2 x+18 x^2-4 x^3\right ) \, dx=e^x-\left (-5+(-1+x)^2-x\right )^2+2 x \]

[Out]

2*x+exp(x)-((-1+x)^2-x-5)^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2225} \[ \int \left (-22+e^x-2 x+18 x^2-4 x^3\right ) \, dx=-x^4+6 x^3-x^2-22 x+e^x \]

[In]

Int[-22 + E^x - 2*x + 18*x^2 - 4*x^3,x]

[Out]

E^x - 22*x - x^2 + 6*x^3 - x^4

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = -22 x-x^2+6 x^3-x^4+\int e^x \, dx \\ & = e^x-22 x-x^2+6 x^3-x^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \left (-22+e^x-2 x+18 x^2-4 x^3\right ) \, dx=e^x-22 x-x^2+6 x^3-x^4 \]

[In]

Integrate[-22 + E^x - 2*x + 18*x^2 - 4*x^3,x]

[Out]

E^x - 22*x - x^2 + 6*x^3 - x^4

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05

method	result	size
default	\(-x^{4}+6 x^{3}-x^{2}-22 x +{\mathrm e}^{x}\)	\(22\)
norman	\(-x^{4}+6 x^{3}-x^{2}-22 x +{\mathrm e}^{x}\)	\(22\)
risch	\(-x^{4}+6 x^{3}-x^{2}-22 x +{\mathrm e}^{x}\)	\(22\)
parallelrisch	\(-x^{4}+6 x^{3}-x^{2}-22 x +{\mathrm e}^{x}\)	\(22\)
parts	\(-x^{4}+6 x^{3}-x^{2}-22 x +{\mathrm e}^{x}\)	\(22\)

[In]

int(exp(x)-4*x^3+18*x^2-2*x-22,x,method=_RETURNVERBOSE)

[Out]

-x^4+6*x^3-x^2-22*x+exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (-22+e^x-2 x+18 x^2-4 x^3\right ) \, dx=-x^{4} + 6 \, x^{3} - x^{2} - 22 \, x + e^{x} \]

[In]

integrate(exp(x)-4*x^3+18*x^2-2*x-22,x, algorithm="fricas")

[Out]

-x^4 + 6*x^3 - x^2 - 22*x + e^x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \left (-22+e^x-2 x+18 x^2-4 x^3\right ) \, dx=- x^{4} + 6 x^{3} - x^{2} - 22 x + e^{x} \]

[In]

integrate(exp(x)-4*x**3+18*x**2-2*x-22,x)

[Out]

-x**4 + 6*x**3 - x**2 - 22*x + exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (-22+e^x-2 x+18 x^2-4 x^3\right ) \, dx=-x^{4} + 6 \, x^{3} - x^{2} - 22 \, x + e^{x} \]

[In]

integrate(exp(x)-4*x^3+18*x^2-2*x-22,x, algorithm="maxima")

[Out]

-x^4 + 6*x^3 - x^2 - 22*x + e^x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (-22+e^x-2 x+18 x^2-4 x^3\right ) \, dx=-x^{4} + 6 \, x^{3} - x^{2} - 22 \, x + e^{x} \]

[In]

integrate(exp(x)-4*x^3+18*x^2-2*x-22,x, algorithm="giac")

[Out]

-x^4 + 6*x^3 - x^2 - 22*x + e^x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (-22+e^x-2 x+18 x^2-4 x^3\right ) \, dx={\mathrm {e}}^x-22\,x-x^2+6\,x^3-x^4 \]

[In]

int(exp(x) - 2*x + 18*x^2 - 4*x^3 - 22,x)

[Out]

exp(x) - 22*x - x^2 + 6*x^3 - x^4