3.89.57 $\int (14+e^{-2x+x^2}(-1+2x-2x^2))dx$

Optimal. Leaf size=13 $$(14-e^{(-2+x)x})x$$

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Rubi [B]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 2.23, number of steps used = 2, number of rules used = 1, integrand size = 22, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.045, Rules used = {2288} $$\begin{array}{c}14x-\frac{e^{x^2-2x}(x-x^2)}{1-x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[14 + E^(-2*x + x^2)*(-1 + 2*x - 2*x^2),x]

[Out]

14*x - (E^(-2*x + x^2)*(x - x^2))/(1 - x)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =14x+\int e^{-2x+x^2}(-1+2x-2x^2)dx\\ & =14x-\frac{e^{-2x+x^2}(x-x^2)}{1-x}\end{array}\end{array}$$

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Mathematica [A]  time = 0.02, size = 12, normalized size = 0.92 $$\begin{array}{c}-\left((-14+e^{(-2+x)x})x\right)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[14 + E^(-2*x + x^2)*(-1 + 2*x - 2*x^2),x]

[Out]

-((-14 + E^((-2 + x)*x))*x)

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fricas [A]  time = 0.53, size = 15, normalized size = 1.15 $$\begin{array}{c}-x{e}^{(x^2-2x)}+14x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+2*x-1)*exp(x^2-2*x)+14,x, algorithm="fricas")

[Out]

-x*e^(x^2 - 2*x) + 14*x

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giac [A]  time = 0.13, size = 15, normalized size = 1.15 $$\begin{array}{c}-x{e}^{(x^2-2x)}+14x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+2*x-1)*exp(x^2-2*x)+14,x, algorithm="giac")

[Out]

-x*e^(x^2 - 2*x) + 14*x

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maple [A]  time = 0.04, size = 14, normalized size = 1.08

method	result	size
risch	$14x-x{\mathrm{e}}^{(x-2)x}$	$14$
default	$14x-x{\mathrm{e}}^{x^2-2x}$	$16$
norman	$14x-x{\mathrm{e}}^{x^2-2x}$	$16$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2+2*x-1)*exp(x^2-2*x)+14,x,method=_RETURNVERBOSE)

[Out]

14*x-x*exp((x-2)*x)

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maxima [C]  time = 0.42, size = 120, normalized size = 9.23 $$\begin{array}{c}\frac{1}{2}i\sqrt{\pi }\mathrm{erf}(ix-i)e^{(-1)}+(\frac{{(x-1)}^3\mathrm{\Gamma }(\frac{3}{2},-{(x-1)}^2)}{{(-{(x-1)}^2)}^{\frac{3}{2}}}-\frac{\sqrt{\pi }(x-1)(\mathrm{erf}\left(\sqrt{-{(x-1)}^2}\right)-1)}{\sqrt{-{(x-1)}^2}}-2e^{\left({(x-1)}^2\right)})e^{(-1)}+(\frac{\sqrt{\pi }(x-1)(\mathrm{erf}\left(\sqrt{-{(x-1)}^2}\right)-1)}{\sqrt{-{(x-1)}^2}}+e^{\left({(x-1)}^2\right)})e^{(-1)}+14x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+2*x-1)*exp(x^2-2*x)+14,x, algorithm="maxima")

[Out]

1/2*I*sqrt(pi)*erf(I*x - I)*e^(-1) + ((x - 1)^3*gamma(3/2, -(x - 1)^2)/(-(x - 1)^2)^(3/2) - sqrt(pi)*(x - 1)*(
erf(sqrt(-(x - 1)^2)) - 1)/sqrt(-(x - 1)^2) - 2*e^((x - 1)^2))*e^(-1) + (sqrt(pi)*(x - 1)*(erf(sqrt(-(x - 1)^2
)) - 1)/sqrt(-(x - 1)^2) + e^((x - 1)^2))*e^(-1) + 14*x

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mupad [B]  time = 5.08, size = 13, normalized size = 1.00 $$\begin{array}{c}-x({\mathrm{e}}^{x^2-2x}-14)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(14 - exp(x^2 - 2*x)*(2*x^2 - 2*x + 1),x)

[Out]

-x*(exp(x^2 - 2*x) - 14)

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sympy [A]  time = 0.09, size = 12, normalized size = 0.92 $$\begin{array}{c}-x{e}^{x^2-2x}+14x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2+2*x-1)*exp(x**2-2*x)+14,x)

[Out]

-x*exp(x**2 - 2*x) + 14*x

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