∫ e4e2x+x2 (16+32x2−2x3−2e5x3+e2x(128x−8x2−8e5x2))_______________________________________

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Rubi [B] time = 0.33, antiderivative size = 95, normalized size of antiderivative = 3.39, number of steps used = 3, number of rules used = 2, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6, 2288} \begin {gather*} \frac {e^{x^2+4 e^{2 x}} \left (-\left (\left (1+e^5\right ) x^3\right )+16 x^2+4 e^{2 x} \left (-e^5 x^2-x^2+16 x\right )\right )}{\left (x+4 e^{2 x}\right ) \left (\left (1+e^{10}\right ) x^2-2 e^5 \left (16 x-x^2\right )-32 x+256\right )} \end {gather*}

Int[(E^(4*E^(2*x) + x^2)*(16 + 32*x^2 - 2*x^3 - 2*E^5*x^3 + E^(2*x)*(128*x - 8*x^2 - 8*E^5*x^2)))/(256 - 32*x

(E^(4*E^(2*x) + x^2)*(16*x^2 - (1 + E^5)*x^3 + 4*E^(2*x)*(16*x - x^2 - E^5*x^2)))/((4*E^(2*x) + x)*(256 - 32*x

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x

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,

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{4 e^{2 x}+x^2} \left (16+32 x^2-2 x^3-2 e^5 x^3+e^{2 x} \left (128 x-8 x^2-8 e^5 x^2\right )\right )}{256-32 x+\left (1+e^{10}\right ) x^2+e^5 \left (-32 x+2 x^2\right )} \, dx\\ &=\int \frac {e^{4 e^{2 x}+x^2} \left (16+32 x^2+\left (-2-2 e^5\right ) x^3+e^{2 x} \left (128 x-8 x^2-8 e^5 x^2\right )\right )}{256-32 x+\left (1+e^{10}\right ) x^2+e^5 \left (-32 x+2 x^2\right )} \, dx\\ &=\frac {e^{4 e^{2 x}+x^2} \left (16 x^2-\left (1+e^5\right ) x^3+4 e^{2 x} \left (16 x-x^2-e^5 x^2\right )\right )}{\left (4 e^{2 x}+x\right ) \left (256-32 x+\left (1+e^{10}\right ) x^2-2 e^5 \left (16 x-x^2\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.53, size = 26, normalized size = 0.93 \begin {gather*} -\frac {e^{4 e^{2 x}+x^2} x}{-16+x+e^5 x} \end {gather*}

Integrate[(E^(4*E^(2*x) + x^2)*(16 + 32*x^2 - 2*x^3 - 2*E^5*x^3 + E^(2*x)*(128*x - 8*x^2 - 8*E^5*x^2)))/(256 -

32*x + x^2 + E^10*x^2 + E^5*(-32*x + 2*x^2)),x]

-((E^(4*E^(2*x) + x^2)*x)/(-16 + x + E^5*x))

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fricas [A] time = 0.55, size = 23, normalized size = 0.82 \begin {gather*} -\frac {x e^{\left (x^{2} + 4 \, e^{\left (2 \, x\right )}\right )}}{x e^{5} + x - 16} \end {gather*}

integrate(((-8*x^2*exp(5)-8*x^2+128*x)*exp(x)^2-2*x^3*exp(5)-2*x^3+32*x^2+16)*exp(4*exp(x)^2+x^2)/(x^2*exp(5)^

2+(2*x^2-32*x)*exp(5)+x^2-32*x+256),x, algorithm="fricas")

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (x^{3} e^{5} + x^{3} - 16 \, x^{2} + 4 \, {\left (x^{2} e^{5} + x^{2} - 16 \, x\right )} e^{\left (2 \, x\right )} - 8\right )} e^{\left (x^{2} + 4 \, e^{\left (2 \, x\right )}\right )}}{x^{2} e^{10} + x^{2} + 2 \, {\left (x^{2} - 16 \, x\right )} e^{5} - 32 \, x + 256}\,{d x} \end {gather*}

integrate(((-8*x^2*exp(5)-8*x^2+128*x)*exp(x)^2-2*x^3*exp(5)-2*x^3+32*x^2+16)*exp(4*exp(x)^2+x^2)/(x^2*exp(5)^

2+(2*x^2-32*x)*exp(5)+x^2-32*x+256),x, algorithm="giac")

integrate(-2*(x^3*e^5 + x^3 - 16*x^2 + 4*(x^2*e^5 + x^2 - 16*x)*e^(2*x) - 8)*e^(x^2 + 4*e^(2*x))/(x^2*e^10 + x

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method	result	size

norman	\(-\frac {x \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}+x^{2}}}{x \,{\mathrm e}^{5}+x -16}\)	\(24\)
risch	\(-\frac {x \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}+x^{2}}}{x \,{\mathrm e}^{5}+x -16}\)	\(24\)

int(((-8*x^2*exp(5)-8*x^2+128*x)*exp(x)^2-2*x^3*exp(5)-2*x^3+32*x^2+16)*exp(4*exp(x)^2+x^2)/(x^2*exp(5)^2+(2*x

^2-32*x)*exp(5)+x^2-32*x+256),x,method=_RETURNVERBOSE)

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maxima [A] time = 0.43, size = 24, normalized size = 0.86 \begin {gather*} -\frac {x e^{\left (x^{2} + 4 \, e^{\left (2 \, x\right )}\right )}}{x {\left (e^{5} + 1\right )} - 16} \end {gather*}

integrate(((-8*x^2*exp(5)-8*x^2+128*x)*exp(x)^2-2*x^3*exp(5)-2*x^3+32*x^2+16)*exp(4*exp(x)^2+x^2)/(x^2*exp(5)^

2+(2*x^2-32*x)*exp(5)+x^2-32*x+256),x, algorithm="maxima")

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mupad [B] time = 7.99, size = 32, normalized size = 1.14 \begin {gather*} -\frac {x\,{\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}+x^2}}{\left (x-\frac {16}{{\mathrm {e}}^5+1}\right )\,\left ({\mathrm {e}}^5+1\right )} \end {gather*}

int(-(exp(4*exp(2*x) + x^2)*(2*x^3*exp(5) + exp(2*x)*(8*x^2*exp(5) - 128*x + 8*x^2) - 32*x^2 + 2*x^3 - 16))/(x

^2*exp(10) - exp(5)*(32*x - 2*x^2) - 32*x + x^2 + 256),x)

-(x*exp(4*exp(2*x) + x^2))/((x - 16/(exp(5) + 1))*(exp(5) + 1))

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sympy [A] time = 0.27, size = 22, normalized size = 0.79 \begin {gather*} - \frac {x e^{x^{2} + 4 e^{2 x}}}{x + x e^{5} - 16} \end {gather*}

integrate(((-8*x**2*exp(5)-8*x**2+128*x)*exp(x)**2-2*x**3*exp(5)-2*x**3+32*x**2+16)*exp(4*exp(x)**2+x**2)/(x**

2*exp(5)**2+(2*x**2-32*x)*exp(5)+x**2-32*x+256),x)

-x*exp(x**2 + 4*exp(2*x))/(x + x*exp(5) - 16)

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3.95.67 \(\int \frac {e^{4 e^{2 x}+x^2} (16+32 x^2-2 x^3-2 e^5 x^3+e^{2 x} (128 x-8 x^2-8 e^5 x^2))}{256-32 x+x^2+e^{10} x^2+e^5 (-32 x+2 x^2)} \, dx\)