\(\int \frac {e^3 (-64 x^2-16 x^3-x^4)+e^{\frac {-96 x+12 x^2+3 x^3+e^3 (96-38 x^2+4 x^3+x^4)}{e^3 (8 x+x^2)}} (192 x^2+48 x^3+3 x^4+e^3 (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5))}{e^3 (64 x^2+16 x^3+x^4)} \, dx\) [432]

Optimal result

Integrand size = 135, antiderivative size = 32 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=e^{(-4+x) \left (x-\frac {3 \left (1-\frac {x}{e^3}+\frac {x}{8+x}\right )}{x}\right )}-x \]

[Out]

exp((x-3*(x/(x+8)+1-x/exp(3))/x)*(x-4))-x

Rubi [F]

\[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=\int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+\exp \left (\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}\right ) \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx \]

[In]

Int[(E^3*(-64*x^2 - 16*x^3 - x^4) + E^((-96*x + 12*x^2 + 3*x^3 + E^3*(96 - 38*x^2 + 4*x^3 + x^4))/(E^3*(8*x +
x^2)))*(192*x^2 + 48*x^3 + 3*x^4 + E^3*(-768 - 192*x - 304*x^2 + 64*x^3 + 28*x^4 + 2*x^5)))/(E^3*(64*x^2 + 16*
x^3 + x^4)),x]

[Out]

-x + ((3 - 4*E^3)*Defer[Int][E^(((-4 + x)*(-24*E^3 + 6*(4 - E^3)*x + (3 + 8*E^3)*x^2 + E^3*x^3))/(E^3*x*(8 + x
))), x])/E^3 - (12*Defer[Int][E^(3 + ((-4 + x)*(-24*E^3 + 6*(4 - E^3)*x + (3 + 8*E^3)*x^2 + E^3*x^3))/(E^3*x*(
8 + x)))/x^2, x])/E^3 + (2*Defer[Int][E^(3 + ((-4 + x)*(-24*E^3 + 6*(4 - E^3)*x + (3 + 8*E^3)*x^2 + E^3*x^3))/
(E^3*x*(8 + x)))*x, x])/E^3 - (36*Defer[Int][E^(3 + ((-4 + x)*(-24*E^3 + 6*(4 - E^3)*x + (3 + 8*E^3)*x^2 + E^3
*x^3))/(E^3*x*(8 + x)))/(8 + x)^2, x])/E^3

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+\exp \left (\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}\right ) \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{64 x^2+16 x^3+x^4} \, dx}{e^3} \\ & = \frac {\int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+\exp \left (\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}\right ) \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{x^2 \left (64+16 x+x^2\right )} \, dx}{e^3} \\ & = \frac {\int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+\exp \left (\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}\right ) \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{x^2 (8+x)^2} \, dx}{e^3} \\ & = \frac {\int \left (-e^3+\frac {\exp \left (\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right ) \left (-768 e^3-192 e^3 x+192 \left (1-\frac {19 e^3}{12}\right ) x^2+48 \left (1+\frac {4 e^3}{3}\right ) x^3+3 \left (1+\frac {28 e^3}{3}\right ) x^4+2 e^3 x^5\right )}{x^2 (8+x)^2}\right ) \, dx}{e^3} \\ & = -x+\frac {\int \frac {\exp \left (\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right ) \left (-768 e^3-192 e^3 x+192 \left (1-\frac {19 e^3}{12}\right ) x^2+48 \left (1+\frac {4 e^3}{3}\right ) x^3+3 \left (1+\frac {28 e^3}{3}\right ) x^4+2 e^3 x^5\right )}{x^2 (8+x)^2} \, dx}{e^3} \\ & = -x+\frac {\int \frac {\exp \left (\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right ) \left (3 x^2 (8+x)^2+2 e^3 \left (-384-96 x-152 x^2+32 x^3+14 x^4+x^5\right )\right )}{x^2 (8+x)^2} \, dx}{e^3} \\ & = -x+\frac {\int \left (3 \exp \left (\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right ) \left (1-\frac {4 e^3}{3}\right )-\frac {12 \exp \left (3+\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right )}{x^2}+2 \exp \left (3+\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right ) x-\frac {36 \exp \left (3+\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right )}{(8+x)^2}\right ) \, dx}{e^3} \\ & = -x+\frac {2 \int \exp \left (3+\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right ) x \, dx}{e^3}-\frac {12 \int \frac {\exp \left (3+\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right )}{x^2} \, dx}{e^3}-\frac {36 \int \frac {\exp \left (3+\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right )}{(8+x)^2} \, dx}{e^3}+\frac {\left (3-4 e^3\right ) \int \exp \left (\frac {(-4+x) \left (-24 e^3+6 \left (4-e^3\right ) x+\left (3+8 e^3\right ) x^2+e^3 x^3\right )}{e^3 x (8+x)}\right ) \, dx}{e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=e^{3-\frac {3 \left (4+3 e^3\right )}{e^3}+\frac {12}{x}-\frac {\left (-3+4 e^3\right ) x}{e^3}+x^2+\frac {36}{8+x}}-x \]

[In]

Integrate[(E^3*(-64*x^2 - 16*x^3 - x^4) + E^((-96*x + 12*x^2 + 3*x^3 + E^3*(96 - 38*x^2 + 4*x^3 + x^4))/(E^3*(
8*x + x^2)))*(192*x^2 + 48*x^3 + 3*x^4 + E^3*(-768 - 192*x - 304*x^2 + 64*x^3 + 28*x^4 + 2*x^5)))/(E^3*(64*x^2
+ 16*x^3 + x^4)),x]

[Out]

E^(3 - (3*(4 + 3*E^3))/E^3 + 12/x - ((-3 + 4*E^3)*x)/E^3 + x^2 + 36/(8 + x)) - x

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59

method	result	size
risch	\(-x +{\mathrm e}^{\frac {\left (x -4\right ) \left (x^{3} {\mathrm e}^{3}+8 x^{2} {\mathrm e}^{3}-6 x \,{\mathrm e}^{3}+3 x^{2}-24 \,{\mathrm e}^{3}+24 x \right ) {\mathrm e}^{-3}}{x \left (x +8\right )}}\)	\(51\)
parallelrisch	\({\mathrm e}^{-3} \left (-x \,{\mathrm e}^{3}+{\mathrm e}^{3} {\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x \left (x +8\right )}}+32 \,{\mathrm e}^{3}\right )\)	\(65\)
parts	\(-x +\frac {x^{2} {\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x^{2}+8 x}}+8 x \,{\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x^{2}+8 x}}}{x \left (x +8\right )}\)	\(116\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x^{2}+8 x}}+64 x -x^{3}+8 x \,{\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x^{2}+8 x}}}{x \left (x +8\right )}\)	\(120\)

[In]

int((((2*x^5+28*x^4+64*x^3-304*x^2-192*x-768)*exp(3)+3*x^4+48*x^3+192*x^2)*exp(((x^4+4*x^3-38*x^2+96)*exp(3)+3
*x^3+12*x^2-96*x)/(x^2+8*x)/exp(3))+(-x^4-16*x^3-64*x^2)*exp(3))/(x^4+16*x^3+64*x^2)/exp(3),x,method=_RETURNVE
RBOSE)

[Out]

-x+exp((x-4)*(x^3*exp(3)+8*x^2*exp(3)-6*x*exp(3)+3*x^2-24*exp(3)+24*x)*exp(-3)/x/(x+8))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=-x + e^{\left (\frac {{\left (3 \, x^{3} + 12 \, x^{2} + {\left (x^{4} + 4 \, x^{3} - 38 \, x^{2} + 96\right )} e^{3} - 96 \, x\right )} e^{\left (-3\right )}}{x^{2} + 8 \, x}\right )} \]

[In]

integrate((((2*x^5+28*x^4+64*x^3-304*x^2-192*x-768)*exp(3)+3*x^4+48*x^3+192*x^2)*exp(((x^4+4*x^3-38*x^2+96)*ex
p(3)+3*x^3+12*x^2-96*x)/(x^2+8*x)/exp(3))+(-x^4-16*x^3-64*x^2)*exp(3))/(x^4+16*x^3+64*x^2)/exp(3),x, algorithm
="fricas")

[Out]

-x + e^((3*x^3 + 12*x^2 + (x^4 + 4*x^3 - 38*x^2 + 96)*e^3 - 96*x)*e^(-3)/(x^2 + 8*x))

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=- x + e^{\frac {3 x^{3} + 12 x^{2} - 96 x + \left (x^{4} + 4 x^{3} - 38 x^{2} + 96\right ) e^{3}}{\left (x^{2} + 8 x\right ) e^{3}}} \]

[In]

integrate((((2*x**5+28*x**4+64*x**3-304*x**2-192*x-768)*exp(3)+3*x**4+48*x**3+192*x**2)*exp(((x**4+4*x**3-38*x
**2+96)*exp(3)+3*x**3+12*x**2-96*x)/(x**2+8*x)/exp(3))+(-x**4-16*x**3-64*x**2)*exp(3))/(x**4+16*x**3+64*x**2)/
exp(3),x)

[Out]

-x + exp((3*x**3 + 12*x**2 - 96*x + (x**4 + 4*x**3 - 38*x**2 + 96)*exp(3))*exp(-3)/(x**2 + 8*x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (30) = 60\).

Time = 0.51 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.50 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=-{\left ({\left (x - \frac {64}{x + 8} - 16 \, \log \left (x + 8\right )\right )} e^{3} + 16 \, {\left (\frac {8}{x + 8} + \log \left (x + 8\right )\right )} e^{3} - \frac {64 \, e^{3}}{x + 8} - e^{\left (x^{2} + 3 \, x e^{\left (-3\right )} - 4 \, x + \frac {36}{x + 8} + \frac {12}{x} - 12 \, e^{\left (-3\right )} - 3\right )}\right )} e^{\left (-3\right )} \]

[In]

integrate((((2*x^5+28*x^4+64*x^3-304*x^2-192*x-768)*exp(3)+3*x^4+48*x^3+192*x^2)*exp(((x^4+4*x^3-38*x^2+96)*ex
p(3)+3*x^3+12*x^2-96*x)/(x^2+8*x)/exp(3))+(-x^4-16*x^3-64*x^2)*exp(3))/(x^4+16*x^3+64*x^2)/exp(3),x, algorithm
="maxima")

[Out]

-((x - 64/(x + 8) - 16*log(x + 8))*e^3 + 16*(8/(x + 8) + log(x + 8))*e^3 - 64*e^3/(x + 8) - e^(x^2 + 3*x*e^(-3
) - 4*x + 36/(x + 8) + 12/x - 12*e^(-3) - 3))*e^(-3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (30) = 60\).

Time = 0.91 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=-{\left (x e^{3} - e^{\left (\frac {x^{4} e^{3} + 4 \, x^{3} e^{3} + 3 \, x^{3} - 38 \, x^{2} e^{3} + 12 \, x^{2} - 96 \, x + 96 \, e^{3}}{x^{2} e^{3} + 8 \, x e^{3}} + 3\right )}\right )} e^{\left (-3\right )} \]

[In]

integrate((((2*x^5+28*x^4+64*x^3-304*x^2-192*x-768)*exp(3)+3*x^4+48*x^3+192*x^2)*exp(((x^4+4*x^3-38*x^2+96)*ex
p(3)+3*x^3+12*x^2-96*x)/(x^2+8*x)/exp(3))+(-x^4-16*x^3-64*x^2)*exp(3))/(x^4+16*x^3+64*x^2)/exp(3),x, algorithm
="giac")

[Out]

-(x*e^3 - e^((x^4*e^3 + 4*x^3*e^3 + 3*x^3 - 38*x^2*e^3 + 12*x^2 - 96*x + 96*e^3)/(x^2*e^3 + 8*x*e^3) + 3))*e^(
-3)

Mupad [B] (verification not implemented)

Time = 8.85 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.44 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx={\mathrm {e}}^{\frac {x^4}{x^2+8\,x}}\,{\mathrm {e}}^{\frac {4\,x^3}{x^2+8\,x}}\,{\mathrm {e}}^{-\frac {38\,x^2}{x^2+8\,x}}\,{\mathrm {e}}^{-\frac {96\,x\,{\mathrm {e}}^{-3}}{x^2+8\,x}}\,{\mathrm {e}}^{\frac {96}{x^2+8\,x}}\,{\mathrm {e}}^{\frac {3\,x^3\,{\mathrm {e}}^{-3}}{x^2+8\,x}}\,{\mathrm {e}}^{\frac {12\,x^2\,{\mathrm {e}}^{-3}}{x^2+8\,x}}-x \]

[In]

int(-(exp(-3)*(exp(3)*(64*x^2 + 16*x^3 + x^4) - exp((exp(-3)*(exp(3)*(4*x^3 - 38*x^2 + x^4 + 96) - 96*x + 12*x
^2 + 3*x^3))/(8*x + x^2))*(192*x^2 - exp(3)*(192*x + 304*x^2 - 64*x^3 - 28*x^4 - 2*x^5 + 768) + 48*x^3 + 3*x^4
)))/(64*x^2 + 16*x^3 + x^4),x)

[Out]

exp(x^4/(8*x + x^2))*exp((4*x^3)/(8*x + x^2))*exp(-(38*x^2)/(8*x + x^2))*exp(-(96*x*exp(-3))/(8*x + x^2))*exp(
96/(8*x + x^2))*exp((3*x^3*exp(-3))/(8*x + x^2))*exp((12*x^2*exp(-3))/(8*x + x^2)) - x