\(\int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} (68-68 x+17 x^2)+e^{-\frac {x}{-2+x}} (-128 x+160 x^2-32 x^3)+e^{-x^2+x^3} (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} (-32-128 x+320 x^2-224 x^3+48 x^4))}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} (64-64 x+16 x^2)+e^{-x^2+x^3} (-128 x^2+128 x^3-32 x^4)} \, dx\) [1400]

Optimal result

Integrand size = 195, antiderivative size = 36 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=-2+\frac {17 x}{16}+\frac {e^{\frac {x}{2-x}}}{-e^{(-1+x) x^2}+x^2} \]

[Out]

exp(x/(2-x))/(x^2-exp((-1+x)*x^2))-2+17/16*x

Rubi [F]

\[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx \]

[In]

Int[(68*x^4 - 68*x^5 + 17*x^6 + E^(-2*x^2 + 2*x^3)*(68 - 68*x + 17*x^2) + (-128*x + 160*x^2 - 32*x^3)/E^(x/(-2
+ x)) + E^(-x^2 + x^3)*(-136*x^2 + 136*x^3 - 34*x^4 + (-32 - 128*x + 320*x^2 - 224*x^3 + 48*x^4)/E^(x/(-2 + x
))))/(64*x^4 - 64*x^5 + 16*x^6 + E^(-2*x^2 + 2*x^3)*(64 - 64*x + 16*x^2) + E^(-x^2 + x^3)*(-128*x^2 + 128*x^3
- 32*x^4)),x]

[Out]

(17*x)/16 - 2*Defer[Int][(E^(-(x/(-2 + x)) + 2*x^2)*x)/(-E^x^3 + E^x^2*x^2)^2, x] - 2*Defer[Int][(E^(-(x/(-2 +
x)) + 2*x^2)*x^3)/(-E^x^3 + E^x^2*x^2)^2, x] + 3*Defer[Int][(E^(-(x/(-2 + x)) + 2*x^2)*x^4)/(-E^x^3 + E^x^2*x
^2)^2, x] + 2*Defer[Int][E^((x*(-1 - 2*x + x^2))/(-2 + x))/((-2 + x)^2*(-E^x^3 + E^x^2*x^2)), x] + 2*Defer[Int
][(E^((x*(-1 - 2*x + x^2))/(-2 + x))*x)/(-E^x^3 + E^x^2*x^2), x] - 3*Defer[Int][(E^((x*(-1 - 2*x + x^2))/(-2 +
x))*x^2)/(-E^x^3 + E^x^2*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2 x^2} \left (68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )\right )}{16 (2-x)^2 \left (e^{x^3}-e^{x^2} x^2\right )^2} \, dx \\ & = \frac {1}{16} \int \frac {e^{2 x^2} \left (68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )\right )}{(2-x)^2 \left (e^{x^3}-e^{x^2} x^2\right )^2} \, dx \\ & = \frac {1}{16} \int \left (17+\frac {16 e^{-\frac {x}{-2+x}+2 x^2} x \left (-2-2 x^2+3 x^3\right )}{\left (-e^{x^3}+e^{x^2} x^2\right )^2}+\frac {16 e^{-\frac {(-1+x)^2 x}{-2+x}+2 x^2} \left (-2-8 x+20 x^2-14 x^3+3 x^4\right )}{(2-x)^2 \left (e^{x^3}-e^{x^2} x^2\right )}\right ) \, dx \\ & = \frac {17 x}{16}+\int \frac {e^{-\frac {x}{-2+x}+2 x^2} x \left (-2-2 x^2+3 x^3\right )}{\left (-e^{x^3}+e^{x^2} x^2\right )^2} \, dx+\int \frac {e^{-\frac {(-1+x)^2 x}{-2+x}+2 x^2} \left (-2-8 x+20 x^2-14 x^3+3 x^4\right )}{(2-x)^2 \left (e^{x^3}-e^{x^2} x^2\right )} \, dx \\ & = \frac {17 x}{16}+\int \frac {e^{\frac {x \left (-1-2 x+x^2\right )}{-2+x}} \left (-2-8 x+20 x^2-14 x^3+3 x^4\right )}{(2-x)^2 \left (e^{x^3}-e^{x^2} x^2\right )} \, dx+\int \left (-\frac {2 e^{-\frac {x}{-2+x}+2 x^2} x}{\left (-e^{x^3}+e^{x^2} x^2\right )^2}-\frac {2 e^{-\frac {x}{-2+x}+2 x^2} x^3}{\left (-e^{x^3}+e^{x^2} x^2\right )^2}+\frac {3 e^{-\frac {x}{-2+x}+2 x^2} x^4}{\left (-e^{x^3}+e^{x^2} x^2\right )^2}\right ) \, dx \\ & = \frac {17 x}{16}-2 \int \frac {e^{-\frac {x}{-2+x}+2 x^2} x}{\left (-e^{x^3}+e^{x^2} x^2\right )^2} \, dx-2 \int \frac {e^{-\frac {x}{-2+x}+2 x^2} x^3}{\left (-e^{x^3}+e^{x^2} x^2\right )^2} \, dx+3 \int \frac {e^{-\frac {x}{-2+x}+2 x^2} x^4}{\left (-e^{x^3}+e^{x^2} x^2\right )^2} \, dx+\int \left (\frac {2 e^{\frac {x \left (-1-2 x+x^2\right )}{-2+x}}}{(-2+x)^2 \left (-e^{x^3}+e^{x^2} x^2\right )}+\frac {2 e^{\frac {x \left (-1-2 x+x^2\right )}{-2+x}} x}{-e^{x^3}+e^{x^2} x^2}-\frac {3 e^{\frac {x \left (-1-2 x+x^2\right )}{-2+x}} x^2}{-e^{x^3}+e^{x^2} x^2}\right ) \, dx \\ & = \frac {17 x}{16}-2 \int \frac {e^{-\frac {x}{-2+x}+2 x^2} x}{\left (-e^{x^3}+e^{x^2} x^2\right )^2} \, dx-2 \int \frac {e^{-\frac {x}{-2+x}+2 x^2} x^3}{\left (-e^{x^3}+e^{x^2} x^2\right )^2} \, dx+2 \int \frac {e^{\frac {x \left (-1-2 x+x^2\right )}{-2+x}}}{(-2+x)^2 \left (-e^{x^3}+e^{x^2} x^2\right )} \, dx+2 \int \frac {e^{\frac {x \left (-1-2 x+x^2\right )}{-2+x}} x}{-e^{x^3}+e^{x^2} x^2} \, dx+3 \int \frac {e^{-\frac {x}{-2+x}+2 x^2} x^4}{\left (-e^{x^3}+e^{x^2} x^2\right )^2} \, dx-3 \int \frac {e^{\frac {x \left (-1-2 x+x^2\right )}{-2+x}} x^2}{-e^{x^3}+e^{x^2} x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {1}{16} \left (17 x-\frac {16 e^{-1-\frac {2}{-2+x}+x^2}}{e^{x^3}-e^{x^2} x^2}\right ) \]

[In]

Integrate[(68*x^4 - 68*x^5 + 17*x^6 + E^(-2*x^2 + 2*x^3)*(68 - 68*x + 17*x^2) + (-128*x + 160*x^2 - 32*x^3)/E^
(x/(-2 + x)) + E^(-x^2 + x^3)*(-136*x^2 + 136*x^3 - 34*x^4 + (-32 - 128*x + 320*x^2 - 224*x^3 + 48*x^4)/E^(x/(
-2 + x))))/(64*x^4 - 64*x^5 + 16*x^6 + E^(-2*x^2 + 2*x^3)*(64 - 64*x + 16*x^2) + E^(-x^2 + x^3)*(-128*x^2 + 12
8*x^3 - 32*x^4)),x]

[Out]

(17*x - (16*E^(-1 - 2/(-2 + x) + x^2))/(E^x^3 - E^x^2*x^2))/16

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.72

method	result	size
parallelrisch	\(\frac {136 x^{3}+408 x^{2}-136 x \,{\mathrm e}^{\left (-1+x \right ) x^{2}}-408 \,{\mathrm e}^{\left (-1+x \right ) x^{2}}+128 \,{\mathrm e}^{-\frac {x}{-2+x}}}{128 x^{2}-128 \,{\mathrm e}^{\left (-1+x \right ) x^{2}}}\)	\(62\)

[In]

int(((17*x^2-68*x+68)*exp(x^3-x^2)^2+((48*x^4-224*x^3+320*x^2-128*x-32)*exp(-x/(-2+x))-34*x^4+136*x^3-136*x^2)
*exp(x^3-x^2)+(-32*x^3+160*x^2-128*x)*exp(-x/(-2+x))+17*x^6-68*x^5+68*x^4)/((16*x^2-64*x+64)*exp(x^3-x^2)^2+(-
32*x^4+128*x^3-128*x^2)*exp(x^3-x^2)+16*x^6-64*x^5+64*x^4),x,method=_RETURNVERBOSE)

[Out]

1/128*(136*x^3+408*x^2-136*x*exp((-1+x)*x^2)-408*exp((-1+x)*x^2)+128*exp(-x/(-2+x)))/(x^2-exp((-1+x)*x^2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {17 \, x^{3} - 17 \, x e^{\left (x^{3} - x^{2}\right )} + 16 \, e^{\left (-\frac {x}{x - 2}\right )}}{16 \, {\left (x^{2} - e^{\left (x^{3} - x^{2}\right )}\right )}} \]

[In]

integrate(((17*x^2-68*x+68)*exp(x^3-x^2)^2+((48*x^4-224*x^3+320*x^2-128*x-32)*exp(-x/(-2+x))-34*x^4+136*x^3-13
6*x^2)*exp(x^3-x^2)+(-32*x^3+160*x^2-128*x)*exp(-x/(-2+x))+17*x^6-68*x^5+68*x^4)/((16*x^2-64*x+64)*exp(x^3-x^2
)^2+(-32*x^4+128*x^3-128*x^2)*exp(x^3-x^2)+16*x^6-64*x^5+64*x^4),x, algorithm="fricas")

[Out]

1/16*(17*x^3 - 17*x*e^(x^3 - x^2) + 16*e^(-x/(x - 2)))/(x^2 - e^(x^3 - x^2))

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {17 x}{16} - \frac {e^{- \frac {x}{x - 2}}}{- x^{2} + e^{x^{3} - x^{2}}} \]

[In]

integrate(((17*x**2-68*x+68)*exp(x**3-x**2)**2+((48*x**4-224*x**3+320*x**2-128*x-32)*exp(-x/(-2+x))-34*x**4+13
6*x**3-136*x**2)*exp(x**3-x**2)+(-32*x**3+160*x**2-128*x)*exp(-x/(-2+x))+17*x**6-68*x**5+68*x**4)/((16*x**2-64
*x+64)*exp(x**3-x**2)**2+(-32*x**4+128*x**3-128*x**2)*exp(x**3-x**2)+16*x**6-64*x**5+64*x**4),x)

[Out]

17*x/16 - exp(-x/(x - 2))/(-x**2 + exp(x**3 - x**2))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {17 \, x^{3} e^{\left (x^{2} + 1\right )} - 17 \, x e^{\left (x^{3} + 1\right )} + 16 \, e^{\left (x^{2} - \frac {2}{x - 2}\right )}}{16 \, {\left (x^{2} e^{\left (x^{2} + 1\right )} - e^{\left (x^{3} + 1\right )}\right )}} \]

[In]

integrate(((17*x^2-68*x+68)*exp(x^3-x^2)^2+((48*x^4-224*x^3+320*x^2-128*x-32)*exp(-x/(-2+x))-34*x^4+136*x^3-13
6*x^2)*exp(x^3-x^2)+(-32*x^3+160*x^2-128*x)*exp(-x/(-2+x))+17*x^6-68*x^5+68*x^4)/((16*x^2-64*x+64)*exp(x^3-x^2
)^2+(-32*x^4+128*x^3-128*x^2)*exp(x^3-x^2)+16*x^6-64*x^5+64*x^4),x, algorithm="maxima")

[Out]

1/16*(17*x^3*e^(x^2 + 1) - 17*x*e^(x^3 + 1) + 16*e^(x^2 - 2/(x - 2)))/(x^2*e^(x^2 + 1) - e^(x^3 + 1))

Giac [F(-2)]

Exception generated. \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((17*x^2-68*x+68)*exp(x^3-x^2)^2+((48*x^4-224*x^3+320*x^2-128*x-32)*exp(-x/(-2+x))-34*x^4+136*x^3-13
6*x^2)*exp(x^3-x^2)+(-32*x^3+160*x^2-128*x)*exp(-x/(-2+x))+17*x^6-68*x^5+68*x^4)/((16*x^2-64*x+64)*exp(x^3-x^2
)^2+(-32*x^4+128*x^3-128*x^2)*exp(x^3-x^2)+16*x^6-64*x^5+64*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{864,[0,15]%%%}+%%%{-10368,[0,14]%%%}+%%%{52992,[0,13]%%%
}+%%%{-1511

Mupad [B] (verification not implemented)

Time = 7.69 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {17\,x}{16}-\frac {{\mathrm {e}}^{-\frac {x}{x-2}}}{{\mathrm {e}}^{x^3-x^2}-x^2} \]

[In]

int(-(exp(-x/(x - 2))*(128*x - 160*x^2 + 32*x^3) - 68*x^4 + 68*x^5 - 17*x^6 + exp(x^3 - x^2)*(exp(-x/(x - 2))*
(128*x - 320*x^2 + 224*x^3 - 48*x^4 + 32) + 136*x^2 - 136*x^3 + 34*x^4) - exp(2*x^3 - 2*x^2)*(17*x^2 - 68*x +
68))/(64*x^4 - exp(x^3 - x^2)*(128*x^2 - 128*x^3 + 32*x^4) - 64*x^5 + 16*x^6 + exp(2*x^3 - 2*x^2)*(16*x^2 - 64
*x + 64)),x)

[Out]

(17*x)/16 - exp(-x/(x - 2))/(exp(x^3 - x^2) - x^2)