3.18.64 \(\int \frac {4096-5632 x-1024 x^2+256 x^3-512 x^4+256 x^5+e^{2 x^2} (-4+x^3)+e^{x^2} (-192-128 x-320 x^2-32 x^3+32 x^4)}{256 x^3+e^{2 x^2} x^3-512 x^4+256 x^5+e^{x^2} (-32 x^3+32 x^4)} \, dx\)

Optimal. Leaf size=24 \[ x+\frac {2+\frac {10}{-1+\frac {e^{x^2}}{16}+x}}{x^2} \]

________________________________________________________________________________________

Rubi [F]  time = 1.49, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {4096-5632 x-1024 x^2+256 x^3-512 x^4+256 x^5+e^{2 x^2} \left (-4+x^3\right )+e^{x^2} \left (-192-128 x-320 x^2-32 x^3+32 x^4\right )}{256 x^3+e^{2 x^2} x^3-512 x^4+256 x^5+e^{x^2} \left (-32 x^3+32 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4096 - 5632*x - 1024*x^2 + 256*x^3 - 512*x^4 + 256*x^5 + E^(2*x^2)*(-4 + x^3) + E^x^2*(-192 - 128*x - 320
*x^2 - 32*x^3 + 32*x^4))/(256*x^3 + E^(2*x^2)*x^3 - 512*x^4 + 256*x^5 + E^x^2*(-32*x^3 + 32*x^4)),x]

[Out]

2/x^2 + x + 5120*Defer[Int][(-16 + E^x^2 + 16*x)^(-2), x] - 2560*Defer[Int][1/(x^2*(-16 + E^x^2 + 16*x)^2), x]
- 5120*Defer[Int][1/(x*(-16 + E^x^2 + 16*x)^2), x] - 320*Defer[Int][1/(x^3*(-16 + E^x^2 + 16*x)), x] - 320*De
fer[Int][1/(x*(-16 + E^x^2 + 16*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x^2} \left (-4+x^3\right )+32 e^{x^2} \left (-6-4 x-10 x^2-x^3+x^4\right )+256 \left (16-22 x-4 x^2+x^3-2 x^4+x^5\right )}{\left (e^{x^2}+16 (-1+x)\right )^2 x^3} \, dx\\ &=\int \left (-\frac {320 \left (1+x^2\right )}{x^3 \left (-16+e^{x^2}+16 x\right )}+\frac {2560 \left (-1-2 x+2 x^2\right )}{x^2 \left (-16+e^{x^2}+16 x\right )^2}+\frac {-4+x^3}{x^3}\right ) \, dx\\ &=-\left (320 \int \frac {1+x^2}{x^3 \left (-16+e^{x^2}+16 x\right )} \, dx\right )+2560 \int \frac {-1-2 x+2 x^2}{x^2 \left (-16+e^{x^2}+16 x\right )^2} \, dx+\int \frac {-4+x^3}{x^3} \, dx\\ &=-\left (320 \int \left (\frac {1}{x^3 \left (-16+e^{x^2}+16 x\right )}+\frac {1}{x \left (-16+e^{x^2}+16 x\right )}\right ) \, dx\right )+2560 \int \left (\frac {2}{\left (-16+e^{x^2}+16 x\right )^2}-\frac {1}{x^2 \left (-16+e^{x^2}+16 x\right )^2}-\frac {2}{x \left (-16+e^{x^2}+16 x\right )^2}\right ) \, dx+\int \left (1-\frac {4}{x^3}\right ) \, dx\\ &=\frac {2}{x^2}+x-320 \int \frac {1}{x^3 \left (-16+e^{x^2}+16 x\right )} \, dx-320 \int \frac {1}{x \left (-16+e^{x^2}+16 x\right )} \, dx-2560 \int \frac {1}{x^2 \left (-16+e^{x^2}+16 x\right )^2} \, dx+5120 \int \frac {1}{\left (-16+e^{x^2}+16 x\right )^2} \, dx-5120 \int \frac {1}{x \left (-16+e^{x^2}+16 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.28, size = 24, normalized size = 1.00 \begin {gather*} \frac {2}{x^2}+x+\frac {160}{x^2 \left (-16+e^{x^2}+16 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4096 - 5632*x - 1024*x^2 + 256*x^3 - 512*x^4 + 256*x^5 + E^(2*x^2)*(-4 + x^3) + E^x^2*(-192 - 128*x
- 320*x^2 - 32*x^3 + 32*x^4))/(256*x^3 + E^(2*x^2)*x^3 - 512*x^4 + 256*x^5 + E^x^2*(-32*x^3 + 32*x^4)),x]

[Out]

2/x^2 + x + 160/(x^2*(-16 + E^x^2 + 16*x))

________________________________________________________________________________________

fricas [B]  time = 0.89, size = 47, normalized size = 1.96 \begin {gather*} \frac {16 \, x^{4} - 16 \, x^{3} + {\left (x^{3} + 2\right )} e^{\left (x^{2}\right )} + 32 \, x + 128}{16 \, x^{3} + x^{2} e^{\left (x^{2}\right )} - 16 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-4)*exp(x^2)^2+(32*x^4-32*x^3-320*x^2-128*x-192)*exp(x^2)+256*x^5-512*x^4+256*x^3-1024*x^2-5632
*x+4096)/(x^3*exp(x^2)^2+(32*x^4-32*x^3)*exp(x^2)+256*x^5-512*x^4+256*x^3),x, algorithm="fricas")

[Out]

(16*x^4 - 16*x^3 + (x^3 + 2)*e^(x^2) + 32*x + 128)/(16*x^3 + x^2*e^(x^2) - 16*x^2)

________________________________________________________________________________________

giac [B]  time = 0.25, size = 51, normalized size = 2.12 \begin {gather*} \frac {16 \, x^{4} + x^{3} e^{\left (x^{2}\right )} - 16 \, x^{3} + 32 \, x + 2 \, e^{\left (x^{2}\right )} + 128}{16 \, x^{3} + x^{2} e^{\left (x^{2}\right )} - 16 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-4)*exp(x^2)^2+(32*x^4-32*x^3-320*x^2-128*x-192)*exp(x^2)+256*x^5-512*x^4+256*x^3-1024*x^2-5632
*x+4096)/(x^3*exp(x^2)^2+(32*x^4-32*x^3)*exp(x^2)+256*x^5-512*x^4+256*x^3),x, algorithm="giac")

[Out]

(16*x^4 + x^3*e^(x^2) - 16*x^3 + 32*x + 2*e^(x^2) + 128)/(16*x^3 + x^2*e^(x^2) - 16*x^2)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 24, normalized size = 1.00

method	result	size
risch	\(\frac {2}{x^{2}}+x +\frac {160}{x^{2} \left (-16+{\mathrm e}^{x^{2}}+16 x \right )}\)	\(24\)
norman	\(\frac {128+x^{3} {\mathrm e}^{x^{2}}-16 x^{3}+32 x +16 x^{4}+2 \,{\mathrm e}^{x^{2}}}{x^{2} \left (-16+{\mathrm e}^{x^{2}}+16 x \right )}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3-4)*exp(x^2)^2+(32*x^4-32*x^3-320*x^2-128*x-192)*exp(x^2)+256*x^5-512*x^4+256*x^3-1024*x^2-5632*x+409
6)/(x^3*exp(x^2)^2+(32*x^4-32*x^3)*exp(x^2)+256*x^5-512*x^4+256*x^3),x,method=_RETURNVERBOSE)

[Out]

2/x^2+x+160/x^2/(-16+exp(x^2)+16*x)

________________________________________________________________________________________

maxima [B]  time = 0.72, size = 47, normalized size = 1.96 \begin {gather*} \frac {16 \, x^{4} - 16 \, x^{3} + {\left (x^{3} + 2\right )} e^{\left (x^{2}\right )} + 32 \, x + 128}{16 \, x^{3} + x^{2} e^{\left (x^{2}\right )} - 16 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-4)*exp(x^2)^2+(32*x^4-32*x^3-320*x^2-128*x-192)*exp(x^2)+256*x^5-512*x^4+256*x^3-1024*x^2-5632
*x+4096)/(x^3*exp(x^2)^2+(32*x^4-32*x^3)*exp(x^2)+256*x^5-512*x^4+256*x^3),x, algorithm="maxima")

[Out]

(16*x^4 - 16*x^3 + (x^3 + 2)*e^(x^2) + 32*x + 128)/(16*x^3 + x^2*e^(x^2) - 16*x^2)

________________________________________________________________________________________

mupad [B]  time = 1.32, size = 28, normalized size = 1.17 \begin {gather*} x+\frac {32\,x+2\,{\mathrm {e}}^{x^2}+128}{x^2\,\left (16\,x+{\mathrm {e}}^{x^2}-16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5632*x - exp(2*x^2)*(x^3 - 4) + exp(x^2)*(128*x + 320*x^2 + 32*x^3 - 32*x^4 + 192) + 1024*x^2 - 256*x^3
+ 512*x^4 - 256*x^5 - 4096)/(x^3*exp(2*x^2) - exp(x^2)*(32*x^3 - 32*x^4) + 256*x^3 - 512*x^4 + 256*x^5),x)

[Out]

x + (32*x + 2*exp(x^2) + 128)/(x^2*(16*x + exp(x^2) - 16))

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 26, normalized size = 1.08 \begin {gather*} x + \frac {160}{16 x^{3} + x^{2} e^{x^{2}} - 16 x^{2}} + \frac {2}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3-4)*exp(x**2)**2+(32*x**4-32*x**3-320*x**2-128*x-192)*exp(x**2)+256*x**5-512*x**4+256*x**3-102
4*x**2-5632*x+4096)/(x**3*exp(x**2)**2+(32*x**4-32*x**3)*exp(x**2)+256*x**5-512*x**4+256*x**3),x)

[Out]

x + 160/(16*x**3 + x**2*exp(x**2) - 16*x**2) + 2/x**2

________________________________________________________________________________________