3.85.22 $\int \frac{e^{2e^4}(512x-1024x^2+768x^3-256x^4+32x^5+e^8(-4+x^2))}{e^{16}{x}^2+4096x^4-8192x^5+6144x^6-2048x^7+256x^8+e^8(-128x^3+128x^4-32x^5)}dx$

Optimal. Leaf size=27 $$\frac{e^{2e^4}}{(-16+\frac{e^8}{(-2+x{)}^{2}x})x^2}$$

________________________________________________________________________________________

Rubi [B]  time = 48.60, antiderivative size = 5694, normalized size of antiderivative = 210.89, number of steps used = 21, number of rules used = 11, integrand size = 96, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.115, Rules used = {12, 2074, 2101, 2081, 2079, 822, 800, 634, 618, 204, 628}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(E^(2*E^4)*(512*x - 1024*x^2 + 768*x^3 - 256*x^4 + 32*x^5 + E^8*(-4 + x^2)))/(E^16*x^2 + 4096*x^4 - 8192*x
^5 + 6144*x^6 - 2048*x^7 + 256*x^8 + E^8*(-128*x^3 + 128*x^4 - 32*x^5)),x]

[Out]

(48*E^(-8 + 2*E^4)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3))/(2^(1/3)*(32*2^(1/3) + (-256 + 27*E^8 +
3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 4*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)*(4 - 3*x)) - (18*2^
(2/3)*E^(-8 + 2*E^4)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(4/3)*(48*E^4*(512 - 27*E^8) - (2^(1/3)*(243
*E^20 + 24576*E^4*(16 + (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 144*E^12*(176 + 9*(-512 + 54*E^8
+ 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + Sqrt[-1536 + 81*E^8]*(27*E^16 + 4096*(8 + (-512 + 54*E^8 + 6*E^4*Sqrt[
-1536 + 81*E^8])^(1/3)) - 16*E^8*(160 + 11*(-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3))))*(4 - 3*x))/(-
256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))/((4608*E^4 - 243*E^12 + 256*Sqrt[-1536 + 81*E^8] - 27*E^8*S
qrt[-1536 + 81*E^8])*(32*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))*((32*2^(2/3))/(-256 + 2
7*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 4*(4 - 3*x))*
(64 - (2048*2^(1/3))/(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3) - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 +
81*E^8])^(2/3) + 4*((32*2^(2/3))/(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-512 + 54*E^8 + 6*E^4*S
qrt[-1536 + 81*E^8])^(1/3))*(4 - 3*x) - 16*(4 - 3*x)^2)) + (4*E^(-8 + 2*E^4))/x + (8*E^(-8 + 2*E^4)*(32 - E^8)
)/(3*(E^8 - 64*x + 64*x^2 - 16*x^3)) + (3*2^(5/6)*E^(-8 + 2*E^4)*Sqrt[3/(729*2^(2/3)*E^16 + 81*2^(2/3)*E^12*Sq
rt[-1536 + 81*E^8] - 192*E^4*Sqrt[-1536 + 81*E^8]*(4*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1
/3)) - 1728*E^8*(8*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + 1024*(32*2^(2/3) + 16*(-256
+ 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))]*
(27*E^16*(112*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 16384*(32*2^(1/3) + (-256 + 27*E
^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 512*E^8*(16*2^(1/3) + 5*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])
^(2/3)) + E^4*Sqrt[-1536 + 81*E^8]*(E^8*(336*2^(1/3) - 3*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) +
256*(16*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))))*ArcTan[(2^(1/3)*(27*E^8 + 3*E^4*Sqrt[
-1536 + 81*E^8] - 32*(8 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3))) - 8*(-256 + 27*E^8 + 3*E^4*Sqrt
[-1536 + 81*E^8])^(2/3)*(4 - 3*x))/Sqrt[6*(729*2^(2/3)*E^16 + 81*2^(2/3)*E^12*Sqrt[-1536 + 81*E^8] - 192*E^4*S
qrt[-1536 + 81*E^8]*(4*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 1728*E^8*(8*2^(2/3) + (
-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + 1024*(32*2^(2/3) + 16*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 +
81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))]])/(27*2^(2/3)*E^8*(-256 + 27*E
^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536
+ 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*
E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))) - (5184*2^(5/6)*E^(2*E^4)*(512 - 27*E^8)*Sqrt[3/(729*2^(2/3)*E^16 +
81*2^(2/3)*E^12*Sqrt[-1536 + 81*E^8] - 192*E^4*Sqrt[-1536 + 81*E^8]*(4*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-
1536 + 81*E^8])^(1/3)) - 1728*E^8*(8*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + 1024*(32*
2^(2/3) + 16*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 +
81*E^8])^(2/3)))]*(2239488*E^24*(16*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 19683*E^32
*(112*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 268435456*(32*2^(1/3) + (-256 + 27*E^8 +
3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 1990656*E^16*(192*2^(1/3) + 25*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^
8])^(2/3)) - 4194304*E^8*(1472*2^(1/3) + 73*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + E^4*Sqrt[-15
36 + 81*E^8]*(20736*E^16*(80*2^(1/3) - 11*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 2187*E^24*(112
*2^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 2097152*(128*2^(1/3) + 5*(-256 + 27*E^8 + 3*E
^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 73728*E^8*(640*2^(1/3) + 47*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/
3))))*ArcTan[(2^(1/3)*(27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8] - 32*(8 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^
8])^(1/3))) - 8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)*(4 - 3*x))/Sqrt[6*(729*2^(2/3)*E^16 + 81*2^
(2/3)*E^12*Sqrt[-1536 + 81*E^8] - 192*E^4*Sqrt[-1536 + 81*E^8]*(4*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536
+ 81*E^8])^(1/3)) - 1728*E^8*(8*2^(2/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) + 1024*(32*2^(2/
3) + 16*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^
8])^(2/3)))]])/((27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536
+ 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 128*(16*2^(1/3) - 2*2^(2/3)*(-256 + 27*E^8 + 3*
E^4*Sqrt[-1536 + 81*E^8])^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*(27*2^(2/3)*E^8*(-256 +
27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-
1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256
+ 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))^3) + (2*E^(-8 + 2*E^4)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E
^8])^(1/3)*(3*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 8*(32*2^(2/3) - 32*(-256 + 27*E^8 + 3*E
^4*Sqrt[-1536 + 81*E^8])^(1/3) + 2^(1/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*Log[2^(1/3)*(32*
2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 4*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])
^(1/3)*(4 - 3*x)])/(27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-15
36 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 +
3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))) - (864*E^(2*E^4)*(512
- 27*E^8)*(256 - 27*E^8 - 3*E^4*Sqrt[-1536 + 81*E^8])*(81*E^16*(144*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-15
36 + 81*E^8])^(2/3)) + 65536*(8*2^(1/3) - 2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 +
27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 768*E^8*(288*2^(1/3) - 9*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1
536 + 81*E^8])^(1/3) + 10*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + E^4*Sqrt[-1536 + 81*E^8]*(9*E^
8*(144*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 768*(16*2^(1/3) - 2^(2/3)*(-256 + 27*E^
8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))))*Log[2^(1/3)*(32*
2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 4*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])
^(1/3)*(4 - 3*x)])/((27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1
536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 128*(16*2^(1/3) - 2*2^(2/3)*(-256 + 27*E^8
+ 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*(27*2^(2/3)*E^8*(-2
56 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sq
rt[-1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-
256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))^3) - (E^(-8 + 2*E^4)*(24*2^(1/3)*E^4*Sqrt[-1536 + 81*E^8] +
3*E^8*(72*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 256*(8*2^(1/3) - 2^(2/3)*(-256 + 27
*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*Log[64*(32*2^(
1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E
^8])^(2/3)) - 9*2^(1/3)*E^8*(16 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 2^(1/3)*E^4*Sqrt[-1536
+ 81*E^8]*(16 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 1024*2^(1/3)*x + 108*2^(1/3)*E^8*x + 12
*2^(1/3)*E^4*Sqrt[3*(-512 + 27*E^8)]*x + 128*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3)*x - 12
8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)*x + 48*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)
*x^2])/(27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]
*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-
1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3))) - (864*E^(2*E^4)*(512 - 27*E^8)*(
2187*E^24*(144*2^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 8388608*(8*2^(1/3) - 2^(2/3)*(-
256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 20736
*E^16*(432*2^(1/3) - 9*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 11*(-256 + 27*E^8 + 3*E^4*
Sqrt[-1536 + 81*E^8])^(2/3)) + 98304*E^8*(648*2^(1/3) - 36*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8]
)^(1/3) + 37*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + E^4*Sqrt[-1536 + 81*E^8]*(243*E^16*(144*2^(
1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) + 196608*(12*2^(1/3) - 2^(2/3)*(-256 + 27*E^8 + 3*E
^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 2304*E^8*(288*2^(1/3) -
9*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 10*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8]
)^(2/3))))*Log[64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + (-256 + 27*E^8
+ 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)) - 9*2^(1/3)*E^8*(16 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3))
- 2^(1/3)*E^4*Sqrt[-1536 + 81*E^8]*(16 - (-512 + 54*E^8 + 6*E^4*Sqrt[-1536 + 81*E^8])^(1/3)) - 1024*2^(1/3)*x
+ 108*2^(1/3)*E^8*x + 12*2^(1/3)*E^4*Sqrt[3*(-512 + 27*E^8)]*x + 128*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-153
6 + 81*E^8])^(1/3)*x - 128*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)*x + 48*(-256 + 27*E^8 + 3*E^4*Sq
rt[-1536 + 81*E^8])^(2/3)*x^2])/((27*2^(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3
)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 128*(16*2^(1/3) - 2*2^(2/3)*(-
256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) - (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))*(27*2^
(2/3)*E^8*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 3*2^(2/3)*E^4*Sqrt[-1536 + 81*E^8]*(-256 + 27*E
^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(1/3) + 64*(32*2^(1/3) - 4*2^(2/3)*(-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8
])^(1/3) + (-256 + 27*E^8 + 3*E^4*Sqrt[-1536 + 81*E^8])^(2/3)))^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 2079

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/
3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r
/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rule 2101

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1)
)/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[P
m, x, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p,
-1]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =e^{2e^4}\int \frac{512x-1024x^2+768x^3-256x^4+32x^5+e^8(-4+x^2)}{e^{16}{x}^2+4096x^4-8192x^5+6144x^6-2048x^7+256x^8+e^8(-128x^3+128x^4-32x^5)}dx\\ & =e^{2e^4}\int (-\frac{4}{e^8{x}^2}+\frac{16384-768e^8+3e^{16}-64(256-9e^8)x+128(32-e^8)x^2}{e^8{(e^8-64x+64x^2-16x^3)}^2}-\frac{2(-128+e^8+32x)}{e^8(e^8-64x+64x^2-16x^3)})dx\\ & =\frac{4e^{-8+2e^4}}{x}+e^{-8+2e^4}\int \frac{16384-768e^8+3e^{16}-64(256-9e^8)x+128(32-e^8)x^2}{{(e^8-64x+64x^2-16x^3)}^2}dx-\left(2e^{-8+2e^4}\right)\int \frac{-128+e^8+32x}{e^8-64x+64x^2-16x^3}dx\\ & =\frac{4e^{-8+2e^4}}{x}+\frac{8e^{-8+2e^4}(32-e^8)}{3(e^8-64x+64x^2-16x^3)}-\frac{1}{48}{e}^{-8+2e^4}\int \frac{-16(32768-1792e^8+9e^{16})+1024(256-11e^8)x}{{(e^8-64x+64x^2-16x^3)}^2}dx-\left(2e^{-8+2e^4}\right)\mathrm{Subst}(\int \frac{\frac{1}{48}(2048+48(-128+e^8))+32x}{\frac{1}{27}(-256+27e^8)+\frac{64x}{3}-16x^3}dx,x,-\frac{4}{3}+x)\\ & =\frac{4e^{-8+2e^4}}{x}+\frac{8e^{-8+2e^4}(32-e^8)}{3(e^8-64x+64x^2-16x^3)}-\frac{1}{48}{e}^{-8+2e^4}\mathrm{Subst}(\int \frac{\frac{1}{48}(65536(256-11e^8)-768(32768-1792e^8+9e^{16}))+1024(256-11e^8)x}{{(\frac{1}{27}(-256+27e^8)+\frac{64x}{3}-16x^3)}^2}dx,x,-\frac{4}{3}+x)-\left(512e^{-8+2e^4}\right)\mathrm{Subst}(\int \frac{\frac{1}{48}(2048+48(-128+e^8))+32x}{(\frac{4}{3}(\frac{322^{2/3}}{\sqrt[3]{-256+27e^8+3e^4\sqrt{-1536+81e^8}}}+\sqrt[3]{-512+54e^8+6e^4\sqrt{-1536+81e^8}})-16x)(-\frac{16}{9}(64-\frac{2048\sqrt[3]{2}}{{(-256+27e^8+3e^4\sqrt{-1536+81e^8})}^{2/3}}-{(-512+54e^8+6e^4\sqrt{-1536+81e^8})}^{2/3})+\frac{64}{3}(\frac{322^{2/3}}{\sqrt[3]{-256+27e^8+3e^4\sqrt{-1536+81e^8}}}+\sqrt[3]{-512+54e^8+6e^4\sqrt{-1536+81e^8}})x+256x^2)}dx,x,-\frac{4}{3}+x)\\ & =\text{Rest of rules removed due to large latex content}\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 33, normalized size = 1.22 $$\begin{array}{c}-\frac{e^{2e^4}(-2+x{)}^2}{x(-e^8+16(-2+x{)}^{2}x)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*E^4)*(512*x - 1024*x^2 + 768*x^3 - 256*x^4 + 32*x^5 + E^8*(-4 + x^2)))/(E^16*x^2 + 4096*x^4 -
8192*x^5 + 6144*x^6 - 2048*x^7 + 256*x^8 + E^8*(-128*x^3 + 128*x^4 - 32*x^5)),x]

[Out]

-((E^(2*E^4)*(-2 + x)^2)/(x*(-E^8 + 16*(-2 + x)^2*x)))

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 38, normalized size = 1.41 $$\begin{array}{c}-\frac{(x^2-4x+4)e^{\left(2e^4\right)}}{16x^4-64x^3+64x^2-x{e}^8}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*exp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^
4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6144*x^6-8192*x^5+4096*x^4),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Exception raised: TypeError}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*exp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^
4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6144*x^6-8192*x^5+4096*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 convert to real -32*exp(8)-8192 Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.38, size = 35, normalized size = 1.30

method	result	size
gosper	$\frac{{(x-2)}^2{\mathrm{e}}^{2{\mathrm{e}}^4}}{x(-16x^3+{\mathrm{e}}^8+64x^2-64x)}$	$35$
risch	$\frac{{\mathrm{e}}^{2{\mathrm{e}}^4}(x^2-4x+4)}{x(-16x^3+{\mathrm{e}}^8+64x^2-64x)}$	$36$
norman	$\frac{x^2{\mathrm{e}}^{2{\mathrm{e}}^4}-4x{\mathrm{e}}^{2{\mathrm{e}}^4}+4{\mathrm{e}}^{2{\mathrm{e}}^4}}{x(-16x^3+{\mathrm{e}}^8+64x^2-64x)}$	$50$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*exp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^4-128*
x^3)*exp(4)^2+256*x^8-2048*x^7+6144*x^6-8192*x^5+4096*x^4),x,method=_RETURNVERBOSE)

[Out]

1/x*(x-2)^2*exp(2*exp(4))/(-16*x^3+exp(4)^2+64*x^2-64*x)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 38, normalized size = 1.41 $$\begin{array}{c}-\frac{(x^2-4x+4)e^{\left(2e^4\right)}}{16x^4-64x^3+64x^2-x{e}^8}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*exp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^
4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6144*x^6-8192*x^5+4096*x^4),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.04 $$\begin{array}{c}\text{Hanged}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*exp(4))*(512*x - 1024*x^2 + 768*x^3 - 256*x^4 + 32*x^5 + exp(8)*(x^2 - 4)))/(x^2*exp(16) - exp(8)*(
128*x^3 - 128*x^4 + 32*x^5) + 4096*x^4 - 8192*x^5 + 6144*x^6 - 2048*x^7 + 256*x^8),x)

[Out]

\text{Hanged}

________________________________________________________________________________________

sympy [B]  time = 1.42, size = 48, normalized size = 1.78 $$\begin{array}{c}\frac{-x^2{e}^{2e^4}+4x{e}^{2e^4}-4e^{2e^4}}{16x^4-64x^3+64x^2-x{e}^8}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-4)*exp(4)**2+32*x**5-256*x**4+768*x**3-1024*x**2+512*x)*exp(2*exp(4))/(x**2*exp(4)**4+(-32*x*
*5+128*x**4-128*x**3)*exp(4)**2+256*x**8-2048*x**7+6144*x**6-8192*x**5+4096*x**4),x)

[Out]

(-x**2*exp(2*exp(4)) + 4*x*exp(2*exp(4)) - 4*exp(2*exp(4)))/(16*x**4 - 64*x**3 + 64*x**2 - x*exp(8))

________________________________________________________________________________________