3.25.94 \(\int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9)}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx\)
Optimal. Leaf size=35 \[ e^x+\frac {1}{4} \left (1-\frac {2 \left (-3-\frac {4}{x}+2 x\right )}{x \left (-3-x+x^3\right )}\right ) \]
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Rubi [B] time = 163.18, antiderivative size = 9556, normalized size of antiderivative =
273.03, number of steps used = 109, number of rules used = 22, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.234,
Rules used = {6741, 12, 6688, 6742, 2194, 2065, 740, 800, 634, 618, 204, 628, 2079, 893, 638, 822,
1646, 6, 27, 21, 1586, 1628}
result too large to display
Warning: Unable to verify antiderivative.
[In]
Int[(24 + 21*x + 6*x^2 - 22*x^3 - 12*x^4 + 6*x^5 + E^x*(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9))/(18
*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9),x]
[Out]
E^x + 93312/((2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3))^3*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))
^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6^(1/3)*(2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(
1/3)) - 6*x)) + (13608*6^(1/3))/((2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3))^2*(6 + 6*3^(1/3)*
(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6^(1/3)*(2*(3/(27 + Sqrt[717]))^(1/3) + (2
*(27 + Sqrt[717]))^(1/3)) - 6*x)) + (648*6^(2/3))/((2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3))
*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6^(1/3)*(2*(3/(27 + Sqrt[7
17]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x)) + 124416/((12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/
3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))
^2*(6^(1/3)*(2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x)) - (2592*(58079*2^(2/3)*3^(5/6)
*Sqrt[239]*(27 + Sqrt[717])^(1/3) - 6480*2^(1/3)*3^(1/6)*Sqrt[239]*(27 + Sqrt[717])^(2/3) + 239*(348474 + 1301
4*Sqrt[717] + 6507*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3) - 242*2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))))/((27 + Sqrt[
717])^(7/3)*(12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2
/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(3*(27 + Sqrt[717])^(5/3) + 2^(1/3)*((484*3^(2/
3) + 54*3^(1/6)*Sqrt[239])*(27 + Sqrt[717])^(1/3) + 2*6^(1/3)*(3267 + 122*Sqrt[717])))*(6^(1/3)*(2*(3/(27 + Sq
rt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x)) - 2/(3*x^2) - 7/(6*x) + (8*(27 + Sqrt[717])^3)/(27*Sqrt[
3]*(9*Sqrt[3] + Sqrt[239])^3*x) + (648*6^(2/3)*(54 + 2^(1/3)*3^(2/3)*(2*2^(1/3)*(3/(27 + Sqrt[717]))^(2/3) + (
27 + Sqrt[717])^(2/3))*x))/((12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(
6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(2*(3/(27 + Sqrt[717]))^(1/3) +
(2*(27 + Sqrt[717]))^(1/3) - 6^(2/3)*x)*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717
]))^(2/3) - 6*3^(1/3)*((6/(27 + Sqrt[717]))^(1/3) + ((27 + Sqrt[717])/2)^(1/3))*x - 18*x^2)) - (7128*6^(2/3)*(
2 + (3^(2/3)*(2/(27 + Sqrt[717]))^(1/3) + ((3*(27 + Sqrt[717]))/2)^(1/3))*x))/((12 - 6*3^(1/3)*(2/(27 + Sqrt[7
17]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 +
Sqrt[717]))^(2/3))*(2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3) - 6^(2/3)*x)*(6 - 6*3^(1/3)*(2/
(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3) - 6*3^(1/3)*((6/(27 + Sqrt[717]))^(1/3) + ((27 +
Sqrt[717])/2)^(1/3))*x - 18*x^2)) - (3*((27 + Sqrt[717])^(2/3)*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(
1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6*3^(1/6)*(9*Sqrt[3] + Sqrt[239])*(2/(27 + Sqrt[717]))^(1/3) + (27 + Sqrt[71
7])*(108 + 2^(2/3)*(3*(27 + Sqrt[717]))^(1/3))) - 72*(2*(27 + Sqrt[717])^(5/3) + 6^(1/3)*((3*(27 + Sqrt[717]))
^(1/3) + 2^(1/3)*(3240 + 121*Sqrt[717])))*x))/((12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + S
qrt[717]))^(2/3))*(3*(27 + Sqrt[717])^(5/3) + 2^(1/3)*((484*3^(2/3) + 54*3^(1/6)*Sqrt[239])*(27 + Sqrt[717])^(
1/3) + 2*6^(1/3)*(3267 + 122*Sqrt[717])))*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[71
7]))^(2/3) - 6*3^(1/3)*((6/(27 + Sqrt[717]))^(1/3) + ((27 + Sqrt[717])/2)^(1/3))*x - 18*x^2)) - (23328*(27 + S
qrt[717])^(2/3)*((270 + 10*Sqrt[717] - 4*3^(2/3)*(2/(27 + Sqrt[717]))^(1/3) - 9*2^(2/3)*3^(5/6)*Sqrt[239]*(27
+ Sqrt[717])^(1/3) - 241*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3))/(27 + Sqrt[717]) - (27 - 4*3^(2/3)*(2/(27 + Sqrt[
717]))^(1/3) - 2*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3))*x))/((2*2^(2/3)*3^(1/3) - 4*(27 + Sqrt[717])^(2/3) + 3^(1
/6)*(9*Sqrt[3] + Sqrt[239])*(2*(27 + Sqrt[717]))^(1/3))*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3
*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6
- 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3) - 3*2^(2/3)*(6^(2/3)/(27 + Sqrt[7
17])^(1/3) + (3*(27 + Sqrt[717]))^(1/3))*x - 18*x^2)) - (181398528*((4*(273*(27 + Sqrt[717])^(1/3)*(58079 + 21
69*Sqrt[717]) - 2^(1/3)*3^(1/6)*(2103804*Sqrt[3] + 235704*Sqrt[239] + 2607075*2^(1/3)*3^(1/6)*(27 + Sqrt[717])
^(2/3) + 97363*2^(1/3)*Sqrt[239]*(3*(27 + Sqrt[717]))^(2/3))))/(27 + Sqrt[717]) - (27 + Sqrt[717])^(1/3)*(8164
80 + 30492*Sqrt[717] - 887*2^(2/3)*3^(5/6)*Sqrt[239]*(27 + Sqrt[717])^(1/3) + 2070*2^(1/3)*3^(1/6)*Sqrt[239]*(
27 + Sqrt[717])^(2/3) - 23751*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3) + 18476*2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*x
))/((27 + Sqrt[717])^(8/3)*(2*2^(2/3)*3^(1/3) - 4*(27 + Sqrt[717])^(2/3) + 3^(1/6)*(9*Sqrt[3] + Sqrt[239])*(2*
(27 + Sqrt[717]))^(1/3))*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3*(6
+ 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6 - 6*3^(1/3)*(2/(27 + Sqrt[71
7]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3) - 3*2^(2/3)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[71
7]))^(1/3))*x - 18*x^2)) + (4536*((3*(126*(27 + Sqrt[717])^(1/3)*(241 + 9*Sqrt[717]) + 2^(1/3)*3^(1/6)*(36952*
Sqrt[3] + 4140*Sqrt[239] - 883*2^(1/3)*3^(1/6)*(27 + Sqrt[717])^(2/3) - 33*2^(1/3)*Sqrt[239]*(3*(27 + Sqrt[717
]))^(2/3))))/(27 + Sqrt[717])^(2/3) - 4*((15*(241 + 9*Sqrt[717]))/(27 + Sqrt[717])^(1/3) - 6^(1/3)*((3*(27 + S
qrt[717]))^(1/3) + 2^(1/3)*(3240 + 121*Sqrt[717])))*x))/((27 + Sqrt[717])*(2*2^(2/3)*3^(1/3) - 4*(27 + Sqrt[71
7])^(2/3) + 3^(1/6)*(9*Sqrt[3] + Sqrt[239])*(2*(27 + Sqrt[717]))^(1/3))*(2 + (6^(1/3)*(4*3^(1/3) + 2^(1/3)*(27
+ Sqrt[717])^(2/3)*(241 + 9*Sqrt[717])))/(27 + Sqrt[717])^(4/3))^2*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3)
- 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3) - 3*2^(2/3)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[717]))^(1/3))
*x - 18*x^2)) - (20155392*Sqrt[2/(2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/
3) - 4*(27 + Sqrt[717])^(1/3))]*(816480 + 30492*Sqrt[717] - 887*2^(2/3)*3^(5/6)*Sqrt[239]*(27 + Sqrt[717])^(1/
3) + 2070*2^(1/3)*3^(1/6)*Sqrt[239]*(27 + Sqrt[717])^(2/3) - 23751*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3) + 18476*
2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*ArcTan[((2*(27 + Sqrt[717]))^(1/6)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(2
7 + Sqrt[717]))^(1/3) + 6*2^(1/3)*x))/(3*Sqrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqr
t[717]))^(1/3) - 4*(27 + Sqrt[717])^(1/3)])])/((27 + Sqrt[717])^(13/6)*(2*2^(2/3)*3^(1/3) - 4*(27 + Sqrt[717])
^(2/3) + 3^(1/6)*(9*Sqrt[3] + Sqrt[239])*(2*(27 + Sqrt[717]))^(1/3))*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3)
- 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717
]))^(2/3))^2) + (171072*2^(5/6)*3^(2/3)*(2^(1/3)*3^(1/6)*(1089*Sqrt[3] + 122*Sqrt[239]) + (27 + Sqrt[717])^(1/
3)*(242 + 9*Sqrt[717]))*ArcTan[((2*(27 + Sqrt[717]))^(1/6)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[717
]))^(1/3) + 6*2^(1/3)*x))/(3*Sqrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/
3) - 4*(27 + Sqrt[717])^(1/3)])])/((27 + Sqrt[717])^(3/2)*Sqrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(
2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(27 + Sqrt[717])^(1/3)]*(12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/
3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))
^3) - (23328*2^(5/6)*((2205*3^(2/3) + 247*3^(1/6)*Sqrt[239])*(27 + Sqrt[717])^(1/3) + 6^(1/3)*(1207 + 45*Sqrt[
717]))*ArcTan[((2*(27 + Sqrt[717]))^(1/6)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[717]))^(1/3) + 6*2^(
1/3)*x))/(3*Sqrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(27 + Sqrt
[717])^(1/3)])])/((27 + Sqrt[717])^(3/2)*Sqrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqr
t[717]))^(1/3) - 4*(27 + Sqrt[717])^(1/3)]*(12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[
717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3) + (3918208204
8*Sqrt[2/(2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(27 + Sqrt[717])^
(1/3))]*(3203438*2^(2/3)*3^(5/6)*Sqrt[239]*(27 + Sqrt[717])^(1/3) + 1025323*2^(1/3)*3^(1/6)*Sqrt[239]*(27 + Sq
rt[717])^(2/3) + 85777997*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3) + 9151642*2^(1/3)*(3*(27 + Sqrt[717]))^(2/3) - 10
7*(1561653 + 58321*Sqrt[717]))*ArcTan[((2*(27 + Sqrt[717]))^(1/6)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + S
qrt[717]))^(1/3) + 6*2^(1/3)*x))/(3*Sqrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717
]))^(1/3) - 4*(27 + Sqrt[717])^(1/3)])])/((27 + Sqrt[717])^(29/6)*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) -
2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^4*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))
^(2/3))^3) - (2592*(27*(27 + Sqrt[717])^(4/3) - (24*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 4*6^(1/3)*(27 + Sqrt[717
])^(5/3))/2^(2/3))*ArcTan[((2*(27 + Sqrt[717]))^(1/6)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[717]))^(
1/3) + 6*2^(1/3)*x))/(3*Sqrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) -
4*(27 + Sqrt[717])^(1/3)])])/((2*2^(2/3)*3^(1/3) - 4*(27 + Sqrt[717])^(2/3) + 3^(1/6)*(9*Sqrt[3] + Sqrt[239])*
(2*(27 + Sqrt[717]))^(1/3))*Sqrt[2^(1/3)*(241*3^(2/3) + 27*3^(1/6)*Sqrt[239]) - 2*(27 + Sqrt[717])^(4/3) + 3^(
1/3)*(2*(27 + Sqrt[717]))^(2/3)]*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3
))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2) - (839808*Sqrt[2/(2^(1/3
)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(27 + Sqrt[717])^(1/3))]*(468*(27
+ Sqrt[717])^(1/3)*(3240 + 121*Sqrt[717]) + 2^(1/3)*3^(1/6)*(1097262*Sqrt[3] + 122934*Sqrt[239] + 28089*2^(1/
3)*3^(1/6)*(27 + Sqrt[717])^(2/3) + 1049*2^(1/3)*Sqrt[239]*(3*(27 + Sqrt[717]))^(2/3)))*ArcTan[((2*(27 + Sqrt[
717]))^(1/6)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[717]))^(1/3) + 6*2^(1/3)*x))/(3*Sqrt[2^(1/3)*3^(1
/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(27 + Sqrt[717])^(1/3)])])/((27 + Sqrt[
717])^(19/6)*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6 + 6*3^(1/3)*
(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3) + (181440*Sqrt[2]*(27 + Sqrt[717])^(1/6)*(
19763*2^(2/3)*3^(5/6)*Sqrt[239]*(27 + Sqrt[717])^(1/3) + 8685*2^(1/3)*3^(1/6)*Sqrt[239]*(27 + Sqrt[717])^(2/3)
+ 529191*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3) + 77519*2^(1/3)*(3*(27 + Sqrt[717]))^(2/3) - 186*(58079 + 2169*Sq
rt[717]))*ArcTan[((2*(27 + Sqrt[717]))^(1/6)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[717]))^(1/3) + 6*
2^(1/3)*x))/(3*Sqrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(27 + S
qrt[717])^(1/3)])])/(Sqrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(
27 + Sqrt[717])^(1/3)]*(2*(27 + Sqrt[717])^(4/3) + 6^(1/3)*(4*3^(1/3) + 2^(1/3)*(27 + Sqrt[717])^(2/3)*(241 +
9*Sqrt[717])))^3) - (2016*(27 + Sqrt[717])^(3/2)*Sqrt[2/(2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(
3/(27 + Sqrt[717]))^(1/3) - 4*(27 + Sqrt[717])^(1/3))]*(121*2^(2/3)*3^(5/6)*Sqrt[239]*(27 + Sqrt[717])^(1/3) +
3240*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3) - 15*(241 + 9*Sqrt[717]))*ArcTan
[((2*(27 + Sqrt[717]))^(1/6)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[717]))^(1/3) + 6*2^(1/3)*x))/(3*S
qrt[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(27 + Sqrt[717])^(1/3)]
)])/((2*2^(2/3)*3^(1/3) - 4*(27 + Sqrt[717])^(2/3) + 3^(1/6)*(9*Sqrt[3] + Sqrt[239])*(2*(27 + Sqrt[717]))^(1/3
))*(2*(27 + Sqrt[717])^(4/3) + 6^(1/3)*(4*3^(1/3) + 2^(1/3)*(27 + Sqrt[717])^(2/3)*(241 + 9*Sqrt[717])))^2) -
(1119744*Sqrt[2/(2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(27 + Sqrt
[717])^(1/3))]*(490*(27 + Sqrt[717])^(2/3)*(6998399 + 261360*Sqrt[717]) + 27*2^(1/3)*(2*(6998399*3^(2/3) + 784
080*3^(1/6)*Sqrt[239])*(27 + Sqrt[717])^(1/3) + 6^(1/3)*(376351893 + 14055119*Sqrt[717])))*ArcTan[((2*(27 + Sq
rt[717]))^(1/6)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[717]))^(1/3) + 6*2^(1/3)*x))/(3*Sqrt[2^(1/3)*3
^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 4*(27 + Sqrt[717])^(1/3)])])/((27 + Sq
rt[717])^(29/6)*(12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3
)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(3*(27 + Sqrt[717])^(5/3) + 2^(1/3)*((484
*3^(2/3) + 54*3^(1/6)*Sqrt[239])*(27 + Sqrt[717])^(1/3) + 2*6^(1/3)*(3267 + 122*Sqrt[717])))) - (31104*3^(2/3)
*(21*2^(2/3) + (30*6^(1/3))/(27 + Sqrt[717])^(2/3) + 5*(3*(27 + Sqrt[717]))^(2/3))*Log[6^(1/3)*(2*(3/(27 + Sqr
t[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x])/((2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/
3))^4*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3) - (3888*2^(2/3)*(9*6^
(2/3) + 9*3^(1/3)*(27 + Sqrt[717])^(2/3) - 4*(2*(27 + Sqrt[717]))^(1/3))*Log[6^(1/3)*(2*(3/(27 + Sqrt[717]))^(
1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x])/((27 + Sqrt[717])^(1/3)*(12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3)
- 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))
^(2/3))^3) - (54432*(9 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*Log[6^(1/3
)*(2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x])/((2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27
+ Sqrt[717]))^(1/3))^3*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3) - (1
944*3^(1/3)*(10*2^(1/3) + (12*3^(1/3))/(27 + Sqrt[717])^(2/3) + (6*(27 + Sqrt[717]))^(2/3))*Log[6^(1/3)*(2*(3/
(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x])/((2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[7
17]))^(1/3))^2*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3) - (14256*(54
- 2^(2/3)*(6^(2/3)/(27 + Sqrt[717])^(1/3) + (3*(27 + Sqrt[717]))^(1/3)))*Log[6^(1/3)*(2*(3/(27 + Sqrt[717]))^
(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x])/((12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqr
t[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3) + (31104*3^
(1/6)*(12*2^(1/3)*(62465040*Sqrt[3] + 6998399*Sqrt[239]) - 1555173*3^(1/6)*(2*(27 + Sqrt[717]))^(2/3) - 58079*
Sqrt[239]*(6*(27 + Sqrt[717]))^(2/3))*Log[6^(1/3)*(2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3))
- 6*x])/((27 + Sqrt[717])^(11/3)*(12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/
3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(3*(27 + Sqrt[717])^(5/3
) + 2^(1/3)*((484*3^(2/3) + 54*3^(1/6)*Sqrt[239])*(27 + Sqrt[717])^(1/3) + 2*6^(1/3)*(3267 + 122*Sqrt[717]))))
- (419904*(90720 + 3388*Sqrt[717] + 6579*2^(2/3)*3^(5/6)*Sqrt[239]*(27 + Sqrt[717])^(1/3) + 1939*2^(1/3)*3^(1
/6)*Sqrt[239]*(27 + Sqrt[717])^(2/3) + 176165*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3) + 17307*2^(1/3)*(3*(27 + Sqrt
[717]))^(2/3))*Log[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 2*(27 + Sq
rt[717])^(1/3) + 2*2^(1/3)*3^(2/3)*x + 3^(1/3)*(2*(27 + Sqrt[717]))^(2/3)*x + 6*(27 + Sqrt[717])^(1/3)*x^2])/(
(27 + Sqrt[717])^3*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6 + 6*3^
(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3) + (42768*(9*(27 + Sqrt[717])^(5/3) -
3^(1/6)*(9*Sqrt[3] + Sqrt[239])*(2*(27 + Sqrt[717]))^(1/3) - 2^(2/3)*3^(1/3)*(241 + 9*Sqrt[717]))*Log[2^(1/3)
*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 2*(27 + Sqrt[717])^(1/3) + 2*2^(1/3)
*3^(2/3)*x + 3^(1/3)*(2*(27 + Sqrt[717]))^(2/3)*x + 6*(27 + Sqrt[717])^(1/3)*x^2])/((27 + Sqrt[717])^(5/3)*(12
- 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717
]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3) - (3888*(4*(27 + Sqrt[717])^(5/3) - 27*(3^(1/6)*(9*Sqrt[3]
+ Sqrt[239])*(2*(27 + Sqrt[717]))^(1/3) + 2^(2/3)*3^(1/3)*(241 + 9*Sqrt[717])))*Log[2^(1/3)*3^(1/6)*(9*Sqrt[3]
+ Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 2*(27 + Sqrt[717])^(1/3) + 2*2^(1/3)*3^(2/3)*x + 3^(1/3
)*(2*(27 + Sqrt[717]))^(2/3)*x + 6*(27 + Sqrt[717])^(1/3)*x^2])/((27 + Sqrt[717])^(5/3)*(12 - 6*3^(1/3)*(2/(27
+ Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)
*(3*(27 + Sqrt[717]))^(2/3))^3) - (93312*2^(1/3)*3^(1/6)*(6718346882*Sqrt[3] + 752703786*Sqrt[239] - 13938719*
2^(1/3)*3^(1/6)*(27 + Sqrt[717])^(2/3) - 520551*2^(1/3)*Sqrt[239]*(3*(27 + Sqrt[717]))^(2/3))*Log[2^(1/3)*3^(1
/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 2*(27 + Sqrt[717])^(1/3) + 2*2^(1/3)*3^(2
/3)*x + 3^(1/3)*(2*(27 + Sqrt[717]))^(2/3)*x + 6*(27 + Sqrt[717])^(1/3)*x^2])/((27 + Sqrt[717])^(14/3)*(12 - 6
*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^
(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(3*(27 + Sqrt[717])^(5/3) + 2^(1/3)*((484*3^(2/3) + 54*3^(1/6)*S
qrt[239])*(27 + Sqrt[717])^(1/3) + 2*6^(1/3)*(3267 + 122*Sqrt[717])))) + (54432*(162*(58079 + 2169*Sqrt[717])
+ (2*(27 + Sqrt[717]))^(1/3)*((182877*3^(2/3) + 20489*3^(1/6)*Sqrt[239])*(27 + Sqrt[717])^(1/3) + 2*6^(1/3)*(1
25878 + 4701*Sqrt[717])))*Log[2^(1/3)*3^(1/6)*(9*Sqrt[3] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) -
2*(27 + Sqrt[717])^(1/3) + 2*2^(1/3)*3^(2/3)*x + 3^(1/3)*(2*(27 + Sqrt[717]))^(2/3)*x + 6*(27 + Sqrt[717])^(1
/3)*x^2])/(2*(27 + Sqrt[717])^(4/3) + 6^(1/3)*(4*3^(1/3) + 2^(1/3)*(27 + Sqrt[717])^(2/3)*(241 + 9*Sqrt[717]))
)^3 + (58773123072*(20*(27 + Sqrt[717])^(2/3)*(58079 + 2169*Sqrt[717]) - 2^(1/3)*((1100261*3^(2/3) + 123270*3^
(1/6)*Sqrt[239])*(27 + Sqrt[717])^(1/3) + 6^(1/3)*(8676210 + 324019*Sqrt[717])))*Log[2^(1/3)*3^(1/6)*(9*Sqrt[3
] + Sqrt[239]) + 2*2^(2/3)*(3/(27 + Sqrt[717]))^(1/3) - 2*(27 + Sqrt[717])^(1/3) + 2*2^(1/3)*3^(2/3)*x + 3^(1/
3)*(2*(27 + Sqrt[717]))^(2/3)*x + 6*(27 + Sqrt[717])^(1/3)*x^2])/((27 + Sqrt[717])^(14/3)*(6 - 6*3^(1/3)*(2/(2
7 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^4*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1
/3)*(3*(27 + Sqrt[717]))^(2/3))^3)
Rule 6
Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] && !FreeQ[v, x]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])
Rule 27
Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
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 638
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Rule 740
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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 893
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))
Rule 1586
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Rule 1628
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Rule 1646
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Rule 2065
Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27
*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[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}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]
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 2194
Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6741
Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
Aborted
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Mathematica [A] time = 0.04, size = 48, normalized size = 1.37 \begin {gather*} \frac {1}{2} \left (2 e^x-\frac {4}{3 x^2}-\frac {5}{9 x}+\frac {-23+12 x+5 x^2}{9 \left (-3-x+x^3\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(24 + 21*x + 6*x^2 - 22*x^3 - 12*x^4 + 6*x^5 + E^x*(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9
))/(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9),x]
[Out]
(2*E^x - 4/(3*x^2) - 5/(9*x) + (-23 + 12*x + 5*x^2)/(9*(-3 - x + x^3)))/2
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fricas [A] time = 0.78, size = 46, normalized size = 1.31 \begin {gather*} -\frac {2 \, x^{2} - 2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )} e^{x} - 3 \, x - 4}{2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*
x^6+2*x^5+12*x^4+18*x^3),x, algorithm="fricas")
[Out]
-1/2*(2*x^2 - 2*(x^5 - x^3 - 3*x^2)*e^x - 3*x - 4)/(x^5 - x^3 - 3*x^2)
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giac [A] time = 0.26, size = 49, normalized size = 1.40 \begin {gather*} \frac {2 \, x^{5} e^{x} - 2 \, x^{3} e^{x} - 6 \, x^{2} e^{x} - 2 \, x^{2} + 3 \, x + 4}{2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*
x^6+2*x^5+12*x^4+18*x^3),x, algorithm="giac")
[Out]
1/2*(2*x^5*e^x - 2*x^3*e^x - 6*x^2*e^x - 2*x^2 + 3*x + 4)/(x^5 - x^3 - 3*x^2)
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maple [A] time = 0.11, size = 28, normalized size = 0.80
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result |
size |
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| risch |
\(\frac {-x^{2}+\frac {3}{2} x +2}{x^{2} \left (x^{3}-x -3\right )}+{\mathrm e}^{x}\) |
\(28\) |
| norman |
\(\frac {2+x^{5} {\mathrm e}^{x}-x^{2}+\frac {3 x}{2}-3 \,{\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x^{3}}{x^{2} \left (x^{3}-x -3\right )}\) |
\(45\) |
| default |
\({\mathrm e}^{x}+\frac {-\frac {539}{1434} x^{2}-\frac {14}{239} x +\frac {917}{1434}}{x^{3}-x -3}-\frac {2}{3 x^{2}}-\frac {5}{18 x}-\frac {6 \left (-\frac {27}{239} x^{2}+\frac {2}{239} x +\frac {18}{239}\right )}{x^{3}-x -3}-\frac {11 \left (\frac {6}{239} x^{2}-\frac {27}{239} x -\frac {4}{239}\right )}{x^{3}-x -3}-\frac {\frac {12}{239} x^{2}-\frac {54}{239} x +\frac {231}{239}}{3 \left (x^{3}-x -3\right )}-\frac {4 \left (-\frac {131}{239} x^{2}+\frac {231}{239} x +\frac {167}{239}\right )}{9 \left (x^{3}-x -3\right )}+\frac {{\mathrm e}^{x} \left (2 x^{2}-9 x -81\right )}{239 x^{3}-239 x -717}+\frac {\frac {6}{239} x^{2}-\frac {27}{239} x -\frac {243}{239}}{x^{3}-x -3}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (239 \textit {\_R}^{2}+12 \textit {\_R} -347\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )}{717}-\frac {6 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (27 \textit {\_R1}^{2}-29 \textit {\_R1} -14\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -x +\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-1}\right )}{239}+\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (2 \textit {\_R1}^{2}-11 \textit {\_R1} -63\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -x +\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-1}\right )}{239}-\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (106 \textit {\_R1}^{2}+612 \textit {\_R1} +963\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -x +\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-1}\right )}{239}-\frac {60 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (\textit {\_R} -9\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )}{239}+\frac {7 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (239 \textit {\_R}^{2}-474 \textit {\_R} -275\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )}{2151}+\frac {9 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (6 \textit {\_R1}^{2}-33 \textit {\_R1} +50\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -x +\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-1}\right )}{239}-\frac {6 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (-27 \textit {\_R} +4\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )}{239}+\frac {2 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (79 \textit {\_R1}^{2}+163 \textit {\_R1} +21\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -x +\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-1}\right )}{239}-\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (239 \textit {\_R}^{2}-609 \textit {\_R} +940\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )}{2151}+\frac {6 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}-\textit {\_Z} -3\right )}{\sum }\frac {\left (9 \textit {\_R1}^{2}+70 \textit {\_R1} +75\right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -x +\textit {\_R1} \right )}{3 \textit {\_R1}^{2}-1}\right )}{239}+\frac {2 \,{\mathrm e}^{x} \left (79 x^{2}+3 x +27\right )}{239 \left (x^{3}-x -3\right )}+\frac {6 \,{\mathrm e}^{x} \left (9 x^{2}+79 x -6\right )}{239 \left (x^{3}-x -3\right )}-\frac {{\mathrm e}^{x} \left (106 x^{2}+240 x +9\right )}{239 \left (x^{3}-x -3\right )}+\frac {9 \,{\mathrm e}^{x} \left (6 x^{2}-27 x -4\right )}{239 \left (x^{3}-x -3\right )}-\frac {6 \,{\mathrm e}^{x} \left (27 x^{2}-2 x -18\right )}{239 \left (x^{3}-x -3\right )}\) |
\(736\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*x^6+2*
x^5+12*x^4+18*x^3),x,method=_RETURNVERBOSE)
[Out]
(-x^2+3/2*x+2)/x^2/(x^3-x-3)+exp(x)
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maxima [A] time = 0.42, size = 46, normalized size = 1.31 \begin {gather*} -\frac {2 \, x^{2} - 2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )} e^{x} - 3 \, x - 4}{2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*
x^6+2*x^5+12*x^4+18*x^3),x, algorithm="maxima")
[Out]
-1/2*(2*x^2 - 2*(x^5 - x^3 - 3*x^2)*e^x - 3*x - 4)/(x^5 - x^3 - 3*x^2)
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mupad [B] time = 1.64, size = 28, normalized size = 0.80 \begin {gather*} {\mathrm {e}}^x-\frac {-x^2+\frac {3\,x}{2}+2}{x^2\,\left (-x^3+x+3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((21*x + 6*x^2 - 22*x^3 - 12*x^4 + 6*x^5 + exp(x)*(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9) + 24)/
(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9),x)
[Out]
exp(x) - ((3*x)/2 - x^2 + 2)/(x^2*(x - x^3 + 3))
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sympy [A] time = 0.21, size = 27, normalized size = 0.77 \begin {gather*} \frac {- 2 x^{2} + 3 x + 4}{2 x^{5} - 2 x^{3} - 6 x^{2}} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x**9-4*x**7-12*x**6+2*x**5+12*x**4+18*x**3)*exp(x)+6*x**5-12*x**4-22*x**3+6*x**2+21*x+24)/(2*x**
9-4*x**7-12*x**6+2*x**5+12*x**4+18*x**3),x)
[Out]
(-2*x**2 + 3*x + 4)/(2*x**5 - 2*x**3 - 6*x**2) + exp(x)
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