3.1.51 \(\int \frac {1}{(3-2 x+x^2)^{21/2} (1+x+2 x^2)^{10}} \, dx\) [51]

Optimal. Leaf size=638 \[ \frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1117646664729238460189-568839749685437871554 x}{31282116146400000000000000 \left (3-2 x+x^2\right )^{7/2}}-\frac {6551405511565449301689-3127298559983309301910 x}{521368602440000000000000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {4179039782398459850819-1886993445589652402694 x}{1042737204880000000000000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {12105495874518671061833-5117656435043679338190 x}{10427372048800000000000000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}+\frac {\sqrt {\frac {1}{70} \left (81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )}} \left (272944589523248381749+191941026386645109841 \sqrt {2}+\left (656826642296538601431+464885615909893491590 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{32282885600000000000000000}-\frac {\sqrt {\frac {1}{70} \left (-81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (-81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )}} \left (272944589523248381749-191941026386645109841 \sqrt {2}+\left (656826642296538601431-464885615909893491590 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{32282885600000000000000000} \]

[Out]

1/1840124479200000000*(37358055634422583-14024622879097678*x)/(x^2-2*x+3)^(19/2)+1/104273720488000000000*(4768
49951294984711-125181871472148210*x)/(x^2-2*x+3)^(17/2)+1/15641058073200000000000*(7851758375483333511+1942164
996204584234*x)/(x^2-2*x+3)^(15/2)-11/406667509903200000000000*(7502325106308201089-7813986379726516886*x)/(x^
2-2*x+3)^(13/2)-3/1147010925368000000000000*(69053268515296359011-44840736195018286006*x)/(x^2-2*x+3)^(11/2)+1
/9384634843920000000000000*(-838519439380295335657+466189390555853643870*x)/(x^2-2*x+3)^(9/2)+1/31282116146400
000000000000*(-1117646664729238460189+568839749685437871554*x)/(x^2-2*x+3)^(7/2)+1/521368602440000000000000000
*(-6551405511565449301689+3127298559983309301910*x)/(x^2-2*x+3)^(5/2)+1/1042737204880000000000000000*(-4179039
782398459850819+1886993445589652402694*x)/(x^2-2*x+3)^(3/2)+1/630*(-1+10*x)/(x^2-2*x+3)^(19/2)/(2*x^2+x+1)^9+1
/88200*(887+2218*x)/(x^2-2*x+3)^(19/2)/(2*x^2+x+1)^8+1/1080450*(14453+29371*x)/(x^2-2*x+3)^(19/2)/(2*x^2+x+1)^
7+1/605052000*(8837931+17459234*x)/(x^2-2*x+3)^(19/2)/(2*x^2+x+1)^6+1/26471025000*(447940041+813432205*x)/(x^2
-2*x+3)^(19/2)/(2*x^2+x+1)^5+1/29647548000000*(592729157441+911061463974*x)/(x^2-2*x+3)^(19/2)/(2*x^2+x+1)^4+1
/12353145000000*(277010166219+310705340015*x)/(x^2-2*x+3)^(19/2)/(2*x^2+x+1)^3+1/276710448000000*(548822129434
9+1384103301166*x)/(x^2-2*x+3)^(19/2)/(2*x^2+x+1)^2+1/2421216420000000*(-37857197792117-146548895467025*x)/(x^
2-2*x+3)^(19/2)/(2*x^2+x+1)+1/10427372048800000000000000000*(-12105495874518671061833+5117656435043679338190*x
)/(x^2-2*x+3)^(1/2)-1/2259801992000000000000000000*arctanh(1/7*(272944589523248381749+x*(656826642296538601431
-464885615909893491590*2^(1/2))-191941026386645109841*2^(1/2))*35^(1/2)/(-810422259212746896054789447978008548
46405+57305922523001707126026363878666500308992*2^(1/2))^(1/2)/(x^2-2*x+3)^(1/2))*(-56729558144892282723835261
35846059839248350+4011414576610119498821845471506655021629440*2^(1/2))^(1/2)+1/2259801992000000000000000000*ar
ctan(1/7*(272944589523248381749+191941026386645109841*2^(1/2)+x*(656826642296538601431+464885615909893491590*2
^(1/2)))*35^(1/2)/(81042225921274689605478944797800854846405+57305922523001707126026363878666500308992*2^(1/2)
)^(1/2)/(x^2-2*x+3)^(1/2))*(5672955814489228272383526135846059839248350+40114145766101194988218454715066550216
29440*2^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.87, antiderivative size = 638, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {988, 1074, 1049, 1043, 212, 210} \begin {gather*} \frac {\sqrt {\frac {1}{70} \left (81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {5}{7 \left (81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )}} \left (\left (656826642296538601431+464885615909893491590 \sqrt {2}\right ) x+191941026386645109841 \sqrt {2}+272944589523248381749\right )}{\sqrt {x^2-2 x+3}}\right )}{32282885600000000000000000}-\frac {12105495874518671061833-5117656435043679338190 x}{10427372048800000000000000000 \sqrt {x^2-2 x+3}}-\frac {146548895467025 x+37857197792117}{2421216420000000 \left (x^2-2 x+3\right )^{19/2} \left (2 x^2+x+1\right )}-\frac {4179039782398459850819-1886993445589652402694 x}{1042737204880000000000000000 \left (x^2-2 x+3\right )^{3/2}}+\frac {1384103301166 x+5488221294349}{276710448000000 \left (x^2-2 x+3\right )^{19/2} \left (2 x^2+x+1\right )^2}-\frac {6551405511565449301689-3127298559983309301910 x}{521368602440000000000000000 \left (x^2-2 x+3\right )^{5/2}}+\frac {310705340015 x+277010166219}{12353145000000 \left (x^2-2 x+3\right )^{19/2} \left (2 x^2+x+1\right )^3}-\frac {1117646664729238460189-568839749685437871554 x}{31282116146400000000000000 \left (x^2-2 x+3\right )^{7/2}}+\frac {911061463974 x+592729157441}{29647548000000 \left (x^2-2 x+3\right )^{19/2} \left (2 x^2+x+1\right )^4}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (x^2-2 x+3\right )^{9/2}}+\frac {813432205 x+447940041}{26471025000 \left (x^2-2 x+3\right )^{19/2} \left (2 x^2+x+1\right )^5}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (x^2-2 x+3\right )^{11/2}}+\frac {17459234 x+8837931}{605052000 \left (x^2-2 x+3\right )^{19/2} \left (2 x^2+x+1\right )^6}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (x^2-2 x+3\right )^{13/2}}+\frac {29371 x+14453}{1080450 \left (x^2-2 x+3\right )^{19/2} \left (2 x^2+x+1\right )^7}+\frac {1942164996204584234 x+7851758375483333511}{15641058073200000000000 \left (x^2-2 x+3\right )^{15/2}}+\frac {2218 x+887}{88200 \left (x^2-2 x+3\right )^{19/2} \left (2 x^2+x+1\right )^8}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (x^2-2 x+3\right )^{17/2}}-\frac {1-10 x}{630 \left (x^2-2 x+3\right )^{19/2} \left (2 x^2+x+1\right )^9}+\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (x^2-2 x+3\right )^{19/2}}-\frac {\sqrt {\frac {1}{70} \left (57305922523001707126026363878666500308992 \sqrt {2}-81042225921274689605478944797800854846405\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (57305922523001707126026363878666500308992 \sqrt {2}-81042225921274689605478944797800854846405\right )}} \left (\left (656826642296538601431-464885615909893491590 \sqrt {2}\right ) x-191941026386645109841 \sqrt {2}+272944589523248381749\right )}{\sqrt {x^2-2 x+3}}\right )}{32282885600000000000000000} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((3 - 2*x + x^2)^(21/2)*(1 + x + 2*x^2)^10),x]

[Out]

(37358055634422583 - 14024622879097678*x)/(1840124479200000000*(3 - 2*x + x^2)^(19/2)) + (476849951294984711 -
125181871472148210*x)/(104273720488000000000*(3 - 2*x + x^2)^(17/2)) + (7851758375483333511 + 194216499620458
4234*x)/(15641058073200000000000*(3 - 2*x + x^2)^(15/2)) - (11*(7502325106308201089 - 7813986379726516886*x))/
(406667509903200000000000*(3 - 2*x + x^2)^(13/2)) - (3*(69053268515296359011 - 44840736195018286006*x))/(11470
10925368000000000000*(3 - 2*x + x^2)^(11/2)) - (838519439380295335657 - 466189390555853643870*x)/(938463484392
0000000000000*(3 - 2*x + x^2)^(9/2)) - (1117646664729238460189 - 568839749685437871554*x)/(3128211614640000000
0000000*(3 - 2*x + x^2)^(7/2)) - (6551405511565449301689 - 3127298559983309301910*x)/(521368602440000000000000
000*(3 - 2*x + x^2)^(5/2)) - (4179039782398459850819 - 1886993445589652402694*x)/(1042737204880000000000000000
*(3 - 2*x + x^2)^(3/2)) - (12105495874518671061833 - 5117656435043679338190*x)/(10427372048800000000000000000*
Sqrt[3 - 2*x + x^2]) - (1 - 10*x)/(630*(3 - 2*x + x^2)^(19/2)*(1 + x + 2*x^2)^9) + (887 + 2218*x)/(88200*(3 -
2*x + x^2)^(19/2)*(1 + x + 2*x^2)^8) + (14453 + 29371*x)/(1080450*(3 - 2*x + x^2)^(19/2)*(1 + x + 2*x^2)^7) +
(8837931 + 17459234*x)/(605052000*(3 - 2*x + x^2)^(19/2)*(1 + x + 2*x^2)^6) + (447940041 + 813432205*x)/(26471
025000*(3 - 2*x + x^2)^(19/2)*(1 + x + 2*x^2)^5) + (592729157441 + 911061463974*x)/(29647548000000*(3 - 2*x +
x^2)^(19/2)*(1 + x + 2*x^2)^4) + (277010166219 + 310705340015*x)/(12353145000000*(3 - 2*x + x^2)^(19/2)*(1 + x
+ 2*x^2)^3) + (5488221294349 + 1384103301166*x)/(276710448000000*(3 - 2*x + x^2)^(19/2)*(1 + x + 2*x^2)^2) -
(37857197792117 + 146548895467025*x)/(2421216420000000*(3 - 2*x + x^2)^(19/2)*(1 + x + 2*x^2)) + (Sqrt[(810422
25921274689605478944797800854846405 + 57305922523001707126026363878666500308992*Sqrt[2])/70]*ArcTan[(Sqrt[5/(7
*(81042225921274689605478944797800854846405 + 57305922523001707126026363878666500308992*Sqrt[2]))]*(2729445895
23248381749 + 191941026386645109841*Sqrt[2] + (656826642296538601431 + 464885615909893491590*Sqrt[2])*x))/Sqrt
[3 - 2*x + x^2]])/32282885600000000000000000 - (Sqrt[(-81042225921274689605478944797800854846405 + 57305922523
001707126026363878666500308992*Sqrt[2])/70]*ArcTanh[(Sqrt[5/(7*(-81042225921274689605478944797800854846405 + 5
7305922523001707126026363878666500308992*Sqrt[2]))]*(272944589523248381749 - 191941026386645109841*Sqrt[2] + (
656826642296538601431 - 464885615909893491590*Sqrt[2])*x))/Sqrt[3 - 2*x + x^2]])/32282885600000000000000000

Rule 210

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

Rule 212

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

Rule 988

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(2*a*
c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
1)*((d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Dist[1/
((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*
x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e
- b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1043

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]

Rule 1049

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x
+ c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e
^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]

Rule 1074

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^
2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e
- 2*a*(c*d - a*f)))*x), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
(b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )^{10}} \, dx &=-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}-\frac {\int \frac {-2960+3060 x-1800 x^2}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )^9} \, dx}{3150}\\ &=-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}-\frac {\int \frac {-8066100+8650900 x-7541200 x^2}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )^8} \, dx}{8820000}\\ &=-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}-\frac {\int \frac {-18577805000+18950890000 x-18797440000 x^2}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )^7} \, dx}{21609000000}\\ &=-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}-\frac {\int \frac {-34422218025000+37067282625000 x-39283276500000 x^2}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )^6} \, dx}{45378900000000}\\ &=-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}-\frac {\int \frac {-47542711206750000+57420932725500000 x-68328305220000000 x^2}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )^5} \, dx}{79413075000000000}\\ &=-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}-\frac {\int \frac {-40751213836916250000+59562989955686250000 x-88828492737465000000 x^2}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )^4} \, dx}{111178305000000000000}\\ &=-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}-\frac {\int \frac {-7802817431641312500000+24109394856584625000000 x-70467971115402000000000 x^2}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )^3} \, dx}{116737220250000000000000}\\ &=-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {\int \frac {16833881379064542187500000-26649709913445904687500000 x-8992346134762856250000000 x^2}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )^2} \, dx}{81716054175000000000000000}\\ &=-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {-6456478383150221671875000000-27827888668333982156250000000 x+34622176554084656250000000000 x^2}{\left (3-2 x+x^2\right )^{21/2} \left (1+x+2 x^2\right )} \, dx}{28600618961250000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {4967712444964210062187500000000-37459045941891614735625000000000 x+29819854396681437847500000000000 x^2}{\left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )} \, dx}{108682352052750000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {14762203931705757912393750000000000-35273795655183209407237500000000000 x+14195624224941607014000000000000000 x^2}{\left (3-2 x+x^2\right )^{17/2} \left (1+x+2 x^2\right )} \, dx}{369519996979350000000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {17682321750123664132939125000000000000-22751442786535433811492750000000000000 x-3854226434967997412373000000000000000 x^2}{\left (3-2 x+x^2\right )^{15/2} \left (1+x+2 x^2\right )} \, dx}{1108559990938050000000000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {13399310005134207269200312500000000000000-7942120150303904451111225000000000000000 x-14620749915106285745394600000000000000000 x^2}{\left (3-2 x+x^2\right )^{13/2} \left (1+x+2 x^2\right )} \, dx}{2882255976438930000000000000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {6370652021675523452523402750000000000000000+1397569502668803779552737500000000000000000 x-14873447992206590376760170000000000000000000 x^2}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )} \, dx}{6340963148165646000000000000000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {1329917982742483936025158935000000000000000000+3615108788879230023346255290000000000000000000 x-9071784474158200631669632800000000000000000000 x^2}{\left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )} \, dx}{11413733666698162800000000000000000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1117646664729238460189-568839749685437871554 x}{31282116146400000000000000 \left (3-2 x+x^2\right )^{7/2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {-420915070441561410503321633100000000000000000000+2181291929685915536692331929800000000000000000000 x-3486830438264921097598322474400000000000000000000 x^2}{\left (3-2 x+x^2\right )^{7/2} \left (1+x+2 x^2\right )} \, dx}{15979227133377427920000000000000000000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1117646664729238460189-568839749685437871554 x}{31282116146400000000000000 \left (3-2 x+x^2\right )^{7/2}}-\frac {6551405511565449301689-3127298559983309301910 x}{521368602440000000000000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {-379110186022081999740742476990000000000000000000000+666051943754340912077121512700000000000000000000000 x-766779031495028265360690083040000000000000000000000 x^2}{\left (3-2 x+x^2\right )^{5/2} \left (1+x+2 x^2\right )} \, dx}{15979227133377427920000000000000000000000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1117646664729238460189-568839749685437871554 x}{31282116146400000000000000 \left (3-2 x+x^2\right )^{7/2}}-\frac {6551405511565449301689-3127298559983309301910 x}{521368602440000000000000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {4179039782398459850819-1886993445589652402694 x}{1042737204880000000000000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {-96896452988962143502089877773000000000000000000000000+91746249937845836577877948114800000000000000000000000 x-69400489538857249491540692270400000000000000000000000 x^2}{\left (3-2 x+x^2\right )^{3/2} \left (1+x+2 x^2\right )} \, dx}{9587536280026456752000000000000000000000000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1117646664729238460189-568839749685437871554 x}{31282116146400000000000000 \left (3-2 x+x^2\right )^{7/2}}-\frac {6551405511565449301689-3127298559983309301910 x}{521368602440000000000000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {4179039782398459850819-1886993445589652402694 x}{1042737204880000000000000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {12105495874518671061833-5117656435043679338190 x}{10427372048800000000000000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {-6260702387317500351852137783268000000000000000000000000+2570011459509637702693150504488000000000000000000000000 x}{\sqrt {3-2 x+x^2} \left (1+x+2 x^2\right )} \, dx}{1917507256005291350400000000000000000000000000000000000000000}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1117646664729238460189-568839749685437871554 x}{31282116146400000000000000 \left (3-2 x+x^2\right )^{7/2}}-\frac {6551405511565449301689-3127298559983309301910 x}{521368602440000000000000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {4179039782398459850819-1886993445589652402694 x}{1042737204880000000000000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {12105495874518671061833-5117656435043679338190 x}{10427372048800000000000000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {296985108420000000000000000000000000 \left (148672670724261260159-105404315061877410477 \sqrt {2}\right )-296985108420000000000000000000000000 \left (62135959399493560795-43268355662383849682 \sqrt {2}\right ) x}{\sqrt {3-2 x+x^2} \left (1+x+2 x^2\right )} \, dx}{19175072560052913504000000000000000000000000000000000000000000 \sqrt {2}}+\frac {\int \frac {296985108420000000000000000000000000 \left (148672670724261260159+105404315061877410477 \sqrt {2}\right )-296985108420000000000000000000000000 \left (62135959399493560795+43268355662383849682 \sqrt {2}\right ) x}{\sqrt {3-2 x+x^2} \left (1+x+2 x^2\right )} \, dx}{19175072560052913504000000000000000000000000000000000000000000 \sqrt {2}}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1117646664729238460189-568839749685437871554 x}{31282116146400000000000000 \left (3-2 x+x^2\right )^{7/2}}-\frac {6551405511565449301689-3127298559983309301910 x}{521368602440000000000000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {4179039782398459850819-1886993445589652402694 x}{1042737204880000000000000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {12105495874518671061833-5117656435043679338190 x}{10427372048800000000000000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}-\frac {\left (3788075362500000 \left (114611845046003414252052727757333000617984-81042225921274689605478944797800854846405 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-617401082362674084274800000000000000000000000000000000000000000000000000 \left (81042225921274689605478944797800854846405-57305922523001707126026363878666500308992 \sqrt {2}\right )-5 x^2} \, dx,x,\frac {296985108420000000000000000000000000 \left (272944589523248381749-191941026386645109841 \sqrt {2}\right )+296985108420000000000000000000000000 \left (656826642296538601431-464885615909893491590 \sqrt {2}\right ) x}{\sqrt {3-2 x+x^2}}\right )}{823543}-\frac {\left (3788075362500000 \left (114611845046003414252052727757333000617984+81042225921274689605478944797800854846405 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-617401082362674084274800000000000000000000000000000000000000000000000000 \left (81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )-5 x^2} \, dx,x,\frac {296985108420000000000000000000000000 \left (272944589523248381749+191941026386645109841 \sqrt {2}\right )+296985108420000000000000000000000000 \left (656826642296538601431+464885615909893491590 \sqrt {2}\right ) x}{\sqrt {3-2 x+x^2}}\right )}{823543}\\ &=\frac {37358055634422583-14024622879097678 x}{1840124479200000000 \left (3-2 x+x^2\right )^{19/2}}+\frac {476849951294984711-125181871472148210 x}{104273720488000000000 \left (3-2 x+x^2\right )^{17/2}}+\frac {7851758375483333511+1942164996204584234 x}{15641058073200000000000 \left (3-2 x+x^2\right )^{15/2}}-\frac {11 (7502325106308201089-7813986379726516886 x)}{406667509903200000000000 \left (3-2 x+x^2\right )^{13/2}}-\frac {3 (69053268515296359011-44840736195018286006 x)}{1147010925368000000000000 \left (3-2 x+x^2\right )^{11/2}}-\frac {838519439380295335657-466189390555853643870 x}{9384634843920000000000000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1117646664729238460189-568839749685437871554 x}{31282116146400000000000000 \left (3-2 x+x^2\right )^{7/2}}-\frac {6551405511565449301689-3127298559983309301910 x}{521368602440000000000000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {4179039782398459850819-1886993445589652402694 x}{1042737204880000000000000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {12105495874518671061833-5117656435043679338190 x}{10427372048800000000000000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{630 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^9}+\frac {887+2218 x}{88200 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^8}+\frac {14453+29371 x}{1080450 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^7}+\frac {8837931+17459234 x}{605052000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^6}+\frac {447940041+813432205 x}{26471025000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^5}+\frac {592729157441+911061463974 x}{29647548000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^4}+\frac {277010166219+310705340015 x}{12353145000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^3}+\frac {5488221294349+1384103301166 x}{276710448000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )^2}-\frac {37857197792117+146548895467025 x}{2421216420000000 \left (3-2 x+x^2\right )^{19/2} \left (1+x+2 x^2\right )}+\frac {\sqrt {\frac {1}{70} \left (81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )}} \left (272944589523248381749+191941026386645109841 \sqrt {2}+\left (656826642296538601431+464885615909893491590 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{32282885600000000000000000}-\frac {\sqrt {\frac {1}{70} \left (-81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (-81042225921274689605478944797800854846405+57305922523001707126026363878666500308992 \sqrt {2}\right )}} \left (272944589523248381749-191941026386645109841 \sqrt {2}+\left (656826642296538601431-464885615909893491590 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{32282885600000000000000000}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 16.85, size = 1431, normalized size = 2.24 \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((3 - 2*x + x^2)^(21/2)*(1 + x + 2*x^2)^10),x]

[Out]

Sqrt[3 - 2*x + x^2]*((1 - x)/(11875000000*(3 - 2*x + x^2)^10) + (265 - 113*x)/(403750000000*(3 - 2*x + x^2)^9)
+ (82361 - 4841*x)/(60562500000000*(3 - 2*x + x^2)^8) + (1062937 + 1642511*x)/(1574625000000000*(3 - 2*x + x^
2)^7) + (7*(-678331 + 833371*x))/(2220625000000000*(3 - 2*x + x^2)^6) + (7*(-73161291 + 43964675*x))/(90843750
000000000*(3 - 2*x + x^2)^5) + (-1340879383 + 430593031*x)/(181687500000000000*(3 - 2*x + x^2)^4) - (11*(16261
25723 + 112950205*x))/(3028125000000000000*(3 - 2*x + x^2)^3) - (11*(3311570647 + 15286717673*x))/(36337500000
000000000*(3 - 2*x + x^2)^2) - (11*(-411521923277 + 484788625685*x))/(363375000000000000000*(3 - 2*x + x^2)) +
(251943 + 221770*x)/(6300000000000*(1 + x + 2*x^2)^9) - (73*(-888423 + 1604678*x))/(882000000000000*(1 + x +
2*x^2)^8) + (-2596903794 - 4965311863*x)/(10804500000000000*(1 + x + 2*x^2)^7) + (-539608494637 - 334647150510
*x)/(1210104000000000000*(1 + x + 2*x^2)^6) + (-40800462989458 + 56711874696335*x)/(264710250000000000000*(1 +
x + 2*x^2)^5) + (42018358198215561 + 129196597088670934*x)/(296475480000000000000000*(1 + x + 2*x^2)^4) + (62
819559864314747 + 169630389653846945*x)/(370594350000000000000000*(1 + x + 2*x^2)^3) + (1082422109196374795 +
4797048907791526114*x)/(8301313440000000000000000*(1 + x + 2*x^2)^2) + (65571203144429922747 + 367152793968978
953465*x)/(363182463000000000000000000*(1 + x + 2*x^2))) + ((232442807954946745795*I + 21634177831191924841*Sq
rt[7])*ArcTan[(-135063738860435016899586558948733259113515 + (188630894626466690216855285995045889396405*I)*Sq
rt[7] - 1506241361872688008559268776761430483700000*x - (105711500937472192718115651350352447938680*I)*Sqrt[7]
*x + 491153540508443587025809789813541985707360*x^2 - (460764064177139993399975100872663310399420*I)*Sqrt[7]*x
^2 - 180084985147246689199448745264977678818020*x^3 + (197868296377913870863837680953446009396860*I)*Sqrt[7]*x
^3 - 176004816500761880926774485599831047775825*x^4 - (207342833228459577163557043035558264835165*I)*Sqrt[7]*x
^4 + (186244248199755548159585682605666126004224*I)*Sqrt[10*(-5 + I*Sqrt[7])]*Sqrt[3 - 2*x + x^2] + (114611845
046003414252052727757333000617984*I)*Sqrt[10*(-5 + I*Sqrt[7])]*x*Sqrt[3 - 2*x + x^2] + (3008560932457589624116
38410362999126622208*I)*Sqrt[10*(-5 + I*Sqrt[7])]*x^2*Sqrt[3 - 2*x + x^2] - (143264806307504267815065909696666
250772480*I)*Sqrt[10*(-5 + I*Sqrt[7])]*x^3*Sqrt[3 - 2*x + x^2])/(2368773290838836979864678493023884746594823*I
+ 423642940259238735473942663180025956729505*Sqrt[7] + (1890613486065620301760074218556745311646936*I)*x + 61
50574559311228258394328777942059796320*Sqrt[7]*x + (2511300259855822962340893027852239157667820*I)*x^2 - 20278
67550801106189867763431094227596320*Sqrt[7]*x^2 - (3134217746230760357128318797499380812303788*I)*x^3 + 634304
31602720043279192866968369397935660*Sqrt[7]*x^3 + (944749064886626467328385369190460703669697*I)*x^4 + 1638131
7765107264789462917221030750634835*Sqrt[7]*x^4)])/(16141442800000000000000000*Sqrt[70*(-5 + I*Sqrt[7])]) - ((I
/16141442800000000000000000)*(-232442807954946745795*I + 21634177831191924841*Sqrt[7])*ArcTan[(35*(43624942906
63946676585186218212607628595*I + 12104084007406821013541218948000741620843*Sqrt[7] - (40919031596617332707196
094500783237405000*I)*x + 175730701694606521668409393655487422752*Sqrt[7]*x + (2648728832926512757773396585336
4310310620*I)*x^2 - 57939072880031605424793240888406502752*Sqrt[7]*x^2 - (152388941497528256839248140210078630
70620*I)*x^3 + 1812298045792001236548367627667697083876*Sqrt[7]*x^3 - (795837271959975808913244203765619963595
*I)*x^4 + 468037650431636136841797634886592875281*Sqrt[7]*x^4))/(135063738860435016899586558948733259113515 +
(188630894626466690216855285995045889396405*I)*Sqrt[7] + 1506241361872688008559268776761430483700000*x - (1057
11500937472192718115651350352447938680*I)*Sqrt[7]*x - 491153540508443587025809789813541985707360*x^2 - (460764
064177139993399975100872663310399420*I)*Sqrt[7]*x^2 + 180084985147246689199448745264977678818020*x^3 + (197868
296377913870863837680953446009396860*I)*Sqrt[7]*x^3 + 176004816500761880926774485599831047775825*x^4 - (207342
833228459577163557043035558264835165*I)*Sqrt[7]*x^4 - (14326480630750426781506590969666625077248*I)*Sqrt[70*(5
+ I*Sqrt[7])]*Sqrt[3 - 2*x + x^2] - (14326480630750426781506590969666625077248*I)*Sqrt[70*(5 + I*Sqrt[7])]*x^
2*Sqrt[3 - 2*x + x^2] + (28652961261500853563013181939333250154496*I)*Sqrt[70*(5 + I*Sqrt[7])]*x^3*Sqrt[3 - 2*
x + x^2])])/Sqrt[70*(5 + I*Sqrt[7])] - ((-232442807954946745795*I + 21634177831191924841*Sqrt[7])*Log[(-I + Sq
rt[7] - (4*I)*x)^2*(I + Sqrt[7] + (4*I)*x)^2])/(32282885600000000000000000*Sqrt[70*(5 + I*Sqrt[7])]) + ((I/322
82885600000000000000000)*(232442807954946745795*I + 21634177831191924841*Sqrt[7])*Log[(-I + Sqrt[7] - (4*I)*x)
^2*(I + Sqrt[7] + (4*I)*x)^2])/Sqrt[70*(-5 + I*Sqrt[7])] - ((I/32282885600000000000000000)*(232442807954946745
795*I + 21634177831191924841*Sqrt[7])*Log[(1 + x + 2*x^2)*(-13*I + 15*Sqrt[7] + (22*I)*x - 10*Sqrt[7]*x + (9*I
)*x^2 + 5*Sqrt[7]*x^2 + I*Sqrt[70*(-5 + I*Sqrt[...

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(86792\) vs. \(2(518)=1036\).
time = 4.55, size = 86793, normalized size = 136.04

method	result	size
risch	\(\frac {3372249001933422237824271360 x^{37}-53502205399640031394796147712 x^{36}+469149394082989701729494575872 x^{35}-2847499220912667753383035299072 x^{34}+13254252261100740556512388253568 x^{33}-49770080058525077628064229832576 x^{32}+156010734937008739388220889457760 x^{31}-417516398850754397130111919794336 x^{30}+971538171913365251873706873353652 x^{29}-1993653213575521837888601204380228 x^{28}+3655553471852957606257345414140031 x^{27}-6054769996581738503753686155104785 x^{26}+9155494158513869230271529746307221 x^{25}-12740106677685048178693605103009787 x^{24}+16442770202470076313197215936814318 x^{23}-19772569734288744720189854470201506 x^{22}+22286437617621909921609206629636086 x^{21}-23584986647560742443188031208946882 x^{20}+23579397211179175240196614296051673 x^{19}-22218747553941794885903840542461607 x^{18}+19912295454080246583636391613811979 x^{17}-16801760806053390242995145349148613 x^{16}+13613407965006475288139078599341572 x^{15}-10279305650733178669223634020962076 x^{14}+7606288378303449524327938977040824 x^{13}-5069838234992751929471190426115248 x^{12}+3507425970596197680016078213030977 x^{11}-1974814483061344405275851094534735 x^{10}+1357002388430055881833293557852283 x^{9}-566969010759169461615951049236597 x^{8}+458426000073846882432457044306894 x^{7}-94704557665253489332536549937026 x^{6}+135183920426913231415208872303230 x^{5}-1023095318901774638403186272874 x^{4}+29398041153524973343917601742151 x^{3}+1933957195570062708781629134823 x^{2}+3397462350398947848063583843461 x -80038710871555316861345369643}{13420027826805600000000000000000 \left (x^{2}-2 x +3\right )^{\frac {19}{2}} \left (2 x^{2}+x +1\right )^{9}}+\frac {\sqrt {4}\, \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \sqrt {2}\, \left (7003218138761840939875 \sqrt {-6050+4280 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-6050+4280 \sqrt {2}}\, \left (40 \sqrt {2}+57\right ) \left (\sqrt {2}-1+x \right )}{49 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-350+280 \sqrt {2}}\, \sqrt {2}+9903469297471243727348 \sqrt {-6050+4280 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-6050+4280 \sqrt {2}}\, \left (40 \sqrt {2}+57\right ) \left (\sqrt {2}-1+x \right )}{49 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-350+280 \sqrt {2}}+321845054725303914701190 \arctanh \left (\frac {7 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}}{\sqrt {-350+280 \sqrt {2}}}\right ) \sqrt {2}-455587903591695621758200 \arctanh \left (\frac {7 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}}{\sqrt {-350+280 \sqrt {2}}}\right )\right )}{63274455776000000000000000000 \sqrt {\frac {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}{\left (\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}+1\right )^{2}}}\, \left (\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}+1\right ) \sqrt {-350+280 \sqrt {2}}}\)	\(552\)
trager	\(\text {Expression too large to display}\)	\(641\)
default	\(\text {Expression too large to display}\)	\(86793\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2-2*x+3)^(21/2)/(2*x^2+x+1)^10,x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2-2*x+3)^(21/2)/(2*x^2+x+1)^10,x, algorithm="maxima")

[Out]

integrate(1/((2*x^2 + x + 1)^10*(x^2 - 2*x + 3)^(21/2)), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2775 vs. \(2 (518) = 1036\).
time = 1.29, size = 2775, normalized size = 4.35 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2-2*x+3)^(21/2)/(2*x^2+x+1)^10,x, algorithm="fricas")

[Out]

1/143446642399233767656421653603627286364552203186085711012681333261811289872032000000000000000000*(3604596077
6272236628083717974972055111190660172853358396135728761934386631817942748278579200*x^38 - 55871239203221966773
5297628612066854223455232679227055140103795809982992793178112598317977600*x^37 + 48121357636323435898491763496
58769357343953133075923345884119789718240615347695356895190323200*x^36 - 2871060775830083647426868136706524189
6063360827677699962522107958880738952242991399003888332800*x^35 + 13150918214713222130798493456693867156629022
4808133871438501689421822367827858786874250861388800*x^34 - 48615412586213217241775435956082154303807288316753
4048023582332295256693402493082467454965081600*x^33 + 15012511859003801791455877071511298942846464125440398838
21859865409233818038611634065993336166400*x^32 - 3960120768072419508193345390732915044310906172694579718477009
206696968949880161377086262197766400*x^31 + 909142000002142860704234021157216421398787616081889741815089464543
5329015717575267031369642428000*x^30 - 18424764929872158270698990044243761821209838303936935404825000845297214
002673057816247491027262800*x^29 + 334130737566736389253336251690111704455988115162219751155902004112933894344
16479555356860509015600*x^28 - 5481653256044929545971751700338269967324241093611430434462984210365662293449024
7108012261346586400*x^27 + 82245983094063518667736627604663547588572840238581597325736701493749880383650749401
133206999014400*x^26 - 113722848067639694402592735862649094093874045443618754295078471234595964240139128161766
283626302000*x^25 + 146086574413322248286514192550522624098477614094095488624493581512454991258074867544318895
241990800*x^24 - 175027094081001021682973752997412023251736305226127144272811232619626419165679723993392477178
363200*x^23 + 196887291605784159433455654443374481739030277196290989156609388218395099469530751149958413044135
200*x^22 - 208068683375682167383215047521697995267539026087882795784482813901791360434798005710722616487282000
*x^21 + 208171444918478482519618165392015730347012009814583465001141378703189206795143605224483243158516400*x^
20 - 196227556184540408353167422341576855508320001795821851558311176995574081069015969836642878534431200*x^19
+ 176534941677723459681422280024952573032106299529482816321219585323399086976471958310981405494523200*x^18 - 1
49136255738011380556954829398929258737007615204074730330565887220730783382923822619571340737358000*x^17 + 1218
90814483587724389011961696733756253105383654426234336150913799569962877883235263704480534144400*x^16 - 9198318
6053222129635537069278588580392985745730700928388526309371776740142438834607398588992195200*x^15 + 69317814132
471559316390137037592557060398996838342232414889371690271398098098738643314402130954400*x^14 - 457430708411325
00247970739727093296878765897323708593659902862883667237249390700654758574610918000*x^13 + 3299696552167639492
9803121509049143329451789049169789455644615129199190308917673348518481311574800*x^12 - 17770083757788737971933
739892049927033484890029804651938270182161740937851280707834822272274354400*x^11 + 135442252674514597019603692
38256374351899362683978498551483729852256655264147093337392596228028800*x^10 - 4813759732728488651728668551069
958186240925466978671799825767568732599092879797201593187475517200*x^9 + 5091181133639025216832620106123280320
347641869015804163342220634415255665812683873707564839486000*x^8 - 4642131185030564007583489945718840607733994
62026537769017971996084095803827142837363184426478400*x^7 + 17712338832647821260422671418114138499869713982650
32235916172889879027134542752439323372429279200*x^6 + 23911503454316320991841103252166564975049644786785360906
9487786445410804754998849116452338787600*x^5 + 79817891129994413353362937273464455099835468*126493875280426512
3815574105117799608149057272418^(1/4)*sqrt(1590558865810545927822094)*sqrt(35)*sqrt(2)*(512*x^38 - 7936*x^37 +
68352*x^36 - 407808*x^35 + 1867968*x^34 - 6905376*x^33 + 21323904*x^32 - 56249904*x^31 + 129135330*x^30 - 261
706983*x^29 + 474602241*x^28 - 778618854*x^27 + 1168229184*x^26 - 1615329345*x^25 + 2075026563*x^24 - 24861002
52*x^23 + 2796604422*x^22 - 2955425895*x^21 + 2956885529*x^20 - 2787233482*x^19 + 2507517852*x^18 - 2118344505
*x^17 + 1731347859*x^16 - 1306537272*x^15 + 984596334*x^14 - 649738605*x^13 + 468691803*x^12 - 252407834*x^11
+ 192383368*x^10 - 68375067*x^9 + 72315585*x^8 - 6593724*x^7 + 25158762*x^6 + 3396411*x^5 + 6720651*x^4 + 1325
322*x^3 + 1023516*x^2 + 137781*x + 59049)*sqrt(81042225921274689605478944797800854846405*sqrt(2) + 11461184504
6003414252052727757333000617984)*arctan(1/54206850781156887023310518673090274966005685838243268724684064391985
05135017594564915473395777024743167351056637371274953501437271981836435236061968*sqrt(795279432905272963911047
)*(9939513250523192816422116593216797292815016511001378462170679301990*sqrt(1100522448786287362112823964249088
8848098)*sqrt(2888868076710542715672947094311)*sqrt(7)*(10*sqrt(2) + 9) + sqrt(1590558865810545927822094)*(5*1
26493875280426512381557410511779960814905727241...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2-2*x+3)**(21/2)/(2*x**2+x+1)**10,x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2-2*x+3)^(21/2)/(2*x^2+x+1)^10,x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (2\,x^2+x+1\right )}^{10}\,{\left (x^2-2\,x+3\right )}^{21/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x + 2*x^2 + 1)^10*(x^2 - 2*x + 3)^(21/2)),x)

[Out]

int(1/((x + 2*x^2 + 1)^10*(x^2 - 2*x + 3)^(21/2)), x)

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