3.48 \(\int \frac {1}{(3-2 x)^{21/2} (1+x+2 x^2)^{10}} \, dx\)

Optimal. Leaf size=648 \[ \frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (2 x^2+x+1\right )}+\frac {451 (14627273 x+28962039)}{283351498752 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^2}+\frac {451 (998691 x+811091)}{10119696384 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^3}+\frac {41 (5677637 x+3436375)}{5059848192 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^4}+\frac {41 (92875 x+47471)}{90354432 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^5}+\frac {5 (47471 x+21409)}{6453888 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^6}+\frac {21409 x+8477}{691488 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^7}+\frac {173 x+53}{7056 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^8}+\frac {x}{63 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^9}-\frac {24229218097975}{22757389978742816768 \sqrt {3-2 x}}-\frac {46601678385075}{11378694989371408384 (3-2 x)^{3/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}+\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}+\frac {11275 \left (9756589235-2148932869 \sqrt {14}\right ) \sqrt {\frac {1}{2} \left (2 \sqrt {14}-7\right )} \log \left (-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3\right )}{637206919404798869504}-\frac {11275 \left (9756589235-2148932869 \sqrt {14}\right ) \sqrt {\frac {1}{2} \left (2 \sqrt {14}-7\right )} \log \left (-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3\right )}{637206919404798869504}+\frac {11275 \sqrt {\frac {1}{2} \left (7+2 \sqrt {14}\right )} \left (9756589235+2148932869 \sqrt {14}\right ) \tan ^{-1}\left (\frac {\sqrt {7+2 \sqrt {14}}-2 \sqrt {3-2 x}}{\sqrt {2 \sqrt {14}-7}}\right )}{318603459702399434752}-\frac {11275 \sqrt {\frac {1}{2} \left (7+2 \sqrt {14}\right )} \left (9756589235+2148932869 \sqrt {14}\right ) \tan ^{-1}\left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {2 \sqrt {14}-7}}\right )}{318603459702399434752} \]

[Out]

4718120139975/351733660450816/(3-2*x)^(19/2)-815900548375/629418129227776/(3-2*x)^(17/2)-3029508823715/1555033
025150976/(3-2*x)^(15/2)-13515743021825/13476952884641792/(3-2*x)^(13/2)-5846828446875/14513641568075776/(3-2*
x)^(11/2)-37283626871975/261245548225363968/(3-2*x)^(9/2)-132355162272575/2844673747342852096/(3-2*x)^(7/2)-11
557581705725/812763927812243456/(3-2*x)^(5/2)-46601678385075/11378694989371408384/(3-2*x)^(3/2)+1/63*x/(3-2*x)
^(19/2)/(2*x^2+x+1)^9+1/7056*(53+173*x)/(3-2*x)^(19/2)/(2*x^2+x+1)^8+1/691488*(8477+21409*x)/(3-2*x)^(19/2)/(2
*x^2+x+1)^7+5/6453888*(21409+47471*x)/(3-2*x)^(19/2)/(2*x^2+x+1)^6+41/90354432*(47471+92875*x)/(3-2*x)^(19/2)/
(2*x^2+x+1)^5+41/5059848192*(3436375+5677637*x)/(3-2*x)^(19/2)/(2*x^2+x+1)^4+451/10119696384*(811091+998691*x)
/(3-2*x)^(19/2)/(2*x^2+x+1)^3+451/283351498752*(28962039+14627273*x)/(3-2*x)^(19/2)/(2*x^2+x+1)^2+11275/396692
0982528*(14627273-35058731*x)/(3-2*x)^(19/2)/(2*x^2+x+1)-24229218097975/22757389978742816768/(3-2*x)^(1/2)+112
75/1274413838809597739008*ln(3-2*x+14^(1/2)-(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(9756589235-2148932869*14^(1/2
))*(-14+4*14^(1/2))^(1/2)-11275/1274413838809597739008*ln(3-2*x+14^(1/2)+(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(
9756589235-2148932869*14^(1/2))*(-14+4*14^(1/2))^(1/2)+11275/637206919404798869504*arctan((-2*(3-2*x)^(1/2)+(7
+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(9756589235+2148932869*14^(1/2))*(14+4*14^(1/2))^(1/2)-11275/637206
919404798869504*arctan((2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(9756589235+2148932869*14
^(1/2))*(14+4*14^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.16, antiderivative size = 648, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {740, 822, 828, 826, 1169, 634, 618, 204, 628} \[ \frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (2 x^2+x+1\right )}+\frac {451 (14627273 x+28962039)}{283351498752 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^2}+\frac {451 (998691 x+811091)}{10119696384 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^3}+\frac {41 (5677637 x+3436375)}{5059848192 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^4}+\frac {41 (92875 x+47471)}{90354432 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^5}+\frac {5 (47471 x+21409)}{6453888 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^6}+\frac {21409 x+8477}{691488 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^7}+\frac {173 x+53}{7056 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^8}+\frac {x}{63 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^9}-\frac {24229218097975}{22757389978742816768 \sqrt {3-2 x}}-\frac {46601678385075}{11378694989371408384 (3-2 x)^{3/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}+\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}+\frac {11275 \left (9756589235-2148932869 \sqrt {14}\right ) \sqrt {\frac {1}{2} \left (2 \sqrt {14}-7\right )} \log \left (-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3\right )}{637206919404798869504}-\frac {11275 \left (9756589235-2148932869 \sqrt {14}\right ) \sqrt {\frac {1}{2} \left (2 \sqrt {14}-7\right )} \log \left (-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3\right )}{637206919404798869504}+\frac {11275 \sqrt {\frac {1}{2} \left (7+2 \sqrt {14}\right )} \left (9756589235+2148932869 \sqrt {14}\right ) \tan ^{-1}\left (\frac {\sqrt {7+2 \sqrt {14}}-2 \sqrt {3-2 x}}{\sqrt {2 \sqrt {14}-7}}\right )}{318603459702399434752}-\frac {11275 \sqrt {\frac {1}{2} \left (7+2 \sqrt {14}\right )} \left (9756589235+2148932869 \sqrt {14}\right ) \tan ^{-1}\left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {2 \sqrt {14}-7}}\right )}{318603459702399434752} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4718120139975/(351733660450816*(3 - 2*x)^(19/2)) - 815900548375/(629418129227776*(3 - 2*x)^(17/2)) - 302950882
3715/(1555033025150976*(3 - 2*x)^(15/2)) - 13515743021825/(13476952884641792*(3 - 2*x)^(13/2)) - 5846828446875
/(14513641568075776*(3 - 2*x)^(11/2)) - 37283626871975/(261245548225363968*(3 - 2*x)^(9/2)) - 132355162272575/
(2844673747342852096*(3 - 2*x)^(7/2)) - 11557581705725/(812763927812243456*(3 - 2*x)^(5/2)) - 46601678385075/(
11378694989371408384*(3 - 2*x)^(3/2)) - 24229218097975/(22757389978742816768*Sqrt[3 - 2*x]) + x/(63*(3 - 2*x)^
(19/2)*(1 + x + 2*x^2)^9) + (53 + 173*x)/(7056*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^8) + (8477 + 21409*x)/(691488*
(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^7) + (5*(21409 + 47471*x))/(6453888*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^6) + (41
*(47471 + 92875*x))/(90354432*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^5) + (41*(3436375 + 5677637*x))/(5059848192*(3
- 2*x)^(19/2)*(1 + x + 2*x^2)^4) + (451*(811091 + 998691*x))/(10119696384*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^3)
+ (451*(28962039 + 14627273*x))/(283351498752*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^2) + (11275*(14627273 - 3505873
1*x))/(3966920982528*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)) + (11275*Sqrt[(7 + 2*Sqrt[14])/2]*(9756589235 + 2148932
869*Sqrt[14])*ArcTan[(Sqrt[7 + 2*Sqrt[14]] - 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]])/318603459702399434752 -
(11275*Sqrt[(7 + 2*Sqrt[14])/2]*(9756589235 + 2148932869*Sqrt[14])*ArcTan[(Sqrt[7 + 2*Sqrt[14]] + 2*Sqrt[3 - 2
*x])/Sqrt[-7 + 2*Sqrt[14]]])/318603459702399434752 + (11275*(9756589235 - 2148932869*Sqrt[14])*Sqrt[(-7 + 2*Sq
rt[14])/2]*Log[3 + Sqrt[14] - Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] - 2*x])/637206919404798869504 - (11275*(97565
89235 - 2148932869*Sqrt[14])*Sqrt[(-7 + 2*Sqrt[14])/2]*Log[3 + Sqrt[14] + Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] -
 2*x])/637206919404798869504

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 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 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 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*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]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), 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] && FractionQ[m] && LtQ[m, -1]

Rule 1169

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

Rubi steps

\begin {align*} \int \frac {1}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )^{10}} \, dx &=\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {\int \frac {1680-1484 x}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )^9} \, dx}{1764}\\ &=\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {\int \frac {2534672-3322984 x}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )^8} \, dx}{2765952}\\ &=\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {\int \frac {3218135760-5287166640 x}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )^7} \, dx}{3794886144}\\ &=\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {\int \frac {3218122918080-6729253503840 x}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )^6} \, dx}{4462786105344}\\ &=\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {\int \frac {2223971291223360-6819728658120000 x}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )^5} \, dx}{4373530383237120}\\ &=\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {\int \frac {602017891719552000-5205664113141824640 x}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )^4} \, dx}{3428847820457902080}\\ &=\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {\int \frac {-644013851165157876480-2602338158011857027840 x}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )^3} \, dx}{2016162518429246423040}\\ &=\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {\int \frac {-781280013553524600192000-460008659488539446208000 x}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )^2} \, dx}{790335707224264597831680}\\ &=\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {-209865664941946247912832000+324150102079841867727744000 x}{(3-2 x)^{21/2} \left (1+x+2 x^2\right )} \, dx}{154905798615955861175009280}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {-2327225523695253718758144000+1105437952711266214715136000 x}{(3-2 x)^{19/2} \left (1+x+2 x^2\right )} \, dx}{4337362361246764112900259840}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {-20828680094984562179495424000-2676274378513417586741760000 x}{(3-2 x)^{17/2} \left (1+x+2 x^2\right )} \, dx}{121446146114909395161207275520}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {-161276892002849662262479872000-99372366651018754238432256000 x}{(3-2 x)^{15/2} \left (1+x+2 x^2\right )} \, dx}{3400492091217463064513803714560}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {-1091470402720759789622974464000-1241341767917511174480513024000 x}{(3-2 x)^{13/2} \left (1+x+2 x^2\right )} \, dx}{95213778554088965806386504007680}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {-6249079685931055968022769664000-11813932218388106205374976000000 x}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )} \, dx}{2665985799514491042578822112215040}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {-26364773050672235333432205312000-95879912054052861104340934656000 x}{(3-2 x)^{9/2} \left (1+x+2 x^2\right )} \, dx}{74647602386405749192207019142021120}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {-19158360297272160458775773184000-680738564527006107959774429184000 x}{(3-2 x)^{7/2} \left (1+x+2 x^2\right )} \, dx}{2090132866819360977381796535976591360}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {1208210246675834932249342672896000-4161064828351125289593749667840000 x}{(3-2 x)^{5/2} \left (1+x+2 x^2\right )} \, dx}{58523720270942107366690303007344558080}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}-\frac {46601678385075}{11378694989371408384 (3-2 x)^{3/2}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {17987811630108930037182240718848000-20133547983403412008565127315456000 x}{(3-2 x)^{3/2} \left (1+x+2 x^2\right )} \, dx}{1638664167586379006267328484205647626240}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}-\frac {46601678385075}{11378694989371408384 (3-2 x)^{3/2}}-\frac {24229218097975}{22757389978742816768 \sqrt {3-2 x}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\int \frac {184169589007678264314588180381696000-48850041379984751902661801017344000 x}{\sqrt {3-2 x} \left (1+x+2 x^2\right )} \, dx}{45882596692418612175485197557758133534720}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}-\frac {46601678385075}{11378694989371408384 (3-2 x)^{3/2}}-\frac {24229218097975}{22757389978742816768 \sqrt {3-2 x}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-221789053875402272921190957711360000-48850041379984751902661801017344000 x^2}{28-14 x^2+2 x^4} \, dx,x,\sqrt {3-2 x}\right )}{22941298346209306087742598778879066767360}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}-\frac {46601678385075}{11378694989371408384 (3-2 x)^{3/2}}-\frac {24229218097975}{22757389978742816768 \sqrt {3-2 x}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-221789053875402272921190957711360000 \sqrt {7+2 \sqrt {14}}-\left (-221789053875402272921190957711360000+48850041379984751902661801017344000 \sqrt {14}\right ) x}{\sqrt {14}-\sqrt {7+2 \sqrt {14}} x+x^2} \, dx,x,\sqrt {3-2 x}\right )}{91765193384837224350970395115516267069440 \sqrt {14 \left (7+2 \sqrt {14}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {-221789053875402272921190957711360000 \sqrt {7+2 \sqrt {14}}+\left (-221789053875402272921190957711360000+48850041379984751902661801017344000 \sqrt {14}\right ) x}{\sqrt {14}+\sqrt {7+2 \sqrt {14}} x+x^2} \, dx,x,\sqrt {3-2 x}\right )}{91765193384837224350970395115516267069440 \sqrt {14 \left (7+2 \sqrt {14}\right )}}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}-\frac {46601678385075}{11378694989371408384 (3-2 x)^{3/2}}-\frac {24229218097975}{22757389978742816768 \sqrt {3-2 x}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {\left (11275 \left (9756589235-2148932869 \sqrt {14}\right )\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {7+2 \sqrt {14}}+2 x}{\sqrt {14}-\sqrt {7+2 \sqrt {14}} x+x^2} \, dx,x,\sqrt {3-2 x}\right )}{91029559914971267072 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {\left (11275 \left (9756589235-2148932869 \sqrt {14}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {7+2 \sqrt {14}}+2 x}{\sqrt {14}+\sqrt {7+2 \sqrt {14}} x+x^2} \, dx,x,\sqrt {3-2 x}\right )}{91029559914971267072 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {\left (11275 \left (30085060166+9756589235 \sqrt {14}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {14}-\sqrt {7+2 \sqrt {14}} x+x^2} \, dx,x,\sqrt {3-2 x}\right )}{1274413838809597739008}-\frac {\left (11275 \left (30085060166+9756589235 \sqrt {14}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {14}+\sqrt {7+2 \sqrt {14}} x+x^2} \, dx,x,\sqrt {3-2 x}\right )}{1274413838809597739008}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}-\frac {46601678385075}{11378694989371408384 (3-2 x)^{3/2}}-\frac {24229218097975}{22757389978742816768 \sqrt {3-2 x}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {11275 \left (9756589235-2148932869 \sqrt {14}\right ) \log \left (3+\sqrt {14}-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}-2 x\right )}{91029559914971267072 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {11275 \left (9756589235-2148932869 \sqrt {14}\right ) \log \left (3+\sqrt {14}+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}-2 x\right )}{91029559914971267072 \sqrt {14 \left (7+2 \sqrt {14}\right )}}+\frac {\left (11275 \left (30085060166+9756589235 \sqrt {14}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{7-2 \sqrt {14}-x^2} \, dx,x,-\sqrt {7+2 \sqrt {14}}+2 \sqrt {3-2 x}\right )}{637206919404798869504}+\frac {\left (11275 \left (30085060166+9756589235 \sqrt {14}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{7-2 \sqrt {14}-x^2} \, dx,x,\sqrt {7+2 \sqrt {14}}+2 \sqrt {3-2 x}\right )}{637206919404798869504}\\ &=\frac {4718120139975}{351733660450816 (3-2 x)^{19/2}}-\frac {815900548375}{629418129227776 (3-2 x)^{17/2}}-\frac {3029508823715}{1555033025150976 (3-2 x)^{15/2}}-\frac {13515743021825}{13476952884641792 (3-2 x)^{13/2}}-\frac {5846828446875}{14513641568075776 (3-2 x)^{11/2}}-\frac {37283626871975}{261245548225363968 (3-2 x)^{9/2}}-\frac {132355162272575}{2844673747342852096 (3-2 x)^{7/2}}-\frac {11557581705725}{812763927812243456 (3-2 x)^{5/2}}-\frac {46601678385075}{11378694989371408384 (3-2 x)^{3/2}}-\frac {24229218097975}{22757389978742816768 \sqrt {3-2 x}}+\frac {x}{63 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^9}+\frac {53+173 x}{7056 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^8}+\frac {8477+21409 x}{691488 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^7}+\frac {5 (21409+47471 x)}{6453888 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^6}+\frac {41 (47471+92875 x)}{90354432 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^5}+\frac {41 (3436375+5677637 x)}{5059848192 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^4}+\frac {451 (811091+998691 x)}{10119696384 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^3}+\frac {451 (28962039+14627273 x)}{283351498752 (3-2 x)^{19/2} \left (1+x+2 x^2\right )^2}+\frac {11275 (14627273-35058731 x)}{3966920982528 (3-2 x)^{19/2} \left (1+x+2 x^2\right )}+\frac {11275 \sqrt {\frac {1}{2} \left (7+2 \sqrt {14}\right )} \left (9756589235+2148932869 \sqrt {14}\right ) \tan ^{-1}\left (\frac {\sqrt {7+2 \sqrt {14}}-2 \sqrt {3-2 x}}{\sqrt {-7+2 \sqrt {14}}}\right )}{318603459702399434752}-\frac {11275 \sqrt {\frac {1}{2} \left (7+2 \sqrt {14}\right )} \left (9756589235+2148932869 \sqrt {14}\right ) \tan ^{-1}\left (\frac {\sqrt {7+2 \sqrt {14}}+2 \sqrt {3-2 x}}{\sqrt {-7+2 \sqrt {14}}}\right )}{318603459702399434752}+\frac {11275 \left (9756589235-2148932869 \sqrt {14}\right ) \log \left (3+\sqrt {14}-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}-2 x\right )}{91029559914971267072 \sqrt {14 \left (7+2 \sqrt {14}\right )}}-\frac {11275 \left (9756589235-2148932869 \sqrt {14}\right ) \log \left (3+\sqrt {14}+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}-2 x\right )}{91029559914971267072 \sqrt {14 \left (7+2 \sqrt {14}\right )}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 6.09, size = 610, normalized size = 0.94 \[ \frac {x}{63 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^9}+\frac {\frac {67816 x+20776}{1568 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^8}+\frac {\frac {117492592 x+46521776}{1372 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^7}+\frac {\frac {164128134240 x+74020332960}{1176 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^6}+\frac {\frac {184316990760000 x+94209549053760}{980 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^5}+\frac {1}{980} \left (\frac {157747397367934080 x+95476201213680000}{784 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^4}+\frac {1}{784} \left (\frac {89735798552133000960 x+72879297583985544960}{588 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^3}+\frac {1}{588} \left (\frac {18400346379541577848320 x+36432734212165998389760}{392 (3-2 x)^{19/2} \left (2 x^2+x+1\right )^2}+\frac {1}{392} \left (\frac {6440121232839552246912000-15435719146659136558464000 x}{196 (3-2 x)^{19/2} \left (2 x^2+x+1\right )}+\frac {1}{196} \left (\frac {39479926882545221954112000}{19 (3-2 x)^{19/2}}+\frac {1}{266} \left (-\frac {908021664138480966930240000}{17 (3-2 x)^{17/2}}+\frac {1}{238} \left (-\frac {19105520493023248582746201600}{(3-2 x)^{15/2}}+\frac {1}{210} \left (-\frac {26849557435537239465884310720000}{13 (3-2 x)^{13/2}}+\frac {1}{182} \left (-\frac {150994423858598796539274120000000}{(3-2 x)^{11/2}}+\frac {1}{154} \left (-\frac {8237718113587514139784976619840000}{(3-2 x)^{9/2}}+\frac {1}{126} \left (-\frac {338389312036560466460044072847040000}{(3-2 x)^{7/2}}+\frac {1}{98} \left (-\frac {10135305528576510550836394515648960000}{(3-2 x)^{5/2}}+\frac {1}{70} \left (-\frac {204334375738495648812805956791073600000}{(3-2 x)^{3/2}}+\frac {1}{42} \left (-\frac {2230994866519889796828561036406228800000}{\sqrt {3-2 x}}+\frac {1}{7} \left (\frac {\sqrt {\frac {1}{2} \left (7-i \sqrt {7}\right )} \left (-31233928131278457155599854509687203200000-71750597240923349846054347713013891200000 i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {3-2 x}}{\sqrt {7-i \sqrt {7}}}\right )}{-14+2 i \sqrt {7}}+\frac {\sqrt {\frac {1}{2} \left (7+i \sqrt {7}\right )} \left (-31233928131278457155599854509687203200000+71750597240923349846054347713013891200000 i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {3-2 x}}{\sqrt {7+i \sqrt {7}}}\right )}{-14-2 i \sqrt {7}}\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )\right )}{1176}}{1372}}{1568}}{1764} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(63*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^9) + ((20776 + 67816*x)/(1568*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^8) + ((4
6521776 + 117492592*x)/(1372*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^7) + ((74020332960 + 164128134240*x)/(1176*(3 -
2*x)^(19/2)*(1 + x + 2*x^2)^6) + ((94209549053760 + 184316990760000*x)/(980*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^5
) + ((95476201213680000 + 157747397367934080*x)/(784*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^4) + ((72879297583985544
960 + 89735798552133000960*x)/(588*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^3) + ((36432734212165998389760 + 184003463
79541577848320*x)/(392*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)^2) + ((6440121232839552246912000 - 1543571914665913655
8464000*x)/(196*(3 - 2*x)^(19/2)*(1 + x + 2*x^2)) + (39479926882545221954112000/(19*(3 - 2*x)^(19/2)) + (-9080
21664138480966930240000/(17*(3 - 2*x)^(17/2)) + (-19105520493023248582746201600/(3 - 2*x)^(15/2) + (-268495574
35537239465884310720000/(13*(3 - 2*x)^(13/2)) + (-150994423858598796539274120000000/(3 - 2*x)^(11/2) + (-82377
18113587514139784976619840000/(3 - 2*x)^(9/2) + (-338389312036560466460044072847040000/(3 - 2*x)^(7/2) + (-101
35305528576510550836394515648960000/(3 - 2*x)^(5/2) + (-204334375738495648812805956791073600000/(3 - 2*x)^(3/2
) + (-2230994866519889796828561036406228800000/Sqrt[3 - 2*x] + ((Sqrt[(7 - I*Sqrt[7])/2]*(-3123392813127845715
5599854509687203200000 - (71750597240923349846054347713013891200000*I)*Sqrt[7])*ArcTanh[(Sqrt[2]*Sqrt[3 - 2*x]
)/Sqrt[7 - I*Sqrt[7]]])/(-14 + (2*I)*Sqrt[7]) + (Sqrt[(7 + I*Sqrt[7])/2]*(-31233928131278457155599854509687203
200000 + (71750597240923349846054347713013891200000*I)*Sqrt[7])*ArcTanh[(Sqrt[2]*Sqrt[3 - 2*x])/Sqrt[7 + I*Sqr
t[7]]])/(-14 - (2*I)*Sqrt[7]))/7)/42)/70)/98)/126)/154)/182)/210)/238)/266)/196)/392)/588)/784)/980)/1176)/137
2)/1568)/1764

________________________________________________________________________________________

fricas [B]  time = 1.72, size = 1563, normalized size = 2.41 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1094755373086200603246995644663447631605361478665641987670016*(4732002380085251586622550100*4787936175075825
342943147314686^(1/4)*sqrt(1169607525756986)*sqrt(14)*sqrt(7)*(524288*x^28 - 5505024*x^27 + 24772608*x^26 - 64
684032*x^25 + 119734272*x^24 - 194052096*x^23 + 295206912*x^22 - 386777088*x^21 + 449261568*x^20 - 515594240*x
^19 + 540503040*x^18 - 496581120*x^17 + 467712000*x^16 - 411828480*x^15 + 303534720*x^14 - 248434368*x^13 + 18
6495624*x^12 - 105219828*x^11 + 83621482*x^10 - 49793667*x^9 + 19105065*x^8 - 20036484*x^7 + 5497632*x^6 - 223
5114*x^5 + 3276126*x^4 + 734832*x^3 + 826686*x^2 + 137781*x + 59049)*sqrt(327571850528462403199*sqrt(14) + 122
6422380928157351936)*arctan(1/36562170851931970248855340113387035354417457241870626866024945379489008832725311
219252*4787936175075825342943147314686^(3/4)*sqrt(2776387167632535361)*sqrt(12865682783326846)*sqrt(1169607525
756986)*sqrt(4787936175075825342943147314686^(1/4)*sqrt(1169607525756986)*sqrt(-2*x + 3)*sqrt(3275718505284624
03199*sqrt(14) + 1226422380928157351936)*(2148932869*sqrt(14) - 9756589235) - 71440233164918992209696826631202
812*x + 28280279689505005187146*sqrt(22335021272086100802556094) + 107160349747378488314545239946804218)*(9756
589235*sqrt(14)*sqrt(7) - 30085060166*sqrt(7))*sqrt(327571850528462403199*sqrt(14) + 1226422380928157351936) -
 1/1023573670806157676669100144258228441327447900096742*4787936175075825342943147314686^(3/4)*sqrt(11696075257
56986)*(9756589235*sqrt(14)*sqrt(7) - 30085060166*sqrt(7))*sqrt(-2*x + 3)*sqrt(327571850528462403199*sqrt(14)
+ 1226422380928157351936) + 2/7*sqrt(14)*sqrt(7) + sqrt(7)) + 4732002380085251586622550100*4787936175075825342
943147314686^(1/4)*sqrt(1169607525756986)*sqrt(14)*sqrt(7)*(524288*x^28 - 5505024*x^27 + 24772608*x^26 - 64684
032*x^25 + 119734272*x^24 - 194052096*x^23 + 295206912*x^22 - 386777088*x^21 + 449261568*x^20 - 515594240*x^19
 + 540503040*x^18 - 496581120*x^17 + 467712000*x^16 - 411828480*x^15 + 303534720*x^14 - 248434368*x^13 + 18649
5624*x^12 - 105219828*x^11 + 83621482*x^10 - 49793667*x^9 + 19105065*x^8 - 20036484*x^7 + 5497632*x^6 - 223511
4*x^5 + 3276126*x^4 + 734832*x^3 + 826686*x^2 + 137781*x + 59049)*sqrt(327571850528462403199*sqrt(14) + 122642
2380928157351936)*arctan(1/39296670234816303076555330542603297083388480635973027797585697454399143598928370335
464344780800*4787936175075825342943147314686^(3/4)*sqrt(2776387167632535361)*sqrt(1169607525756986)*sqrt(-1486
2107440409842545228890767360000*4787936175075825342943147314686^(1/4)*sqrt(1169607525756986)*sqrt(-2*x + 3)*sq
rt(327571850528462403199*sqrt(14) + 1226422380928157351936)*(2148932869*sqrt(14) - 9756589235) - 1061752420864
956548109093061495542399038192585561809435358469816320000*x + 420304555190263689316852795001664341102416628348
354560000*sqrt(22335021272086100802556094) + 15926286312974348221636395922433135985572888783427141530377047244
80000)*(9756589235*sqrt(14)*sqrt(7) - 30085060166*sqrt(7))*sqrt(327571850528462403199*sqrt(14) + 1226422380928
157351936) - 1/1023573670806157676669100144258228441327447900096742*4787936175075825342943147314686^(3/4)*sqrt
(1169607525756986)*(9756589235*sqrt(14)*sqrt(7) - 30085060166*sqrt(7))*sqrt(-2*x + 3)*sqrt(3275718505284624031
99*sqrt(14) + 1226422380928157351936) - 2/7*sqrt(14)*sqrt(7) - sqrt(7)) + 271150425*47879361750758253429431473
14686^(1/4)*sqrt(1169607525756986)*(642998537252061761731821568*x^28 - 6751484641146648498184126464*x^27 + 303
81680885159918241828569088*x^26 - 79329944533473119853663485952*x^25 + 146844790944939604835504750592*x^24 - 2
37989833600419359560990457856*x^23 + 362048363881489025715123781632*x^22 - 474352077153419437787597242368*x^21
 + 550984441886077267281495195648*x^20 - 632336315413643784471854448640*x^19 + 662885025215707070319757885440*
x^18 - 609018199514371017360613048320*x^17 + 573612464628670331388690432000*x^16 - 505075664975624031448627937
280*x^15 + 372261773996761581935835217920*x^14 - 304685469106942025132773736448*x^13 + 22872240721876240451949
1928064*x^12 - 129043951976611196927641387008*x^11 + 102555257051181053298083889152*x^10 - 6106806763728381810
5902989312*x^9 + 23430879305087206538965155840*x^8 - 24573192412708929931548033024*x^7 + 674241892690682755903
8615552*x^6 - 2741193833525857491515080704*x^5 + 4017914249140640432768679936*x^4 + 90121441102219972323783475
2*x^3 + 1013866212399974688642564096*x^2 - 327571850528462403199*sqrt(14)*(524288*x^28 - 5505024*x^27 + 247726
08*x^26 - 64684032*x^25 + 119734272*x^24 - 194052096*x^23 + 295206912*x^22 - 386777088*x^21 + 449261568*x^20 -
 515594240*x^19 + 540503040*x^18 - 496581120*x^17 + 467712000*x^16 - 411828480*x^15 + 303534720*x^14 - 2484343
68*x^13 + 186495624*x^12 - 105219828*x^11 + 83621482*x^10 - 49793667*x^9 + 19105065*x^8 - 20036484*x^7 + 54976
32*x^6 - 2235114*x^5 + 3276126*x^4 + 734832*x^3 + 826686*x^2 + 137781*x + 59049) + 168977702066662448107094016
*x + 72419015171426763474468864)*sqrt(327571850528462403199*sqrt(14) + 1226422380928157351936)*log(14862107440
409842545228890767360000/2776387167632535361*4787936175075825342943147314686^(1/4)*sqrt(1169607525756986)*sqrt
(-2*x + 3)*sqrt(327571850528462403199*sqrt(14) + 1226422380928157351936)*(2148932869*sqrt(14) - 9756589235) -
382422319640069460132720868272698184789257093120000*x + 151385426388014656165701481356328960000*sqrt(223350212
72086100802556094) + 573633479460104190199081302409047277183885639680000) - 271150425*478793617507582534294314
7314686^(1/4)*sqrt(1169607525756986)*(642998537252061761731821568*x^28 - 6751484641146648498184126464*x^27 + 3
0381680885159918241828569088*x^26 - 79329944533473119853663485952*x^25 + 146844790944939604835504750592*x^24 -
 237989833600419359560990457856*x^23 + 362048363881489025715123781632*x^22 - 474352077153419437787597242368*x^
21 + 550984441886077267281495195648*x^20 - 632336315413643784471854448640*x^19 + 66288502521570707031975788544
0*x^18 - 609018199514371017360613048320*x^17 + 573612464628670331388690432000*x^16 - 5050756649756240314486279
37280*x^15 + 372261773996761581935835217920*x^14 - 304685469106942025132773736448*x^13 + 228722407218762404519
491928064*x^12 - 129043951976611196927641387008*x^11 + 102555257051181053298083889152*x^10 - 61068067637283818
105902989312*x^9 + 23430879305087206538965155840*x^8 - 24573192412708929931548033024*x^7 + 6742418926906827559
038615552*x^6 - 2741193833525857491515080704*x^5 + 4017914249140640432768679936*x^4 + 901214411022199723237834
752*x^3 + 1013866212399974688642564096*x^2 - 327571850528462403199*sqrt(14)*(524288*x^28 - 5505024*x^27 + 2477
2608*x^26 - 64684032*x^25 + 119734272*x^24 - 194052096*x^23 + 295206912*x^22 - 386777088*x^21 + 449261568*x^20
 - 515594240*x^19 + 540503040*x^18 - 496581120*x^17 + 467712000*x^16 - 411828480*x^15 + 303534720*x^14 - 24843
4368*x^13 + 186495624*x^12 - 105219828*x^11 + 83621482*x^10 - 49793667*x^9 + 19105065*x^8 - 20036484*x^7 + 549
7632*x^6 - 2235114*x^5 + 3276126*x^4 + 734832*x^3 + 826686*x^2 + 137781*x + 59049) + 1689777020666624481070940
16*x + 72419015171426763474468864)*sqrt(327571850528462403199*sqrt(14) + 1226422380928157351936)*log(-14862107
440409842545228890767360000/2776387167632535361*4787936175075825342943147314686^(1/4)*sqrt(1169607525756986)*s
qrt(-2*x + 3)*sqrt(327571850528462403199*sqrt(14) + 1226422380928157351936)*(2148932869*sqrt(14) - 9756589235)
 - 382422319640069460132720868272698184789257093120000*x + 151385426388014656165701481356328960000*sqrt(223350
21272086100802556094) + 573633479460104190199081302409047277183885639680000) + 1272935063665829315736416183610
522832*(240031204937714427494400*x^27 - 2621948941596237063782400*x^26 + 12365045055896811105484800*x^25 - 339
69890064381284111155200*x^24 + 65360120291258796757811200*x^23 - 106701725825102321939251200*x^22 + 1622903072
23249502039654400*x^21 - 216634228326470609547509760*x^20 + 253788172995391086570485760*x^19 - 287279159180291
305208156160*x^18 + 304010591010966811155955200*x^17 - 282644664539994827031006720*x^16 + 25881925681516324984
5447936*x^15 - 229408132984166521977166336*x^14 + 172649692294614969274168896*x^13 - 1333125413772463861158902
40*x^12 + 102031573634317834547976132*x^11 - 59791102681494117572149176*x^10 + 41613884937255303086792337*x^9
- 27246604251076689552043953*x^8 + 10718131725916893151555068*x^7 - 8685973988079840377705700*x^6 + 3673303058
277822225386926*x^5 - 809990362095044210054958*x^4 + 1362587089603925431664856*x^3 + 111926768697602999806116*
x^2 + 205702452014540322797289*x - 4884417100172357749737)*sqrt(-2*x + 3))/(524288*x^28 - 5505024*x^27 + 24772
608*x^26 - 64684032*x^25 + 119734272*x^24 - 194052096*x^23 + 295206912*x^22 - 386777088*x^21 + 449261568*x^20
- 515594240*x^19 + 540503040*x^18 - 496581120*x^17 + 467712000*x^16 - 411828480*x^15 + 303534720*x^14 - 248434
368*x^13 + 186495624*x^12 - 105219828*x^11 + 83621482*x^10 - 49793667*x^9 + 19105065*x^8 - 20036484*x^7 + 5497
632*x^6 - 2235114*x^5 + 3276126*x^4 + 734832*x^3 + 826686*x^2 + 137781*x + 59049)

________________________________________________________________________________________

giac [A]  time = 4.19, size = 972, normalized size = 1.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11275/2283749599146799148302336*sqrt(7)*(240680481328*14^(3/4)*sqrt(2)*(sqrt(14) + 4)^(3/2) + 722041443984*14^
(3/4)*sqrt(2)*sqrt(sqrt(14) + 4)*(sqrt(14) - 4) - 51574388856*14^(3/4)*sqrt(7)*(sqrt(14) + 4)*sqrt(-8*sqrt(14)
 + 32) + 2148932869*14^(3/4)*sqrt(7)*(-8*sqrt(14) + 32)^(3/2) + 8741903954560*14^(1/4)*sqrt(2)*sqrt(sqrt(14) +
 4) - 624421711040*14^(1/4)*sqrt(7)*sqrt(-8*sqrt(14) + 32))*arctan(1/28*14^(3/4)*(14^(1/4)*sqrt(1/2)*sqrt(sqrt
(14) + 4) + 2*sqrt(-2*x + 3))/sqrt(-1/8*sqrt(14) + 1/2)) + 11275/2283749599146799148302336*sqrt(7)*(2406804813
28*14^(3/4)*sqrt(2)*(sqrt(14) + 4)^(3/2) + 722041443984*14^(3/4)*sqrt(2)*sqrt(sqrt(14) + 4)*(sqrt(14) - 4) - 5
1574388856*14^(3/4)*sqrt(7)*(sqrt(14) + 4)*sqrt(-8*sqrt(14) + 32) + 2148932869*14^(3/4)*sqrt(7)*(-8*sqrt(14) +
 32)^(3/2) + 8741903954560*14^(1/4)*sqrt(2)*sqrt(sqrt(14) + 4) - 624421711040*14^(1/4)*sqrt(7)*sqrt(-8*sqrt(14
) + 32))*arctan(-1/28*14^(3/4)*(14^(1/4)*sqrt(1/2)*sqrt(sqrt(14) + 4) - 2*sqrt(-2*x + 3))/sqrt(-1/8*sqrt(14) +
 1/2)) - 11275/4567499198293598296604672*sqrt(7)*(34382925904*14^(3/4)*sqrt(7)*sqrt(2)*(sqrt(14) + 4)^(3/2) +
103148777712*14^(3/4)*sqrt(7)*sqrt(2)*sqrt(sqrt(14) + 4)*(sqrt(14) - 4) + 361020721992*14^(3/4)*(sqrt(14) + 4)
*sqrt(-8*sqrt(14) + 32) - 15042530083*14^(3/4)*(-8*sqrt(14) + 32)^(3/2) + 1248843422080*14^(1/4)*sqrt(7)*sqrt(
2)*sqrt(sqrt(14) + 4) + 4370951977280*14^(1/4)*sqrt(-8*sqrt(14) + 32))*log(14^(1/4)*sqrt(1/2)*sqrt(-2*x + 3)*s
qrt(sqrt(14) + 4) - 2*x + sqrt(14) + 3) + 11275/4567499198293598296604672*sqrt(7)*(34382925904*14^(3/4)*sqrt(7
)*sqrt(2)*(sqrt(14) + 4)^(3/2) + 103148777712*14^(3/4)*sqrt(7)*sqrt(2)*sqrt(sqrt(14) + 4)*(sqrt(14) - 4) + 361
020721992*14^(3/4)*(sqrt(14) + 4)*sqrt(-8*sqrt(14) + 32) - 15042530083*14^(3/4)*(-8*sqrt(14) + 32)^(3/2) + 124
8843422080*14^(1/4)*sqrt(7)*sqrt(2)*sqrt(sqrt(14) + 4) + 4370951977280*14^(1/4)*sqrt(-8*sqrt(14) + 32))*log(-1
4^(1/4)*sqrt(1/2)*sqrt(-2*x + 3)*sqrt(sqrt(14) + 4) - 2*x + sqrt(14) + 3) + 1/204816509808685350912*(232787883
652335*(2*x - 3)^17*sqrt(-2*x + 3) + 13820106668010555*(2*x - 3)^16*sqrt(-2*x + 3) + 389618236717151904*(2*x -
 3)^15*sqrt(-2*x + 3) + 6925854690067471092*(2*x - 3)^14*sqrt(-2*x + 3) + 86924717622268515682*(2*x - 3)^13*sq
rt(-2*x + 3) + 817308030405306394458*(2*x - 3)^12*sqrt(-2*x + 3) + 5960699611609964201316*(2*x - 3)^11*sqrt(-2
*x + 3) + 34438539253455396724476*(2*x - 3)^10*sqrt(-2*x + 3) + 159569809573892673649239*(2*x - 3)^9*sqrt(-2*x
 + 3) + 596312099501239401271299*(2*x - 3)^8*sqrt(-2*x + 3) + 1797250621001927736488676*(2*x - 3)^7*sqrt(-2*x
+ 3) + 4343978582610098069631672*(2*x - 3)^6*sqrt(-2*x + 3) + 8317212692450176764092592*(2*x - 3)^5*sqrt(-2*x
+ 3) + 12350951282904546626644288*(2*x - 3)^4*sqrt(-2*x + 3) + 13738697725192288735303872*(2*x - 3)^3*sqrt(-2*
x + 3) + 10788479661863702869789824*(2*x - 3)^2*sqrt(-2*x + 3) - 5340653236079401357791744*(-2*x + 3)^(3/2) +
1255138952440667471476992*sqrt(-2*x + 3))/((2*x - 3)^2 + 14*x - 7)^9 + 1/3280733202692679552*(235862511885*(2*
x - 3)^9 - 107316677325*(2*x - 3)^8 + 80348352084*(2*x - 3)^7 - 64554208290*(2*x - 3)^6 + 49954696792*(2*x - 3
)^5 - 35035280280*(2*x - 3)^4 + 21058773120*(2*x - 3)^3 - 10093321056*(2*x - 3)^2 + 6831901440*x - 10859127552
)/((2*x - 3)^9*sqrt(-2*x + 3))

________________________________________________________________________________________

maple [A]  time = 0.07, size = 719, normalized size = 1.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/86812553324672*(-1352841099712333/8192*(-2*x+3)^(31/2)+4606702222670185/786432*(-2*x+3)^(33/2)-2586532040581
5/262144*(-2*x+3)^(35/2)-3476987783905860258979/1536*(-2*x+3)^(3/2)+9364999706478908741137/2048*(-2*x+3)^(5/2)
-23851905772903279054347/4096*(-2*x+3)^(7/2)+192983613795383541041317/36864*(-2*x+3)^(9/2)-5775842147534844975
0643/16384*(-2*x+3)^(11/2)+60333035869584695411551/32768*(-2*x+3)^(13/2)-149770885083493978040723/196608*(-2*x
+3)^(15/2)+66256899944582155696811/262144*(-2*x+3)^(17/2)-17729978841543630405471/262144*(-2*x+3)^(19/2)+28698
78271121283060373/196608*(-2*x+3)^(21/2)-165574989211387894481/65536*(-2*x+3)^(23/2)+45406001689183688581/1310
72*(-2*x+3)^(25/2)-43462358811134257841/1179648*(-2*x+3)^(27/2)+192384852501874197/65536*(-2*x+3)^(29/2)+54476
5170330150812273/1024*(-2*x+3)^(1/2))/(14*x+(-2*x+3)^2-7)^9-206922416016525/1274413838809597739008*(7+2*14^(1/
2))^(1/2)*14^(1/2)*ln(-2*x+3+14^(1/2)-(-2*x+3)^(1/2)*(7+2*14^(1/2))^(1/2))+389615613935075/6372069194047988695
04*(7+2*14^(1/2))^(1/2)*ln(-2*x+3+14^(1/2)-(-2*x+3)^(1/2)*(7+2*14^(1/2))^(1/2))-206922416016525/63720691940479
8869504/(-7+2*14^(1/2))^(1/2)*(7+2*14^(1/2))*14^(1/2)*arctan((2*(-2*x+3)^(1/2)-(7+2*14^(1/2))^(1/2))/(-7+2*14^
(1/2))^(1/2))+389615613935075/318603459702399434752/(-7+2*14^(1/2))^(1/2)*(7+2*14^(1/2))*arctan((2*(-2*x+3)^(1
/2)-(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))-110005543624625/318603459702399434752/(-7+2*14^(1/2))^(1/2)*1
4^(1/2)*arctan((2*(-2*x+3)^(1/2)-(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))+206922416016525/1274413838809597
739008*(7+2*14^(1/2))^(1/2)*14^(1/2)*ln(-2*x+3+14^(1/2)+(-2*x+3)^(1/2)*(7+2*14^(1/2))^(1/2))-389615613935075/6
37206919404798869504*(7+2*14^(1/2))^(1/2)*ln(-2*x+3+14^(1/2)+(-2*x+3)^(1/2)*(7+2*14^(1/2))^(1/2))-206922416016
525/637206919404798869504/(-7+2*14^(1/2))^(1/2)*(7+2*14^(1/2))*14^(1/2)*arctan((2*(-2*x+3)^(1/2)+(7+2*14^(1/2)
)^(1/2))/(-7+2*14^(1/2))^(1/2))+389615613935075/318603459702399434752/(-7+2*14^(1/2))^(1/2)*(7+2*14^(1/2))*arc
tan((2*(-2*x+3)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))-110005543624625/318603459702399434752/(-7+2
*14^(1/2))^(1/2)*14^(1/2)*arctan((2*(-2*x+3)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))+1/5367029731/(
-2*x+3)^(19/2)+5/4802079233/(-2*x+3)^(17/2)+73/23727920916/(-2*x+3)^(15/2)+165/25705247659/(-2*x+3)^(13/2)+236
5/221460595216/(-2*x+3)^(11/2)+30349/1993145356944/(-2*x+3)^(9/2)+854095/43406276662336/(-2*x+3)^(7/2)+75933/3
100448333024/(-2*x+3)^(5/2)+8519225/260437659974016/(-2*x+3)^(3/2)+891605/12401793332096/(-2*x+3)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (2 \, x^{2} + x + 1\right )}^{10} {\left (-2 \, x + 3\right )}^{\frac {21}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.56, size = 567, normalized size = 0.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((184192*(2*x - 3)^2)/47481 - (18944*x)/2261 - (15552*(2*x - 3)^3)/4199 + (5666272*(2*x - 3)^4)/1440257 - (634
90768*(2*x - 3)^5)/12962313 + (533495672*(2*x - 3)^6)/70572593 - (1111521492*(2*x - 3)^7)/70572593 + (78007323
158*(2*x - 3)^8)/1482024453 - (250239440467*(2*x - 3)^9)/494008151 + (1118693654785651073*(2*x - 3)^10)/453254
454575104 + (1624300450152249301*(2*x - 3)^11)/97125954551808 + (35048653520674948897*(2*x - 3)^12)/9065089091
50208 + (95527511967437577915*(2*x - 3)^13)/1813017818300416 + (5640662999731415610547*(2*x - 3)^14)/114220122
552926208 + (1737142288764447500149*(2*x - 3)^15)/50764498912411648 + (12971210667229097601055*(2*x - 3)^16)/7
10702984773763072 + (32723441206946795665235*(2*x - 3)^17)/4264217908642578432 + (102645797034777710681325*(2*
x - 3)^18)/39799367147330732032 + (1460931787430200665315*(2*x - 3)^19)/2094703534070038528 + (687618468821894
139745*(2*x - 3)^20)/4528256169239642112 + (39968995676603847725*(2*x - 3)^21)/1509418723079880704 + (59401329
43613849875*(2*x - 3)^22)/1625527855624486912 + (5717978503620010375*(2*x - 3)^23)/14629750700620382208 + (178
056995818325525*(2*x - 3)^24)/5689347494685704192 + (179665281323275*(2*x - 3)^25)/101595490976530432 + (14332
37383402275*(2*x - 3)^26)/22757389978742816768 + (24229218097975*(2*x - 3)^27)/22757389978742816768 + 37120/22
61)/(20661046784*(3 - 2*x)^(19/2) - 92974710528*(3 - 2*x)^(21/2) + 199231522560*(3 - 2*x)^(23/2) - 27006939724
8*(3 - 2*x)^(25/2) + 259475340096*(3 - 2*x)^(27/2) - 187609683744*(3 - 2*x)^(29/2) + 105782451264*(3 - 2*x)^(3
1/2) - 47554666992*(3 - 2*x)^(33/2) + 17278167438*(3 - 2*x)^(35/2) - 5111496103*(3 - 2*x)^(37/2) + 1234154817*
(3 - 2*x)^(39/2) - 242625852*(3 - 2*x)^(41/2) + 38550456*(3 - 2*x)^(43/2) - 4883634*(3 - 2*x)^(45/2) + 482454*
(3 - 2*x)^(47/2) - 35868*(3 - 2*x)^(49/2) + 1890*(3 - 2*x)^(51/2) - 63*(3 - 2*x)^(53/2) + (3 - 2*x)^(55/2)) -
(atan(((- 7^(1/2)*30540258843957888971i - 2293002953699236822393)^(1/2)*(3 - 2*x)^(1/2)*4377461803582914433031
6520640625i)/(330008698047761583560870082619263806430093600589158123831296*((7^(1/2)*4270909670946074738724274
49424977178671875i)/165004349023880791780435041309631903215046800294579061915648 + 803365829195061345550676106
938401175484375/23572049860554398825776434472804557602149542899225580273664)) + (43774618035829144330316520640
625*7^(1/2)*(- 7^(1/2)*30540258843957888971i - 2293002953699236822393)^(1/2)*(3 - 2*x)^(1/2))/(330008698047761
583560870082619263806430093600589158123831296*((7^(1/2)*427090967094607473872427449424977178671875i)/165004349
023880791780435041309631903215046800294579061915648 + 803365829195061345550676106938401175484375/2357204986055
4398825776434472804557602149542899225580273664)))*(- 7^(1/2)*30540258843957888971i - 2293002953699236822393)^(
1/2)*11275i)/318603459702399434752 + (atan(((7^(1/2)*30540258843957888971i - 2293002953699236822393)^(1/2)*(3
- 2*x)^(1/2)*43774618035829144330316520640625i)/(330008698047761583560870082619263806430093600589158123831296*
((7^(1/2)*427090967094607473872427449424977178671875i)/1650043490238807917804350413096319032150468002945790619
15648 - 803365829195061345550676106938401175484375/23572049860554398825776434472804557602149542899225580273664
)) - (43774618035829144330316520640625*7^(1/2)*(7^(1/2)*30540258843957888971i - 2293002953699236822393)^(1/2)*
(3 - 2*x)^(1/2))/(330008698047761583560870082619263806430093600589158123831296*((7^(1/2)*427090967094607473872
427449424977178671875i)/165004349023880791780435041309631903215046800294579061915648 - 80336582919506134555067
6106938401175484375/23572049860554398825776434472804557602149542899225580273664)))*(7^(1/2)*305402588439578889
71i - 2293002953699236822393)^(1/2)*11275i)/318603459702399434752

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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