3.1.50 \(\int \frac {1}{(3-2 x+x^2)^{11/2} (1+x+2 x^2)^5} \, dx\) [50]

Optimal. Leaf size=378 \[ -\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {4878869-2578034 x}{411600000 \left (3-2 x+x^2\right )^{7/2}}-\frac {30316369-15043110 x}{6860000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {63043297-29625922 x}{41160000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {31 (7434109-3088870 x)}{411600000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}+\frac {\sqrt {\frac {1}{70} \left (151363871237318045+110320475741093888 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (308108167+312239803 \sqrt {2}+\left (932587773+620347970 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{137200000000}-\frac {\sqrt {\frac {1}{70} \left (-151363871237318045+110320475741093888 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (-151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (308108167-312239803 \sqrt {2}+\left (932587773-620347970 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{137200000000} \]

[Out]

1/123480000*(-3450497+2004270*x)/(x^2-2*x+3)^(9/2)+1/411600000*(-4878869+2578034*x)/(x^2-2*x+3)^(7/2)+1/686000
0000*(-30316369+15043110*x)/(x^2-2*x+3)^(5/2)+1/41160000000*(-63043297+29625922*x)/(x^2-2*x+3)^(3/2)+1/280*(-1
+10*x)/(x^2-2*x+3)^(9/2)/(2*x^2+x+1)^4+1/1050*(28+67*x)/(x^2-2*x+3)^(9/2)/(2*x^2+x+1)^3+1/117600*(5485+8878*x)
/(x^2-2*x+3)^(9/2)/(2*x^2+x+1)^2+3/343000*(8822+8233*x)/(x^2-2*x+3)^(9/2)/(2*x^2+x+1)-31/411600000000*(7434109
-3088870*x)/(x^2-2*x+3)^(1/2)-1/9604000000000*arctanh(1/7*(308108167+x*(932587773-620347970*2^(1/2))-312239803
*2^(1/2))*35^(1/2)/(-151363871237318045+110320475741093888*2^(1/2))^(1/2)/(x^2-2*x+3)^(1/2))*(-105954709866122
63150+7722433301876572160*2^(1/2))^(1/2)+1/9604000000000*arctan(1/7*(308108167+312239803*2^(1/2)+x*(932587773+
620347970*2^(1/2)))*35^(1/2)/(151363871237318045+110320475741093888*2^(1/2))^(1/2)/(x^2-2*x+3)^(1/2))*(1059547
0986612263150+7722433301876572160*2^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.49, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 14, 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 (151363871237318045+110320475741093888 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {5}{7 \left (151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (\left (932587773+620347970 \sqrt {2}\right ) x+312239803 \sqrt {2}[phone]\right )}{\sqrt {x^2-2 x+3}}\right )}{137200000000}-\frac {63043297-29625922 x}{41160000000 \left (x^2-2 x+3\right )^{3/2}}-\frac {31 (7434109-3088870 x)}{411600000000 \sqrt {x^2-2 x+3}}+\frac {3 (8233 x+8822)}{343000 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}+\frac {8878 x+5485}{117600 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}-\frac {30316369-15043110 x}{6860000000 \left (x^2-2 x+3\right )^{5/2}}+\frac {67 x+28}{1050 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}-\frac {4878869-2578034 x}{411600000 \left (x^2-2 x+3\right )^{7/2}}-\frac {1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}-\frac {3450497-2004270 x}{123480000 \left (x^2-2 x+3\right )^{9/2}}-\frac {\sqrt {\frac {1}{70} \left (110320475741093888 \sqrt {2}-151363871237318045\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (110320475741093888 \sqrt {2}-151363871237318045\right )}} \left (\left (932587773-620347970 \sqrt {2}\right ) x-312239803 \sqrt {2}[phone]\right )}{\sqrt {x^2-2 x+3}}\right )}{137200000000} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((3 - 2*x + x^2)^(11/2)*(1 + x + 2*x^2)^5),x]

[Out]

-1/123480000*(3450497 - 2004270*x)/(3 - 2*x + x^2)^(9/2) - (4878869 - 2578034*x)/(411600000*(3 - 2*x + x^2)^(7
/2)) - (30316369 - 15043110*x)/(6860000000*(3 - 2*x + x^2)^(5/2)) - (63043297 - 29625922*x)/(41160000000*(3 -
2*x + x^2)^(3/2)) - (31*(7434109 - 3088870*x))/(411600000000*Sqrt[3 - 2*x + x^2]) - (1 - 10*x)/(280*(3 - 2*x +
x^2)^(9/2)*(1 + x + 2*x^2)^4) + (28 + 67*x)/(1050*(3 - 2*x + x^2)^(9/2)*(1 + x + 2*x^2)^3) + (5485 + 8878*x)/
(117600*(3 - 2*x + x^2)^(9/2)*(1 + x + 2*x^2)^2) + (3*(8822 + 8233*x))/(343000*(3 - 2*x + x^2)^(9/2)*(1 + x +
2*x^2)) + (Sqrt[(151363871237318045 + 110320475741093888*Sqrt[2])/70]*ArcTan[(Sqrt[5/(7*(151363871237318045 +
110320475741093888*Sqrt[2]))]*(308108167 + 312239803*Sqrt[2] + (932587773 + 620347970*Sqrt[2])*x))/Sqrt[3 - 2*
x + x^2]])/137200000000 - (Sqrt[(-151363871237318045 + 110320475741093888*Sqrt[2])/70]*ArcTanh[(Sqrt[5/(7*(-15
1363871237318045 + 110320475741093888*Sqrt[2]))]*(308108167 - 312239803*Sqrt[2] + (932587773 - 620347970*Sqrt[
2])*x))/Sqrt[3 - 2*x + x^2]])/137200000000

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 )^{11/2} \left (1+x+2 x^2\right )^5} \, dx &=-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}-\frac {\int \frac {-1235+1335 x-800 x^2}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^4} \, dx}{1400}\\ &=-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}-\frac {\int \frac {-1015350+1334900 x-1313200 x^2}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^3} \, dx}{1470000}\\ &=-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}-\frac {\int \frac {-333716250+619001250 x-932190000 x^2}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )^2} \, dx}{1029000000}\\ &=-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {127736962500-7441875000 x-259339500000 x^2}{\left (3-2 x+x^2\right )^{11/2} \left (1+x+2 x^2\right )} \, dx}{360150000000}\\ &=-\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {32819267250000+52489111500000 x-168358680000000 x^2}{\left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )} \, dx}{648270000000000}\\ &=-\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {4878869-2578034 x}{411600000 \left (3-2 x+x^2\right )^{7/2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {-4101557985000000+36919386630000000 x-68214779640000000 x^2}{\left (3-2 x+x^2\right )^{7/2} \left (1+x+2 x^2\right )} \, dx}{907578000000000000}\\ &=-\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {4878869-2578034 x}{411600000 \left (3-2 x+x^2\right )^{7/2}}-\frac {30316369-15043110 x}{6860000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {-6061741906500000000+12245707845000000000 x-15921627624000000000 x^2}{\left (3-2 x+x^2\right )^{5/2} \left (1+x+2 x^2\right )} \, dx}{907578000000000000000}\\ &=-\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {4878869-2578034 x}{411600000 \left (3-2 x+x^2\right )^{7/2}}-\frac {30316369-15043110 x}{6860000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {63043297-29625922 x}{41160000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {-1654460252550000000000+1868769331380000000000 x-1567803792240000000000 x^2}{\left (3-2 x+x^2\right )^{3/2} \left (1+x+2 x^2\right )} \, dx}{544546800000000000000000}\\ &=-\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {4878869-2578034 x}{411600000 \left (3-2 x+x^2\right )^{7/2}}-\frac {30316369-15043110 x}{6860000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {63043297-29625922 x}{41160000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {31 (7434109-3088870 x)}{411600000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {-105286925935800000000000+71284514842800000000000 x}{\sqrt {3-2 x+x^2} \left (1+x+2 x^2\right )} \, dx}{108909360000000000000000000}\\ &=-\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {4878869-2578034 x}{411600000 \left (3-2 x+x^2\right )^{7/2}}-\frac {30316369-15043110 x}{6860000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {63043297-29625922 x}{41160000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {31 (7434109-3088870 x)}{411600000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}-\frac {\int \frac {3969000000000000 \left (222438197-132636591 \sqrt {2}\right )-3969000000000000 \left (42834985-89801606 \sqrt {2}\right ) x}{\sqrt {3-2 x+x^2} \left (1+x+2 x^2\right )} \, dx}{1089093600000000000000000000 \sqrt {2}}+\frac {\int \frac {3969000000000000 \left (222438197+132636591 \sqrt {2}\right )-3969000000000000 \left (42834985+89801606 \sqrt {2}\right ) x}{\sqrt {3-2 x+x^2} \left (1+x+2 x^2\right )} \, dx}{1089093600000000000000000000 \sqrt {2}}\\ &=-\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {4878869-2578034 x}{411600000 \left (3-2 x+x^2\right )^{7/2}}-\frac {30316369-15043110 x}{6860000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {63043297-29625922 x}{41160000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {31 (7434109-3088870 x)}{411600000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}-\frac {1}{7} \left (101250 \left (220640951482187776-151363871237318045 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-110270727000000000000000000000000 \left (151363871237318045-110320475741093888 \sqrt {2}\right )-5 x^2} \, dx,x,\frac {3969000000000000 \left (308108167-312239803 \sqrt {2}\right )+3969000000000000 \left (932587773-620347970 \sqrt {2}\right ) x}{\sqrt {3-2 x+x^2}}\right )-\frac {1}{7} \left (101250 \left (220640951482187776+151363871237318045 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-110270727000000000000000000000000 \left (151363871237318045+110320475741093888 \sqrt {2}\right )-5 x^2} \, dx,x,\frac {3969000000000000 \left (308108167+312239803 \sqrt {2}\right )+3969000000000000 \left (932587773+620347970 \sqrt {2}\right ) x}{\sqrt {3-2 x+x^2}}\right )\\ &=-\frac {3450497-2004270 x}{123480000 \left (3-2 x+x^2\right )^{9/2}}-\frac {4878869-2578034 x}{411600000 \left (3-2 x+x^2\right )^{7/2}}-\frac {30316369-15043110 x}{6860000000 \left (3-2 x+x^2\right )^{5/2}}-\frac {63043297-29625922 x}{41160000000 \left (3-2 x+x^2\right )^{3/2}}-\frac {31 (7434109-3088870 x)}{411600000000 \sqrt {3-2 x+x^2}}-\frac {1-10 x}{280 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}+\frac {28+67 x}{1050 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^3}+\frac {5485+8878 x}{117600 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^2}+\frac {3 (8822+8233 x)}{343000 \left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )}+\frac {\sqrt {\frac {1}{70} \left (151363871237318045+110320475741093888 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (308108167+312239803 \sqrt {2}+\left (932587773+620347970 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{137200000000}-\frac {\sqrt {\frac {1}{70} \left (-151363871237318045+110320475741093888 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (-151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (308108167-312239803 \sqrt {2}+\left (932587773-620347970 \sqrt {2}\right ) x\right )}{\sqrt {3-2 x+x^2}}\right )}{137200000000}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.67, size = 733, normalized size = 1.94 \begin {gather*} \frac {\frac {-53205422447+261702502714 x-266966654968 x^2+1002897791524 x^3-1409335257371 x^4+2503427226914 x^5-3359813871472 x^6+4591320676952 x^7-5134334619701 x^8+5380603084494 x^9-4915797913008 x^{10}[phone] x^{11}-2679143870481 x^{12}[phone] x^{13}-606785954952 x^{14}[phone] x^{15}-38639385552 x^{16}[phone] x^{17}}{\left (3-2 x+x^2\right )^{9/2} \left (1+x+2 x^2\right )^4}-49392 \text {RootSum}\left [14+7 \text {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-6014 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-10727 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+3229 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-56448 \text {RootSum}\left [14+7 \text {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {73781 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-60407 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+13104 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-504 \text {RootSum}\left [14+7 \text {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {275935046 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-208696097 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}[phone] \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]+1440 \text {RootSum}\left [14+7 \text {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {3276009822 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-2447831621 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}[phone] \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-18 \text {RootSum}\left [14+7 \text {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {254137663854 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-189631531133 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}[phone] \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{1234800000000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((3 - 2*x + x^2)^(11/2)*(1 + x + 2*x^2)^5),x]

[Out]

((-53205422447 + 261702502714*x - 266966654968*x^2 + 1002897791524*x^3 - 1409335257371*x^4 + 2503427226914*x^5
- 3359813871472*x^6 + 4591320676952*x^7 - 5134334619701*x^8 + 5380603084494*x^9 - 4915797913008*x^10 + 399965
6132532*x^11 - 2679143870481*x^12 + 1459208021718*x^13 - 606785954952*x^14 + 188603773872*x^15 - 38639385552*x
^16 + 4596238560*x^17)/((3 - 2*x + x^2)^(9/2)*(1 + x + 2*x^2)^4) - 49392*RootSum[14 + 7*#1 - 5*#1^2 - #1^3 + #
1^4 & , (-6014*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 10727*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1 + 3229*Log[-x
+ Sqrt[3 - 2*x + x^2] - #1]*#1^2)/(7 - 10*#1 - 3*#1^2 + 4*#1^3) & ] - 56448*RootSum[14 + 7*#1 - 5*#1^2 - #1^3
+ #1^4 & , (73781*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 60407*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1 + 13104*Log
[-x + Sqrt[3 - 2*x + x^2] - #1]*#1^2)/(7 - 10*#1 - 3*#1^2 + 4*#1^3) & ] - 504*RootSum[14 + 7*#1 - 5*#1^2 - #1^
3 + #1^4 & , (275935046*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 208696097*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1 +
50007219*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1^2)/(7 - 10*#1 - 3*#1^2 + 4*#1^3) & ] + 1440*RootSum[14 + 7*#1
- 5*#1^2 - #1^3 + #1^4 & , (3276009822*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 2447831621*Log[-x + Sqrt[3 - 2*x +
x^2] - #1]*#1 + 590084719*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1^2)/(7 - 10*#1 - 3*#1^2 + 4*#1^3) & ] - 18*Roo
tSum[14 + 7*#1 - 5*#1^2 - #1^3 + #1^4 & , (254137663854*Log[-x + Sqrt[3 - 2*x + x^2] - #1] - 189631531133*Log[
-x + Sqrt[3 - 2*x + x^2] - #1]*#1 + 45801521671*Log[-x + Sqrt[3 - 2*x + x^2] - #1]*#1^2)/(7 - 10*#1 - 3*#1^2 +
4*#1^3) & ])/1234800000000

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(21027\) vs. \(2(298)=596\).
time = 1.52, size = 21028, normalized size = 55.63

method	result	size
risch	\(\frac {4596238560 x^{17}-38639385552 x^{16}[phone] x^{15}-606785954952 x^{14}[phone] x^{13}-2679143870481 x^{12}[phone] x^{11}-4915797913008 x^{10}[phone] x^{9}-5134334619701 x^{8}[phone] x^{7}-3359813871472 x^{6}[phone] x^{5}-1409335257371 x^{4}[phone] x^{3}-266966654968 x^{2}[phone] x -53205422447}{1234800000000 \left (x^{2}-2 x +3\right )^{\frac {9}{2}} \left (2 x^{2}+x +1\right )^{4}}+\frac {\sqrt {4}\, \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \sqrt {2}\, \left (9625722625 \sqrt {-6050+4280 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-6050+4280 \sqrt {2}}\, \left (40 \sqrt {2}+57\right ) \left (\sqrt {2}-1+x \right )}{49 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-350+280 \sqrt {2}}\, \sqrt {2}[phone] \sqrt {-6050+4280 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-6050+4280 \sqrt {2}}\, \left (40 \sqrt {2}+57\right ) \left (\sqrt {2}-1+x \right )}{49 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-350+280 \sqrt {2}}[phone] \arctanh \left (\frac {7 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}}{\sqrt {-350+280 \sqrt {2}}}\right ) \sqrt {2}-607941010600 \arctanh \left (\frac {7 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}}{\sqrt {-350+280 \sqrt {2}}}\right )\right )}{268912000000000 \sqrt {\frac {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}{\left (\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}+1\right )^{2}}}\, \left (\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}+1\right ) \sqrt {-350+280 \sqrt {2}}}\)	\(452\)
trager	\(\text {Expression too large to display}\)	\(541\)
default	\(\text {Expression too large to display}\)	\(21028\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2-2*x+3)^(11/2)/(2*x^2+x+1)^5,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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)^(11/2)/(2*x^2+x+1)^5,x, algorithm="maxima")

[Out]

integrate(1/((2*x^2 + x + 1)^5*(x^2 - 2*x + 3)^(11/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1873 vs. \(2 (298) = 596\).
time = 1.13, size = 1873, normalized size = 4.96 \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)^(11/2)/(2*x^2+x+1)^5,x, algorithm="fricas")

[Out]

1/7108652444723216758028075295024000000000*(26460206086876512301981559146074412800*x^18 - 21168164869501209841
5852473168595302400*x^17 + 1018717934344745723626290027123864892800*x^16 - 32149150395554962446907594362480411
55200*x^15 + 7688343631118056605744516779381246569200*x^14 - 13980911391153377187559506313807067863200*x^13 +
20977982138251784909414754860497120398000*x^12 - 25712705264922250829450580100197810638400*x^11 + 287572827277
93479526197333249442997761200*x^10 - 27283780001330543747380735174495978898400*x^9 + 2556221284280314066573305
9982554512415600*x^8 - 18045860551249781389951423337622749529600*x^7 + 152063496855518456635450272717596391060
00*x^6 - 7266634096608462190931685680490685615200*x^5 - 3602042876982878244*337802213083473608^(1/4)*sqrt(2054
87899)*sqrt(35)*sqrt(2)*(16*x^18 - 128*x^17 + 616*x^16 - 1944*x^15 + 4649*x^14 - 8454*x^13 + 12685*x^12 - 1554
8*x^11 + 17389*x^10 - 16498*x^9 + 15457*x^8 - 10912*x^7 + 9195*x^6 - 4394*x^5 + 4407*x^4 - 396*x^3 + 1647*x^2
+ 162*x + 243)*sqrt(151363871237318045*sqrt(2) + 220640951482187776)*arctan(1/96439362234996391967746783551420
5441102895152270484353118304*sqrt(205487899)*(12071210867722009415131100925112940*sqrt(41672947348129)*sqrt(7)
*sqrt(2)*(10*sqrt(2) + 9) + sqrt(205487899)*(5*337802213083473608^(3/4)*sqrt(41672947348129)*sqrt(35)*(5346780
00*sqrt(2) - 573381349) + 2876830586*337802213083473608^(1/4)*sqrt(41672947348129)*sqrt(35)*(201502465*sqrt(2)
+ 108532744))*sqrt(151363871237318045*sqrt(2) + 220640951482187776) + 2414242173544401883026220185022588*sqrt
(41672947348129)*sqrt(7)*(125*sqrt(2) + 172))*sqrt(164483605088694913184970968*x^2 + sqrt(205487899)*(33780221
3083473608^(1/4)*sqrt(35)*sqrt(7)*sqrt(x^2 - 2*x + 3)*(89801606*sqrt(2) - 42834985) - 337802213083473608^(1/4)
*sqrt(35)*sqrt(7)*(sqrt(2)*(89801606*x - 132636591) - 42834985*x + 222438197))*sqrt(151363871237318045*sqrt(2)
+ 220640951482187776) - 41120901272173728296242742*sqrt(x^2 - 2*x + 3)*(4*x + 1) - 12336270381652118488872822
6*x + 205604506360868641481213710*sqrt(2) + 287846308905216098073699194) + 5/476*sqrt(7)*sqrt(2)*(sqrt(2)*(10*
x - 19) + 9*x - 29) + 1/1149179274607135296320480808070751888*sqrt(205487899)*(5*337802213083473608^(3/4)*sqrt
(35)*(sqrt(2)*(534678000*x + 38703349) - 573381349*x - 495974651) + 2876830586*337802213083473608^(1/4)*sqrt(3
5)*(sqrt(2)*(201502465*x - 310035209) + 108532744*x - 511537674) - (5*337802213083473608^(3/4)*sqrt(35)*(53467
8000*sqrt(2) - 573381349) + 2876830586*337802213083473608^(1/4)*sqrt(35)*(201502465*sqrt(2) + 108532744))*sqrt
(x^2 - 2*x + 3))*sqrt(151363871237318045*sqrt(2) + 220640951482187776) - 1/476*sqrt(x^2 - 2*x + 3)*(5*sqrt(7)*
sqrt(2)*(10*sqrt(2) + 9) + sqrt(7)*(125*sqrt(2) + 172)) + 1/476*sqrt(7)*(25*sqrt(2)*(5*x - 1) + 172*x - 82)) -
3602042876982878244*337802213083473608^(1/4)*sqrt(205487899)*sqrt(35)*sqrt(2)*(16*x^18 - 128*x^17 + 616*x^16
- 1944*x^15 + 4649*x^14 - 8454*x^13 + 12685*x^12 - 15548*x^11 + 17389*x^10 - 16498*x^9 + 15457*x^8 - 10912*x^7
+ 9195*x^6 - 4394*x^5 + 4407*x^4 - 396*x^3 + 1647*x^2 + 162*x + 243)*sqrt(151363871237318045*sqrt(2) + 220640
951482187776)*arctan(-1/964393622349963919677467835514205441102895152270484353118304*sqrt(205487899)*(12071210
867722009415131100925112940*sqrt(41672947348129)*sqrt(7)*sqrt(2)*(10*sqrt(2) + 9) - sqrt(205487899)*(5*3378022
13083473608^(3/4)*sqrt(41672947348129)*sqrt(35)*(534678000*sqrt(2) - 573381349) + 2876830586*33780221308347360
8^(1/4)*sqrt(41672947348129)*sqrt(35)*(201502465*sqrt(2) + 108532744))*sqrt(151363871237318045*sqrt(2) + 22064
0951482187776) + 2414242173544401883026220185022588*sqrt(41672947348129)*sqrt(7)*(125*sqrt(2) + 172))*sqrt(164
483605088694913184970968*x^2 - sqrt(205487899)*(337802213083473608^(1/4)*sqrt(35)*sqrt(7)*sqrt(x^2 - 2*x + 3)*
(89801606*sqrt(2) - 42834985) - 337802213083473608^(1/4)*sqrt(35)*sqrt(7)*(sqrt(2)*(89801606*x - 132636591) -
42834985*x + 222438197))*sqrt(151363871237318045*sqrt(2) + 220640951482187776) - 41120901272173728296242742*sq
rt(x^2 - 2*x + 3)*(4*x + 1) - 123362703816521184888728226*x + 205604506360868641481213710*sqrt(2) + 2878463089
05216098073699194) - 5/476*sqrt(7)*sqrt(2)*(sqrt(2)*(10*x - 19) + 9*x - 29) + 1/114917927460713529632048080807
0751888*sqrt(205487899)*(5*337802213083473608^(3/4)*sqrt(35)*(sqrt(2)*(534678000*x + 38703349) - 573381349*x -
495974651) + 2876830586*337802213083473608^(1/4)*sqrt(35)*(sqrt(2)*(201502465*x - 310035209) + 108532744*x -
511537674) - (5*337802213083473608^(3/4)*sqrt(35)*(534678000*sqrt(2) - 573381349) + 2876830586*337802213083473
608^(1/4)*sqrt(35)*(201502465*sqrt(2) + 108532744))*sqrt(x^2 - 2*x + 3))*sqrt(151363871237318045*sqrt(2) + 220
640951482187776) + 1/476*sqrt(x^2 - 2*x + 3)*(5*sqrt(7)*sqrt(2)*(10*sqrt(2) + 9) + sqrt(7)*(125*sqrt(2) + 172)
) - 1/476*sqrt(7)*(25*sqrt(2)*(5*x - 1) + 172*x - 82)) + 9*337802213083473608^(1/4)*sqrt(205487899)*sqrt(35)*s
qrt(7)*(3530255223715004416*x^18 - 28242041789720035328*x^17 + 135914826113027670016*x^16 - 428926009681373036
544*x^15 + 1025759783440690970624*x^14 - 186529...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x^{2} - 2 x + 3\right )^{\frac {11}{2}} \left (2 x^{2} + x + 1\right )^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2-2*x+3)**(11/2)/(2*x**2+x+1)**5,x)

[Out]

Integral(1/((x**2 - 2*x + 3)**(11/2)*(2*x**2 + x + 1)**5), x)

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Giac [C] Result contains complex when optimal does not.
time = 34.88, size = 2509, normalized size = 6.64 \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)^(11/2)/(2*x^2+x+1)^5,x, algorithm="giac")

[Out]

1/19208000000000*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*log(3136*(24743015359883014513591422
6638091465128017779071251327216101236181293485559300330785024470114864584026604284622700*sqrt(7)*sqrt(2)*sqrt(
7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 1443
34256265984251329283298872220021324677037791563274209392387772421199909591859624597607567004340682185832696575
0*sqrt(7)*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 2886685125319685026585665977444400426493540755
831265484187847755448423998191837192491952151340086813643716653931500*sqrt(2)*(110320475741093888*sqrt(2) - 15
1363871237318045)^3 + 2061917946656917876132618555317428876066814825593761060134176968177445712994169423208537
25095720486688836903852250*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2
) - 151363871237318045)^3 - 1049138541122969625735220807296230414367334042656225923210840932892592514150275756
86933144355006438004151420024881229481000*sqrt(7)*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045)^2
- 104913854112296962573522080729623041436733404265622592321084093289259251415027575686933144355006438004151420
02488122948100*sqrt(7)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) -
151363871237318045)^2 - 20982770822459392514704416145924608287346680853124518464216818657851850283005515137386
628871001287600830284004976245896200*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(1103204
75741093888*sqrt(2) - 151363871237318045)^2 - 1223994964643464563357757608512268816761889716432263577079314421
70802459984198838301422001747507511004843323362361434394500*(110320475741093888*sqrt(2) - 151363871237318045)^
3 + 7245039962569580166831441103036590485272265790998347306296004020158434652823450341545126027545398381938404
5801613291562600472200700*sqrt(7)*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(1103204757
41093888*sqrt(2) - 151363871237318045) + 633940996724838266247285453841235968367418100966298490154352212238871
880229393479427155097805557896986440201529880194683449362574125*sqrt(7)*(110320475741093888*sqrt(2) - 15136387
1237318045)^2 + 1267881993449676532082187318351088361508312490869111205095341459358991548431951565218821053012
281909331172952868319415989224917443750*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 12678819
93449676535381256033002156963320502179376997406802245180309009244646634714302734193802952986464832554399847203
01061537907975*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 15136387
1237318045)^2 - 2126452822068698508278415644474982440028614132240433950807302144189930664852200062263413008378
0322911432744853704088500185306368048860002880*sqrt(7)*sqrt(2)*(110320475741093888*sqrt(2) - 15136387123731804
5) - 354408803678116418845746257791902502469699160932438355128709627960208195159816911628734559796548815662417
3235998790978918611634665760818080*sqrt(7)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475
741093888*sqrt(2) - 151363871237318045) - 70881760735623283749165660298895364765649912047511853609221277160494
47858985212646112610216597117728735334198568887900615291459333783931760*sqrt(2)*sqrt(7722433301876572160*sqrt(
2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045) - 6202154064367037342040509157792
9394267973114014403593823713156536262791860899401705831007948594881042345033377486353135919026969012248710900*
(110320475741093888*sqrt(2) - 151363871237318045)^2 + 32899118880973852710380417819638428242533122398715783525
42445952179742344286835588398026084412498768879366524177397299680463206375263895127106986700*sqrt(7)*sqrt(2)*s
qrt(7722433301876572160*sqrt(2) - 10595470986612263150) + 5757345804170424253296732968715045594687603850659596
1640397621471290299770258470645587747906056941016215068838775424495834136105150442722719839966690*sqrt(7)*(110
320475741093888*sqrt(2) - 151363871237318045) + 11514691608340848499348425974860511029613199558249177418119093
5500623259371276266538520099917051160383062730419039816825148360727220332176154717624880*sqrt(2)*(110320475741
093888*sqrt(2) - 151363871237318045) + 19191152680568080928847909459028587443440938343434315974646074253344559
590752575102628626876766807523046716011108365196910710221166349350116972946360*sqrt(7722433301876572160*sqrt(2
) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045) - 42190834681344113572044118299115
75030935087599894152681923125521189411158761429808471398986722740737297060377427265233396895882194727754358575
99079592655320*sqrt(7)*sqrt(2) - 21095417340672056867029785704555913528740324247840131983737454865741140167685
6640700014506909797905726420309651469993318236639490549988690120558780166576108*sqrt(7)*sqrt(77224333018765721
60*sqrt(2) - 10595470986612263150) - 4219083468134411371380763977036231747195786343738197611526042097212408620
85099587557106516194625617695932035781267647104...

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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 )}^5\,{\left (x^2-2\,x+3\right )}^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x + 2*x^2 + 1)^5*(x^2 - 2*x + 3)^(11/2)),x)

[Out]

int(1/((x + 2*x^2 + 1)^5*(x^2 - 2*x + 3)^(11/2)), x)

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