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 = 0.51, 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 {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 (151363871237318045+110320475741093888 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{7 \left (151363871237318045+110320475741093888 \sqrt {2}\right )}} \left (\left (932587773+620347970 \sqrt {2}\right ) x+312239803 \sqrt {2}+308108167\right )}{\sqrt {x^2-2 x+3}}\right )}{137200000000}-\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}+308108167\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*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.58, 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}+3999656132532 x^{11}-2679143870481 x^{12}+1459208021718 x^{13}-606785954952 x^{14}+188603773872 x^{15}-38639385552 x^{16}+4596238560 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 {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-6014 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-10727 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+3229 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-56448 \text {RootSum}\left [14+7 \text {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {73781 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-60407 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+13104 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-504 \text {RootSum}\left [14+7 \text {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {275935046 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-208696097 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+50007219 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]+1440 \text {RootSum}\left [14+7 \text {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {3276009822 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-2447831621 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+590084719 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-18 \text {RootSum}\left [14+7 \text {$\#$1}-5 \text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {254137663854 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right )-189631531133 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+45801521671 \log \left (-x+\sqrt {3-2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{7-10 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$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
________________________________________________________________________________________
Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
mathics('Integrate[1/((1 + x + 2*x^2)^5*(3 - 2*x + x^2)^(11/2)),x]')
[Out]
Timed out
________________________________________________________________________________________
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 |
\(\frac {4596238560 x^{17}-38639385552 x^{16}+188603773872 x^{15}-606785954952 x^{14}+1459208021718 x^{13}-2679143870481 x^{12}+3999656132532 x^{11}-4915797913008 x^{10}+5380603084494 x^{9}-5134334619701 x^{8}+4591320676952 x^{7}-3359813871472 x^{6}+2503427226914 x^{5}-1409335257371 x^{4}+1002897791524 x^{3}-266966654968 x^{2}+261702502714 x -53205422447}{1234800000000 \left (x^{2}-2 x +3\right )^{\frac {9}{2}} \left (2 x^{2}+x +1\right )^{4}}+\frac {\sqrt {4}\, \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \sqrt {2}\, \left (9625722625 \sqrt {-6050+4280 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-6050+4280 \sqrt {2}}\, \left (40 \sqrt {2}+57\right ) \left (\sqrt {2}-1+x \right )}{49 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-350+280 \sqrt {2}}\, \sqrt {2}+13664181884 \sqrt {-6050+4280 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-6050+4280 \sqrt {2}}\, \left (40 \sqrt {2}+57\right ) \left (\sqrt {2}-1+x \right )}{49 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}\, \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-350+280 \sqrt {2}}+456968008770 \arctanh \left (\frac {7 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}}{\sqrt {-350+280 \sqrt {2}}}\right ) \sqrt {2}-607941010600 \arctanh \left (\frac {7 \sqrt {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}}{\sqrt {-350+280 \sqrt {2}}}\right )\right )}{268912000000000 \sqrt {\frac {\frac {\left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+1}{\left (\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}+1\right )^{2}}}\, \left (\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}+1\right ) \sqrt {-350+280 \sqrt {2}}}\) |
\(452\) |
| 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
________________________________________________________________________________________
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)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1873 vs.
\(2 (298) = 596\).
time = 0.38, size = 1873, normalized size = 4.96
result too large to display
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 - 1865298603830415458304*x^13 + 2798830469551551938560*x^12 - 343052551
3645055541248*x^11 + 3836725505323763236864*x^10 - 3640134417553133928448*x^9 + 3410447187060176453632*x^8 - 2
407634062573633011712*x^7 + 2028793548878716600320*x^6 - 969496340812733087744*x^5 + 972364673182001528832*x^4
- 87373816786946359296*x^3 + 363395647091163267072*x^2 - 151363871237318045*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 - 109
12*x^7 + 9195*x^6 - 4394*x^5 + 4407*x^4 - 396*x^3 + 1647*x^2 + 162*x + 243) + 35743834140114419712*x + 5361575
1210171629568)*sqrt(151363871237318045*sqrt(2) + 220640951482187776)*log(1908351235261833493759852130293986099
2*x^2 + 236911417693579806112743424/2041974420058321*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)*(8980
1606*x - 132636591) - 42834985*x + 222438197))*sqrt(151363871237318045*sqrt(2) + 220640951482187776) - 4770878
088154583734399630325734965248*sqrt(x^2 - 2*x + 3)*(4*x + 1) - 14312634264463751203198890977204895744*x + 2385
4390440772918671998151628674826240*sqrt(2) + 33396146617082086140797412280144756736) - 9*337802213083473608^(1
/4)*sqrt(205487899)*sqrt(35)*sqrt(7)*(3530255223715004416*x^18 - 28242041789720035328*x^17 + 13591482611302767
0016*x^16 - 428926009681373036544*x^15 + 1025759783440690970624*x^14 - 1865298603830415458304*x^13 + 279883046
9551551938560*x^12 - 3430525513645055541248*x^11 + 3836725505323763236864*x^10 - 3640134417553133928448*x^9 +
3410447187060176453632*x^8 - 2407634062573633011712*x^7 + 2028793548878716600320*x^6 - 969496340812733087744*x
^5 + 972364673182001528832*x^4 - 87373816786946359296*x^3 + 363395647091163267072*x^2 - 151363871237318045*sqr
t(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) + 35
743834140114419712*x + 53615751210171629568)*sqrt(151363871237318045*sqrt(2) + 220640951482187776)*log(1908351
2352618334937598521302939860992*x^2 - 236911417693579806112743424/2041974420058321*sqrt(205487899)*(3378022130
83473608^(1/4)*sqrt(35)*sqrt(7)*sqrt(x^2 - 2*x + 3)*(89801606*sqrt(2) - 42834985) - 337802213083473608^(1/4)*s
qrt(35)*sqrt(7)*(sqrt(2)*(89801606*x - 132636591) - 42834985*x + 222438197))*sqrt(151363871237318045*sqrt(2) +
220640951482187776) - 4770878088154583734399630325734965248*sqrt(x^2 - 2*x + 3)*(4*x + 1) - 14312634264463751
203198890977204895744*x + 23854390440772918671998151628674826240*sqrt(2) + 33396146617082086140797412280144756
736) + 7288133014054049357177045697296871075600*x^4 - 654890100650193679474043588865341716800*x^3 + 2723747464
067850985085226744599034867600*x^2 + 5756926178104321961473983880*(4596238560*x^17 - 38639385552*x^16 + 188603
773872*x^15 - 606785954952*x^14 + 1459208021718*x^13 - 2679143870481*x^12 + 3999656132532*x^11 - 4915797913008
*x^10 + 5380603084494*x^9 - 5134334619701*x^8 + 4591320676952*x^7 - 3359813871472*x^6 + 2503427226914*x^5 - 14
09335257371*x^4 + 1002897791524*x^3 - 266966654968*x^2 + 261702502714*x - 53205422447)*sqrt(x^2 - 2*x + 3) + 2
67909586629624687057563286354003429600*x + 401864379944437030586344929531005144400)/(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 - 1091
2*x^7 + 9195*x^6 - 4394*x^5 + 4407*x^4 - 396*x^3 + 1647*x^2 + 162*x + 243)
________________________________________________________________________________________
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)
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 26.73, size = 18547, normalized size = 49.07
result too large to display
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]
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
85099587557106516194625617695932035781267647104895808573044339426070450150150098280*sqrt(2)*sqrt(7722433301876
572160*sqrt(2) - 10595470986612263150) - 920589952140153149540860681042500826098422635875714185943723383762178
09267922249720111767473806366273424910555828722901509465001692379638157996477568878185171912773*sqrt(7) - 8145
39767127070039237583513123589170024107600674987295687521641346791604050146707308458885605400724949845902669298
065086595442158392477166657536193932267555915756504095096650*sqrt(2) - 920589952140153111158555319906339171263
72266976678439922113048655699796659517564449092929141850974541546613899009112709389970185858812421123596605881
876967985540965*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150) + 11532882145479371991155511150683805
49180924836937115919743481151633860416743449381465954829260459683046848466114871616567450209145337077878522828
152145687185666015740023320320)^2 + 3136*(34124314806601555041367954040995026009203193083759390918637565727519
13121135591082568720686452963161513000*sqrt(7)*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150
)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 199058503038509071074646398572470985053685292988596447
02538580007719493206624281314984204004308951775492500*sqrt(7)*(110320475741093888*sqrt(2) - 151363871237318045
)^3 + 39811700607701814214929279714494197010737058597719289405077160015438986413248562629968408008617903550985
000*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 28436929005501295867806628367495855007669327
56979949243219797143959927600946325902140600572044135967927500*sqrt(7722433301876572160*sqrt(2) - 105954709866
12263150)*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 7685814726400200227147651553044706147214968426
720207048666477678525633293871852840661789826971901179214250*sqrt(7)*sqrt(2)*(110320475741093888*sqrt(2) - 151
363871237318045)^2 + 76858147264002002271476515530447061472149684267202070486664776785256332938718528406617898
2697190117921425*sqrt(7)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2)
- 151363871237318045)^2 + 153716294528004004542953031060894122944299368534404140973329553570512665877437056813
2357965394380235842850*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sq
rt(2) - 151363871237318045)^2 + 896678384746690026500559347855215717175079649784024155677755729161323884285049
4980772088131467218042416625*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 976510829351336985318426092
720643328053917556999006303431427669091931152434985476815319185951875160339805775571594750014000*sqrt(7)*sqrt(
2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045)
+ 854446975682419864428577151570666581471708131773798916116045416039399152956450007381346368464951487009810511
2604541806042500*sqrt(7)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 1708893951364839728288415723031
3072455872836961976807320787042806828084572689405859707072167406377935915009081120811676230000*sqrt(2)*(110320
475741093888*sqrt(2) - 151363871237318045)^2 + 170889395136483973283832436391151458443634423499701753330579669
1850727246420616016306591378254759237016461823382698716307000*sqrt(7722433301876572160*sqrt(2) - 1059547098661
2263150)*(110320475741093888*sqrt(2) - 151363871237318045)^2 - 16156629103216429859182505251559538530528459085
62683937530778095691283476808165943145292528342610931516231547450242131454340*sqrt(7)*sqrt(2)*(110320475741093
888*sqrt(2) - 151363871237318045) - 26927715172027383040078920360978657724993896167132685456256065962700766147
4391262636548122809722557868949842520038360635590*sqrt(7)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263
150)*(110320475741093888*sqrt(2) - 151363871237318045) - 53855430344054766094797487819862458739792842911493382
6214930241667733552447191623911679736931289931750505205553827219922880*sqrt(2)*sqrt(7722433301876572160*sqrt(2
) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045) - 47123501551047920230470272157043
64836868338350696062783094015045632392519024289804526353306319649781934534112633453268706200*(1103204757410938
88*sqrt(2) - 151363871237318045)^2 + 4193859254705817563786545204110979417249615218275883808083699006997908745
9179211592863757285640930322993492863343001050374777654585752870920*sqrt(7)*sqrt(2)*sqrt(7722433301876572160*s
qrt(2) - 10595470986612263150) + 73392536957351807756868872812476933721242879824990105229931845076953384941467
6062708934630055035134045830109366690499889371007117918795898700*sqrt(7)*(110320475741093888*sqrt(2) - 1513638
71237318045) + 14678507391470361541608666268982016900814119268249174920123751742827764242323014015762064196361
58097776887638820246148744285006222232316501400*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045) + 24
46417898578393603288255436181495977780897934081649365296599297616538778164584609859198885592495876512187088988
51314243292243938339792092100*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqr
t(2) - 151363871237318045) - 129207775331269401934589343014991032633136481739826341489743140048131430668820437
812159214718582829755008425909786359842144150949633159991150*sqrt(7)*sqrt(2) + 7587629160082400034098191013754
10993692930390503123728189868562072536874773198417845474059096092571231597614031305995383233253822344649520984
786771597244159013334688*x - 646038876656347010288437347578438205687950441405951033530044590248134676986715874
11875073203121114557903957207825219859233524843054466661575*sqrt(7)*sqrt(7722433301876572160*sqrt(2) - 1059547
0986612263150) - 129207775331269402042300203703100565068677528624357666238053675568548792622685475800752550765
348440250164638699943637093422466523499441710582*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 105954709866122631
50) - 36018223914820167914680328062277541844579926085292035680717885777055540822317688033834995139282160837267
30307366377332828337773366071083598072772386051930660*sqrt(7) - 2566532515528593027457553740848822251225795581
35538546802966417819334438408482678055766092810745610958399639440237969215404840097857114692088861260268773750
000*sqrt(2) - 758762916008240003409819101375410993692930390503123728189868562072536874773198417845474059096092
571231597614031305995383233253822344649520984786771597244159013334688*sqrt(x^2 - 2*x + 3) - 360182239148201674
25396748346598823921159161032736404931012591722796526817745688546882730709582846841129178670301992545389090569
56640775330160129944904845080*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150) + 189691081878191176772
04446029749420693848707484584497610859189744893664882215461159303816661032890098995841242558407986974293025662
4900190943273280656848430181850997)^2) - 1/19208000000000*sqrt(7722433301876572160*sqrt(2) - 10595470986612263
150)*log(3136*(24743015329451280770116633045664126608658995463429055851824680263463919214336932408224360731263
4021080788865625700*sqrt(7)*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(1103204757410938
88*sqrt(2) - 151363871237318045)^2 + 1443342560884658044923470260997074052171774735366694924689773015368728620
836321057146421042657031789637935049483250*sqrt(7)*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 28866
85121769316089846940521994148104343549470733389849379546030737457241672642114292842085314063579275870098966500
*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 20619179441209400641763860871386772173882496219
5242132098539002195532660119474436735203006093861684233990721354750*sqrt(7722433301876572160*sqrt(2) - 1059547
0986612263150)*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 10491385490060546359808826122527787179224
3668954806319376337814136029745746425269205752115419545432491274148221669945001000*sqrt(7)*sqrt(2)*(1103204757
41093888*sqrt(2) - 151363871237318045)^2 + 1049138549006054635980882612252778717922436689548063193763378141360
2974574642526920575211541954543249127414822166994500100*sqrt(7)*sqrt(7722433301876572160*sqrt(2) - 10595470986
612263150)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 209827709801210927196176522450555743584487337
90961263875267562827205949149285053841150423083909086498254829644333989000200*sqrt(2)*sqrt(7722433301876572160
*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 12239949738403970753110
2971429490850424284280447274039272394116492034703370829480740044134656136337906486506258614935834500*(11032047
5741093888*sqrt(2) - 151363871237318045)^3 + 72450399556079180248329015146999235208892617070047340278059551131
392900835267393877060925502058563859855540246511032336003698229700*sqrt(7)*sqrt(2)*sqrt(7722433301876572160*sq
rt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045) + 6339409961156928288224132378
32995359418919269073856001343620769961624939096901713988544379098807927655360025907865006811958130347875*sqrt(
7)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 12678819922313856572324428868418027060025613207199765
59209591615532765613996725423586023437958666981840314039628006645155934817986250*sqrt(2)*(110320475741093888*s
qrt(2) - 151363871237318045)^2 + 12678819922313856605315115974353068086847790601418601070307910106566397275733
4945871454599987025796960356213712979359289978635966225*sqrt(7722433301876572160*sqrt(2) - 1059547098661226315
0)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 21264527927126900961164377132668373311820329495714368
356539982937044757067834357827140293364137684307991178111246209024321545469979809839680*sqrt(7)*sqrt(2)*(11032
0475741093888*sqrt(2) - 151363871237318045) + 3544087987854483501520832752633716588014874818830708767403800505
111750511407655000601839046131539324151071812411163027198287919028630782880*sqrt(7)*sqrt(7722433301876572160*s
qrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045) + 708817597570896700104330636
43035195836852113286085056911458168396557616061231611605173972209025973300973414263547940069437349195492626133
60*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237
318045) + 6202153978745346139901407055512974782136328075902825385385078928672967322202807900256688667813141891
2687891373331690882456544841615974534900*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 328991188476301
52211015411144055054747532701564810653878870517591440238490285735651535789555031009928566367922372193172392300
96498179223616314209700*sqrt(7)*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150) + 57573457983
35276665907856772641306665068371869928396900097820597735224485555484099581070415601388699164524304214143405050
3060360394862431867078254790*sqrt(7)*(110320475741093888*sqrt(2) - 151363871237318045) + 115146915966705533245
70673589674695312373275176428664785053988930084036831493297504145438133911660943041218690890125397893446827788
8412836527277030080*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045) + 191911526611175889708849887242
30066021393869653903156391556097575709573739753723383708361939864243785854901671586306846977795770326107328538
508811760*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 1513638712373
18045) + 42190832671754364192435882068986125477387841164893037938190180929920275078233100150837480351787062815
1016355380370573185797674929411754956761480005114821080*sqrt(7)*sqrt(2) + 210954163358771821772256664711669711
25635837748319235420777168987856867089453384049969627290987030043155180365618048585416083969716512306115822804
6039881708*sqrt(7)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150) + 42190832671754364334199401583165
46516252963276902594609801197158120119916596969628857249046996928079015439356752814246211982667749167166247357
20821529103720*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150) - 9205899640875603209167615810
42015283905433780645215921762515407245365322347939602047491610965392490295654216539646083128661631912861645825
99975902782992478976911083*sqrt(7) + 8145397279615270466948378330621458645845906344529745650843197576833185158
54195479301757363227685529077360827463499277712322108218563975398844449113459715493032206173237469850*sqrt(2)
- 920589964087560282534456258806837544996575089206789003174950888192444472823378371605758126780610256882687440
52098593389738274301105516868817358383880901522380694835*sqrt(7722433301876572160*sqrt(2) - 105954709866122631
50) - 11532881607348687037747868321346713994558911627676219472353978719356293894710618931600085322372043380298
05266313956553667567456432846022880089884089461103241448700675305025280)^2 + 3136*(341243148066015550413679540
4099502600920319308375939091863756572751913121135591082568720686452963161513000*sqrt(7)*sqrt(2)*sqrt(772243330
1876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 1990585030385
0907107464639857247098505368529298859644702538580007719493206624281314984204004308951775492500*sqrt(7)*(110320
475741093888*sqrt(2) - 151363871237318045)^3 + 398117006077018142149292797144941970107370585977192894050771600
15438986413248562629968408008617903550985000*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 284
3692900550129586780662836749585500766932756979949243219797143959927600946325902140600572044135967927500*sqrt(7
722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045)^3 + 76858
14726400200227147651553044706147214968426720207048666477678525633293871852840661789826971901179214250*sqrt(7)*
sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 768581472640020022714765155304470614721496842672
020704866647767852563329387185284066178982697190117921425*sqrt(7)*sqrt(7722433301876572160*sqrt(2) - 105954709
86612263150)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 1537162945280040045429530310608941229442993
685344041409733295535705126658774370568132357965394380235842850*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 105
95470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 8966783847466900265005593478552157171
750796497840241556777557291613238842850494980772088131467218042416625*(110320475741093888*sqrt(2) - 1513638712
37318045)^3 + 976510829351336985318426092720643328053917556999006303431427669091931152434985476815319185951875
160339805775571594750014000*sqrt(7)*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(11032047
5741093888*sqrt(2) - 151363871237318045) + 8544469756824198644285771515706665814717081317737989161160454160393
991529564500073813463684649514870098105112604541806042500*sqrt(7)*(110320475741093888*sqrt(2) - 15136387123731
8045)^2 + 1708893951364839728288415723031307245587283696197680732078704280682808457268940585970707216740637793
5915009081120811676230000*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 1708893951364839732838
324363911514584436344234997017533305796691850727246420616016306591378254759237016461823382698716307000*sqrt(77
22433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045)^2 - 161566
29103216429859182505251559538530528459085626839375307780956912834768081659431452925283426109315162315474502421
31454340*sqrt(7)*sqrt(2)*(110320475741093888*sqrt(2) - 151363871237318045) - 269277151720273830400789203609786
577249938961671326854562560659627007661474391262636548122809722557868949842520038360635590*sqrt(7)*sqrt(772243
3301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045) - 538554303440
54766094797487819862458739792842911493382621493024166773355244719162391167973693128993175050520555382721992288
0*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*(110320475741093888*sqrt(2) - 1513638712373
18045) - 47123501551047920230470272157043648368683383506960627830940150456323925190242898045263533063196497819
34534112633453268706200*(110320475741093888*sqrt(2) - 151363871237318045)^2 + 41938592547058175637865452041109
794172496152182758838080836990069979087459179211592863757285640930322993492863343001050374777654585752870920*s
qrt(7)*sqrt(2)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150) + 733925369573518077568688728124769337
212428798249901052299318450769533849414676062708934630055035134045830109366690499889371007117918795898700*sqrt
(7)*(110320475741093888*sqrt(2) - 151363871237318045) + 146785073914703615416086662689820169008141192682491749
2012375174282776424232301401576206419636158097776887638820246148744285006222232316501400*sqrt(2)*(110320475741
093888*sqrt(2) - 151363871237318045) + 24464178985783936032882554361814959777808979340816493652965992976165387
7816458460985919888559249587651218708898851314243292243938339792092100*sqrt(7722433301876572160*sqrt(2) - 1059
5470986612263150)*(110320475741093888*sqrt(2) - 151363871237318045) - 1292077753312694019345893430149910326331
36481739826341489743140048131430668820437812159214718582829755008425909786359842144150949633159991150*sqrt(7)*
sqrt(2) - 7587629160082400034098191013754109936929303905031237281898685620725368747731984178454740590960925712
31597614031305995383233253822344649520984786771597244159013334688*x - 6460388766563470102884373475784382056879
5044140595103353004459024813467698671587411875073203121114557903957207825219859233524843054466661575*sqrt(7)*s
qrt(7722433301876572160*sqrt(2) - 10595470986612263150) - 1292077753312694020423002037031005650686775286243576
66238053675568548792622685475800752550765348440250164638699943637093422466523499441710582*sqrt(2)*sqrt(7722433
301876572160*sqrt(2) - 10595470986612263150) - 360182239148201679146803280622775418445799260852920356807178857
7705554082231768803383499513928216083726730307366377332828337773366071083598072772386051930660*sqrt(7) - 25665
32515528593027457553740848822251225795581355385468029664178193344384084826780557660928107456109583996394402379
69215404840097857114692088861260268773750000*sqrt(2) + 7587629160082400034098191013754109936929303905031237281
89868562072536874773198417845474059096092571231597614031305995383233253822344649520984786771597244159013334688
*sqrt(x^2 - 2*x + 3) - 360182239148201674253967483465988239211591610327364049310125917227965268177456885468827
3070958284684112917867030199254538909056956640775330160129944904845080*sqrt(7722433301876572160*sqrt(2) - 1059
5470986612263150) - 189690376125928824932865090390211289907978120405716887986342383587331788564444597329698862
937717384625840394590068917821873696654547424569549120105141773649324816347)^2) + 41672947348129/28000000000*s
qrt(7)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*arctan(-75876291600824000340981910137541099369
29303905031237281898685620725368747731984178454740590960925712315976140313059953832332538223446495209847867715
97244159013334688*(4*x - 4*sqrt(x^2 - 2*x + 3) - I*sqrt(20*sqrt(2) - 25) + 1)/(2404569034002236590242120764218
575948765964811188059545345978971635505204657923525607623616276623751473316663*(sqrt(7) + 2*sqrt(2) + sqrt(772
2433301876572160*sqrt(2) - 10595470986612263150))^7 - 14273993756775096948778514384982726726086177451105114601
50803990330057842381327560366437338163352898015665578569812646*(sqrt(7) + 2*sqrt(2) + sqrt(7722433301876572160
*sqrt(2) - 10595470986612263150))^6 + 103500570893851144987969410044573593528466831057025915469516609398945671
350435784286059468190138006696033871436116351184730996386*(sqrt(7) + 2*sqrt(2) + sqrt(7722433301876572160*sqrt
(2) - 10595470986612263150))^5 - 50629829096873773899573990740257783361032907165973215049765624930757608699456
368874374542973258344666404687384663029077837715816904194414*(sqrt(7) + 2*sqrt(2) + sqrt(7722433301876572160*s
qrt(2) - 10595470986612263150))^4 + 10966372960324617518376520492953903536419826007364881536262517747994521927
58375873916572220994961819156070680914794516208869024576339316465291922847*(sqrt(7) + 2*sqrt(2) + sqrt(7722433
301876572160*sqrt(2) - 10595470986612263150))^3 - 421908346813441134201546310084944284823056999047600551666136
893929797531708026880070061835013010220191747748244162706578121799670727647784642769438205254146*(sqrt(7) + 2*
sqrt(2) + sqrt(7722433301876572160*sqrt(2) - 10595470986612263150))^2 - 36823598085606129381210044942331473355
86129428951003980433292453331368231815800247214763542815020667923568605086834785543043094755407710336721654538
13191045495483200*sqrt(7) - 7364719617121225876242008988466294671172258857902007960866584906662736463631600494
42952708563004133584713721017366957108608618951081542067344330907626382090990966400*sqrt(2) - 3682359808560612
93812100449423314733558612942895100398043329245333136823181580024721476354281502066792356860508683478554304309
475540771033672165453813191045495483200*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150) + 14283521052
03887288279038536932322893004029548936486004185262156116702645830414445699158147702914990438794861101657539550
23507391431740254479476659166189278113070264611396886))/(110320475741093888*sqrt(2) - 151363871237318045) - 41
672947348129/28000000000*sqrt(7)*sqrt(7722433301876572160*sqrt(2) - 10595470986612263150)*arctan(-758762916008
24000340981910137541099369293039050312372818986856207253687477319841784547405909609257123159761403130599538323
3253822344649520984786771597244159013334688*(4*x - 4*sqrt(x^2 - 2*x + 3) - I*sqrt(20*sqrt(2) - 25) + 1)/(24045
69031044828063179458993747728533397375652422648770828443174291926065533229582917819313047949670367238733*(sqrt
(7) + 2*sqrt(2) + sqrt(7722433301876572160*sqrt(2) - 10595470986612263150))^7 + 142739938640279542310324164932
3508459758417264691242440494392029061629193828915227289144427476808605323457798934284966*(sqrt(7) + 2*sqrt(2)
+ sqrt(7722433301876572160*sqrt(2) - 10595470986612263150))^6 + 1035005707943988286737045590931491036645748300
42270418676217772397611737399109743666577656693380270736279743910695330805730930506*(sqrt(7) + 2*sqrt(2) + sqr
t(7722433301876572160*sqrt(2) - 10595470986612263150))^5 + 506298283979211926576221344309055178192100417667234
56843213902617311000457764306932616604186968701543769353918396995134839582865594841454*(sqrt(7) + 2*sqrt(2) +
sqrt(7722433301876572160*sqrt(2) - 10595470986612263150))^4 + 109663729492100506852548516508189371952187779338
1198945537497268383001795642074241605363210271082182350299185943188313831810524294377442378547095677*(sqrt(7)
+ 2*sqrt(2) + sqrt(7722433301876572160*sqrt(2) - 10595470986612263150))^3 + 4219083267175436404054639687522254
73761982330345651759435786313555664940163096802031617835780482986046113584519114737539776217075850964725785432
795347393026*(sqrt(7) + 2*sqrt(2) + sqrt(7722433301876572160*sqrt(2) - 10595470986612263150))^2 - 368235985635
02416236246077496796333400424389779330357869555041050779919584456719690428590074630983105994437988538748147431
2223533832259969580105714665875528461138380*sqrt(7) - 73647197127004832472492154993592666800848779558660715739
1100821015598391689134393808571801492619662119888759770774962948624447067664519939160211429331751056922276760*
sqrt(2) - 3682359856350241623624607749679633340042438977933035786955504105077991958445671969042859007463098310
59944379885387481474312223533832259969580105714665875528461138380*sqrt(7722433301876572160*sqrt(2) - 105954709
86612263150) - 14283520819361359291533003672226045153345857894604895779030735125826227723975160464191638450688
9981140070748629723613145062430716033621311266813074832334518174840024828894486))/(110320475741093888*sqrt(2)
- 151363871237318045) + 1/205800000000*(108121281*(x - sqrt(x^2 - 2*x + 3))^15 + 135317265*(x - sqrt(x^2 - 2*x
+ 3))^14 - 2309618731*(x - sqrt(x^2 - 2*x + 3))^13 - 4089866767*(x - sqrt(x^2 - 2*x + 3))^12 + 23951599406*(x
- sqrt(x^2 - 2*x + 3))^11 + 45641347654*(x - sqrt(x^2 - 2*x + 3))^10 - 149568395690*(x - sqrt(x^2 - 2*x + 3))
^9 - 288215430978*(x - sqrt(x^2 - 2*x + 3))^8 + 660704292769*(x - sqrt(x^2 - 2*x + 3))^7 + 1062639157153*(x -
sqrt(x^2 - 2*x + 3))^6 - 2094971437979*(x - sqrt(x^2 - 2*x + 3))^5 - 2301192104575*(x - sqrt(x^2 - 2*x + 3))^4
+ 4977175786352*(x - sqrt(x^2 - 2*x + 3))^3 + 1302994004424*(x - sqrt(x^2 - 2*x + 3))^2 - 6052879270032*x + 6
052879270032*sqrt(x^2 - 2*x + 3) + 2841437414928)/((x - sqrt(x^2 - 2*x + 3))^4 + (x - sqrt(x^2 - 2*x + 3))^3 -
5*(x - sqrt(x^2 - 2*x + 3))^2 - 7*x + 7*sqrt(x^2 - 2*x + 3) + 14)^4 + 1/3150000000*(3*((((((((29420*x - 33258
9)*x + 1860912)*x - 6743744)*x + 17167416)*x - 31960026)*x + 43362368)*x - 42014736)*x + 26516604)*x - 2719986
7)/(x^2 - 2*x + 3)^(9/2)
________________________________________________________________________________________
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)
________________________________________________________________________________________