3.50 \(\int \frac{1}{(3-2 x+x^2)^{11/2} (1+x+2 x^2)^5} \, dx\)
Optimal. Leaf size=378 \[ -\frac{63043297-29625922 x}{41160000000 \left (x^2-2 x+3\right )^{3/2}}-\frac{31 (7434109-3088870 x)}{411600000000 \sqrt{x^2-2 x+3}}+\frac{3 (8233 x+8822)}{343000 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}+\frac{8878 x+5485}{117600 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}-\frac{30316369-15043110 x}{6860000000 \left (x^2-2 x+3\right )^{5/2}}+\frac{67 x+28}{1050 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}-\frac{4878869-2578034 x}{411600000 \left (x^2-2 x+3\right )^{7/2}}-\frac{1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}-\frac{3450497-2004270 x}{123480000 \left (x^2-2 x+3\right )^{9/2}}+\frac{\sqrt{\frac{1}{70} \left (151363871237318045+110320475741093888 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{7 \left (151363871237318045+110320475741093888 \sqrt{2}\right )}} \left (\left (932587773+620347970 \sqrt{2}\right ) x+312239803 \sqrt{2}+308108167\right )}{\sqrt{x^2-2 x+3}}\right )}{137200000000}-\frac{\sqrt{\frac{1}{70} \left (110320475741093888 \sqrt{2}-151363871237318045\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{5}{7 \left (110320475741093888 \sqrt{2}-151363871237318045\right )}} \left (\left (932587773-620347970 \sqrt{2}\right ) x-312239803 \sqrt{2}+308108167\right )}{\sqrt{x^2-2 x+3}}\right )}{137200000000} \]
[Out]
-(3450497 - 2004270*x)/(123480000*(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
________________________________________________________________________________________
Rubi [A] time = 0.773819, antiderivative size = 378, normalized size of antiderivative = 1.,
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
= {974, 1060, 1035, 1029, 206, 204} \[ -\frac{63043297-29625922 x}{41160000000 \left (x^2-2 x+3\right )^{3/2}}-\frac{31 (7434109-3088870 x)}{411600000000 \sqrt{x^2-2 x+3}}+\frac{3 (8233 x+8822)}{343000 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )}+\frac{8878 x+5485}{117600 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^2}-\frac{30316369-15043110 x}{6860000000 \left (x^2-2 x+3\right )^{5/2}}+\frac{67 x+28}{1050 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^3}-\frac{4878869-2578034 x}{411600000 \left (x^2-2 x+3\right )^{7/2}}-\frac{1-10 x}{280 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4}-\frac{3450497-2004270 x}{123480000 \left (x^2-2 x+3\right )^{9/2}}+\frac{\sqrt{\frac{1}{70} \left (151363871237318045+110320475741093888 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{7 \left (151363871237318045+110320475741093888 \sqrt{2}\right )}} \left (\left (932587773+620347970 \sqrt{2}\right ) x+312239803 \sqrt{2}+308108167\right )}{\sqrt{x^2-2 x+3}}\right )}{137200000000}-\frac{\sqrt{\frac{1}{70} \left (110320475741093888 \sqrt{2}-151363871237318045\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{5}{7 \left (110320475741093888 \sqrt{2}-151363871237318045\right )}} \left (\left (932587773-620347970 \sqrt{2}\right ) x-312239803 \sqrt{2}+308108167\right )}{\sqrt{x^2-2 x+3}}\right )}{137200000000} \]
Antiderivative was successfully verified.
[In]
Int[1/((3 - 2*x + x^2)^(11/2)*(1 + x + 2*x^2)^5),x]
[Out]
-(3450497 - 2004270*x)/(123480000*(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 974
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 1060
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)*((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))/((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[(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]
Rule 1035
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 1029
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
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 ) \operatorname{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 ) \operatorname{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] time = 6.02736, size = 342, normalized size = 0.9 \[ \frac{560 \left (4596238560 x^{17}-38639385552 x^{16}+188603773872 x^{15}-606785954952 x^{14}+1459208021718 x^{13}-2679143870481 x^{12}+3999656132532 x^{11}-4915797913008 x^{10}+5380603084494 x^9-5134334619701 x^8+4591320676952 x^7-3359813871472 x^6+2503427226914 x^5-1409335257371 x^4+1002897791524 x^3-266966654968 x^2+261702502714 x-53205422447\right )-9 i \sqrt{50+10 i \sqrt{7}} \left (932587773 \sqrt{7}-299844895 i\right ) \sqrt{x^2-2 x+3} \left (2 x^4-3 x^3+5 x^2+x+3\right )^4 \tanh ^{-1}\left (\frac{\left (-5-i \sqrt{7}\right ) x+i \sqrt{7}+13}{\sqrt{50+10 i \sqrt{7}} \sqrt{x^2-2 x+3}}\right )+9 \sqrt{50-10 i \sqrt{7}} \left (299844895-932587773 i \sqrt{7}\right ) \sqrt{x^2-2 x+3} \left (2 x^4-3 x^3+5 x^2+x+3\right )^4 \tanh ^{-1}\left (\frac{\left (5-i \sqrt{7}\right ) x+i \sqrt{7}-13}{\sqrt{50-10 i \sqrt{7}} \sqrt{x^2-2 x+3}}\right )}{691488000000000 \left (x^2-2 x+3\right )^{9/2} \left (2 x^2+x+1\right )^4} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((3 - 2*x + x^2)^(11/2)*(1 + x + 2*x^2)^5),x]
[Out]
(560*(-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 + 39
99656132532*x^11 - 2679143870481*x^12 + 1459208021718*x^13 - 606785954952*x^14 + 188603773872*x^15 - 386393855
52*x^16 + 4596238560*x^17) - (9*I)*Sqrt[50 + (10*I)*Sqrt[7]]*(-299844895*I + 932587773*Sqrt[7])*Sqrt[3 - 2*x +
x^2]*(3 + x + 5*x^2 - 3*x^3 + 2*x^4)^4*ArcTanh[(13 + I*Sqrt[7] + (-5 - I*Sqrt[7])*x)/(Sqrt[50 + (10*I)*Sqrt[7
]]*Sqrt[3 - 2*x + x^2])] + 9*Sqrt[50 - (10*I)*Sqrt[7]]*(299844895 - (932587773*I)*Sqrt[7])*Sqrt[3 - 2*x + x^2]
*(3 + x + 5*x^2 - 3*x^3 + 2*x^4)^4*ArcTanh[(-13 + I*Sqrt[7] + (5 - I*Sqrt[7])*x)/(Sqrt[50 - (10*I)*Sqrt[7]]*Sq
rt[3 - 2*x + x^2])])/(691488000000000*(3 - 2*x + x^2)^(9/2)*(1 + x + 2*x^2)^4)
________________________________________________________________________________________
Maple [B] time = 0.533, size = 21028, normalized size = 55.6 \begin{align*} \text{output too large to display} \end{align*}
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)
[Out]
result too large to display
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, x^{2} + x + 1\right )}^{5}{\left (x^{2} - 2 \, x + 3\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
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] time = 4.00831, size = 11537, normalized size = 30.52 \begin{align*} \text{result too large to display} \end{align*}
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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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]
Timed out
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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]
Timed out