3.124 \(\int \frac {x^{11} (A+B x^2)}{(a+b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=365 \[ -\frac {x^4 \left (x^2 \left (20 a^2 B c^2+10 a A b c^2-20 a b^2 B c-A b^3 c+3 b^4 B\right )+a \left (16 a A c^2-18 a b B c-A b^2 c+3 b^3 B\right )\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x^2 \left (30 a^2 B c^2+7 a A b c^2-21 a b^2 B c-A b^3 c+3 b^4 B\right )}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {\left (-60 a^3 B c^3-30 a^2 A b c^3+90 a^2 b^2 B c^2+10 a A b^3 c^2-30 a b^4 B c-A b^5 c+3 b^6 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {x^8 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {(3 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^4} \]

[Out]

1/2*(7*A*a*b*c^2-A*b^3*c+30*B*a^2*c^2-21*B*a*b^2*c+3*B*b^4)*x^2/c^3/(-4*a*c+b^2)^2-1/4*x^8*(a*(-2*A*c+B*b)+(-A
*b*c-2*B*a*c+B*b^2)*x^2)/c/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2-1/4*x^4*(a*(16*A*a*c^2-A*b^2*c-18*B*a*b*c+3*B*b^3)+(
10*A*a*b*c^2-A*b^3*c+20*B*a^2*c^2-20*B*a*b^2*c+3*B*b^4)*x^2)/c^2/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)-1/2*(-30*A*a^2
*b*c^3+10*A*a*b^3*c^2-A*b^5*c-60*B*a^3*c^3+90*B*a^2*b^2*c^2-30*B*a*b^4*c+3*B*b^6)*arctanh((2*c*x^2+b)/(-4*a*c+
b^2)^(1/2))/c^4/(-4*a*c+b^2)^(5/2)-1/4*(-A*c+3*B*b)*ln(c*x^4+b*x^2+a)/c^4

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Rubi [A]  time = 1.45, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1251, 818, 773, 634, 618, 206, 628} \[ -\frac {x^4 \left (x^2 \left (20 a^2 B c^2+10 a A b c^2-20 a b^2 B c-A b^3 c+3 b^4 B\right )+a \left (16 a A c^2-18 a b B c-A b^2 c+3 b^3 B\right )\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x^2 \left (30 a^2 B c^2+7 a A b c^2-21 a b^2 B c-A b^3 c+3 b^4 B\right )}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {\left (-30 a^2 A b c^3+90 a^2 b^2 B c^2-60 a^3 B c^3+10 a A b^3 c^2-30 a b^4 B c-A b^5 c+3 b^6 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {x^8 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {(3 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^4} \]

Antiderivative was successfully verified.

[In]

Int[(x^11*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

((3*b^4*B - A*b^3*c - 21*a*b^2*B*c + 7*a*A*b*c^2 + 30*a^2*B*c^2)*x^2)/(2*c^3*(b^2 - 4*a*c)^2) - (x^8*(a*(b*B -
2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(4*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x^4*(a*(3*b^3*B - A*b^2
*c - 18*a*b*B*c + 16*a*A*c^2) + (3*b^4*B - A*b^3*c - 20*a*b^2*B*c + 10*a*A*b*c^2 + 20*a^2*B*c^2)*x^2))/(4*c^2*
(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((3*b^6*B - A*b^5*c - 30*a*b^4*B*c + 10*a*A*b^3*c^2 + 90*a^2*b^2*B*c^2
- 30*a^2*A*b*c^3 - 60*a^3*B*c^3)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^4*(b^2 - 4*a*c)^(5/2)) - ((3*b
*B - A*c)*Log[a + b*x^2 + c*x^4])/(4*c^4)

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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 773

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

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^5 (A+B x)}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {x^8 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (4 a (b B-2 A c)+\left (3 b^2 B-A b c-10 a B c\right ) x\right )}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 c \left (b^2-4 a c\right )}\\ &=-\frac {x^8 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x^4 \left (a \left (3 b^3 B-A b^2 c-18 a b B c+16 a A c^2\right )+\left (3 b^4 B-A b^3 c-20 a b^2 B c+10 a A b c^2+20 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {x \left (2 a \left (3 b^3 B-A b^2 c-18 a b B c+16 a A c^2\right )+2 \left (3 b^4 B-A b^3 c-21 a b^2 B c+7 a A b c^2+30 a^2 B c^2\right ) x\right )}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2}\\ &=\frac {\left (3 b^4 B-A b^3 c-21 a b^2 B c+7 a A b c^2+30 a^2 B c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {x^8 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x^4 \left (a \left (3 b^3 B-A b^2 c-18 a b B c+16 a A c^2\right )+\left (3 b^4 B-A b^3 c-20 a b^2 B c+10 a A b c^2+20 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {-2 a \left (3 b^4 B-A b^3 c-21 a b^2 B c+7 a A b c^2+30 a^2 B c^2\right )+\left (2 a c \left (3 b^3 B-A b^2 c-18 a b B c+16 a A c^2\right )-2 b \left (3 b^4 B-A b^3 c-21 a b^2 B c+7 a A b c^2+30 a^2 B c^2\right )\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3 \left (b^2-4 a c\right )^2}\\ &=\frac {\left (3 b^4 B-A b^3 c-21 a b^2 B c+7 a A b c^2+30 a^2 B c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {x^8 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x^4 \left (a \left (3 b^3 B-A b^2 c-18 a b B c+16 a A c^2\right )+\left (3 b^4 B-A b^3 c-20 a b^2 B c+10 a A b c^2+20 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {(3 b B-A c) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4}+\frac {\left (3 b^6 B-A b^5 c-30 a b^4 B c+10 a A b^3 c^2+90 a^2 b^2 B c^2-30 a^2 A b c^3-60 a^3 B c^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4 \left (b^2-4 a c\right )^2}\\ &=\frac {\left (3 b^4 B-A b^3 c-21 a b^2 B c+7 a A b c^2+30 a^2 B c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {x^8 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x^4 \left (a \left (3 b^3 B-A b^2 c-18 a b B c+16 a A c^2\right )+\left (3 b^4 B-A b^3 c-20 a b^2 B c+10 a A b c^2+20 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {(3 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^4}-\frac {\left (3 b^6 B-A b^5 c-30 a b^4 B c+10 a A b^3 c^2+90 a^2 b^2 B c^2-30 a^2 A b c^3-60 a^3 B c^3\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^4 \left (b^2-4 a c\right )^2}\\ &=\frac {\left (3 b^4 B-A b^3 c-21 a b^2 B c+7 a A b c^2+30 a^2 B c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {x^8 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x^4 \left (a \left (3 b^3 B-A b^2 c-18 a b B c+16 a A c^2\right )+\left (3 b^4 B-A b^3 c-20 a b^2 B c+10 a A b c^2+20 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^6 B-A b^5 c-30 a b^4 B c+10 a A b^3 c^2+90 a^2 b^2 B c^2-30 a^2 A b c^3-60 a^3 B c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {(3 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^4}\\ \end {align*}

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Mathematica [A]  time = 0.66, size = 435, normalized size = 1.19 \[ \frac {\frac {a^3 c^2 \left (2 c \left (A+B x^2\right )-5 b B\right )+a^2 b c \left (-b c \left (4 A+9 B x^2\right )+5 A c^2 x^2+5 b^2 B\right )+a b^3 \left (b c \left (A+6 B x^2\right )-5 A c^2 x^2+b^2 (-B)\right )+b^5 x^2 (A c-b B)}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {2 c \left (60 a^3 B c^3+30 a^2 A b c^3-90 a^2 b^2 B c^2-10 a A b^3 c^2+30 a b^4 B c+A b^5 c-3 b^6 B\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac {4 a^3 c^4 \left (8 A+9 B x^2\right )-3 a^2 b^2 c^3 \left (13 A+34 B x^2\right )+2 a^2 b c^3 \left (25 A c x^2-39 a B\right )+2 b^5 c \left (2 A c x^2-7 a B\right )+a b^4 c^2 \left (11 A+48 B x^2\right )+a b^3 c^2 \left (61 a B-30 A c x^2\right )-b^6 c \left (A+6 B x^2\right )+b^7 B}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+c (A c-3 b B) \log \left (a+b x^2+c x^4\right )+2 B c^2 x^2}{4 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

(2*B*c^2*x^2 + (b^7*B - b^6*c*(A + 6*B*x^2) + 4*a^3*c^4*(8*A + 9*B*x^2) - 3*a^2*b^2*c^3*(13*A + 34*B*x^2) + a*
b^4*c^2*(11*A + 48*B*x^2) + a*b^3*c^2*(61*a*B - 30*A*c*x^2) + 2*b^5*c*(-7*a*B + 2*A*c*x^2) + 2*a^2*b*c^3*(-39*
a*B + 25*A*c*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b^5*(-(b*B) + A*c)*x^2 + a^3*c^2*(-5*b*B + 2*c*(A
+ B*x^2)) + a*b^3*(-(b^2*B) - 5*A*c^2*x^2 + b*c*(A + 6*B*x^2)) + a^2*b*c*(5*b^2*B + 5*A*c^2*x^2 - b*c*(4*A + 9
*B*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (2*c*(-3*b^6*B + A*b^5*c + 30*a*b^4*B*c - 10*a*A*b^3*c^2 - 9
0*a^2*b^2*B*c^2 + 30*a^2*A*b*c^3 + 60*a^3*B*c^3)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2
) + c*(-3*b*B + A*c)*Log[a + b*x^2 + c*x^4])/(4*c^5)

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fricas [B]  time = 1.27, size = 3196, normalized size = 8.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/4*(2*(B*b^6*c^3 - 12*B*a*b^4*c^4 + 48*B*a^2*b^2*c^5 - 64*B*a^3*c^6)*x^10 - 5*B*a^2*b^7 - 96*A*a^5*c^4 + 4*(
B*b^7*c^2 - 12*B*a*b^5*c^3 + 48*B*a^2*b^3*c^4 - 64*B*a^3*b*c^5)*x^8 - 2*(2*B*b^8*c + 100*(2*B*a^4 + A*a^3*b)*c
^5 - (254*B*a^3*b^2 + 85*A*a^2*b^3)*c^4 + (123*B*a^2*b^4 + 23*A*a*b^5)*c^3 - 2*(13*B*a*b^6 + A*b^7)*c^2)*x^6 -
(5*B*b^9 + 128*A*a^4*c^5 + 4*(22*B*a^4*b + 3*A*a^3*b^2)*c^4 - (314*B*a^3*b^3 + 87*A*a^2*b^4)*c^3 + (225*B*a^2
*b^5 + 31*A*a*b^6)*c^2 - (58*B*a*b^7 + 3*A*b^8)*c)*x^4 + 4*(58*B*a^5*b + 27*A*a^4*b^2)*c^3 - (202*B*a^4*b^3 +
33*A*a^3*b^4)*c^2 - 2*(5*B*a*b^8 + 4*(30*B*a^5 + 31*A*a^4*b)*c^4 - (346*B*a^4*b^2 + 119*A*a^3*b^3)*c^3 + (235*
B*a^3*b^4 + 34*A*a^2*b^5)*c^2 - (59*B*a^2*b^6 + 3*A*a*b^7)*c)*x^2 - (3*B*a^2*b^6 + (3*B*b^6*c^2 - 30*(2*B*a^3
+ A*a^2*b)*c^5 + 10*(9*B*a^2*b^2 + A*a*b^3)*c^4 - (30*B*a*b^4 + A*b^5)*c^3)*x^8 + 2*(3*B*b^7*c - 30*(2*B*a^3*b
+ A*a^2*b^2)*c^4 + 10*(9*B*a^2*b^3 + A*a*b^4)*c^3 - (30*B*a*b^5 + A*b^6)*c^2)*x^6 + (3*B*b^8 - 60*(2*B*a^4 +
A*a^3*b)*c^4 + 10*(12*B*a^3*b^2 - A*a^2*b^3)*c^3 + 2*(15*B*a^2*b^4 + 4*A*a*b^5)*c^2 - (24*B*a*b^6 + A*b^7)*c)*
x^4 - 30*(2*B*a^5 + A*a^4*b)*c^3 + 10*(9*B*a^4*b^2 + A*a^3*b^3)*c^2 + 2*(3*B*a*b^7 - 30*(2*B*a^4*b + A*a^3*b^2
)*c^3 + 10*(9*B*a^3*b^3 + A*a^2*b^4)*c^2 - (30*B*a^2*b^5 + A*a*b^6)*c)*x^2 - (30*B*a^3*b^4 + A*a^2*b^5)*c)*sqr
t(b^2 - 4*a*c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)
) + (56*B*a^3*b^5 + 3*A*a^2*b^6)*c - (3*B*a^2*b^7 + 64*A*a^5*c^4 + (3*B*b^7*c^2 + 64*A*a^3*c^6 - 48*(4*B*a^3*b
+ A*a^2*b^2)*c^5 + 12*(12*B*a^2*b^3 + A*a*b^4)*c^4 - (36*B*a*b^5 + A*b^6)*c^3)*x^8 + 2*(3*B*b^8*c + 64*A*a^3*
b*c^5 - 48*(4*B*a^3*b^2 + A*a^2*b^3)*c^4 + 12*(12*B*a^2*b^4 + A*a*b^5)*c^3 - (36*B*a*b^6 + A*b^7)*c^2)*x^6 + (
3*B*b^9 + 128*A*a^4*c^5 - 32*(12*B*a^4*b + A*a^3*b^2)*c^4 + 24*(4*B*a^3*b^3 - A*a^2*b^4)*c^3 + 2*(36*B*a^2*b^5
+ 5*A*a*b^6)*c^2 - (30*B*a*b^7 + A*b^8)*c)*x^4 - 48*(4*B*a^5*b + A*a^4*b^2)*c^3 + 12*(12*B*a^4*b^3 + A*a^3*b^
4)*c^2 + 2*(3*B*a*b^8 + 64*A*a^4*b*c^4 - 48*(4*B*a^4*b^2 + A*a^3*b^3)*c^3 + 12*(12*B*a^3*b^4 + A*a^2*b^5)*c^2
- (36*B*a^2*b^6 + A*a*b^7)*c)*x^2 - (36*B*a^3*b^5 + A*a^2*b^6)*c)*log(c*x^4 + b*x^2 + a))/(a^2*b^6*c^4 - 12*a^
3*b^4*c^5 + 48*a^4*b^2*c^6 - 64*a^5*c^7 + (b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*x^8 + 2*(b^7*
c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*x^6 + (b^8*c^4 - 10*a*b^6*c^5 + 24*a^2*b^4*c^6 + 32*a^3*b^
2*c^7 - 128*a^4*c^8)*x^4 + 2*(a*b^7*c^4 - 12*a^2*b^5*c^5 + 48*a^3*b^3*c^6 - 64*a^4*b*c^7)*x^2), 1/4*(2*(B*b^6*
c^3 - 12*B*a*b^4*c^4 + 48*B*a^2*b^2*c^5 - 64*B*a^3*c^6)*x^10 - 5*B*a^2*b^7 - 96*A*a^5*c^4 + 4*(B*b^7*c^2 - 12*
B*a*b^5*c^3 + 48*B*a^2*b^3*c^4 - 64*B*a^3*b*c^5)*x^8 - 2*(2*B*b^8*c + 100*(2*B*a^4 + A*a^3*b)*c^5 - (254*B*a^3
*b^2 + 85*A*a^2*b^3)*c^4 + (123*B*a^2*b^4 + 23*A*a*b^5)*c^3 - 2*(13*B*a*b^6 + A*b^7)*c^2)*x^6 - (5*B*b^9 + 128
*A*a^4*c^5 + 4*(22*B*a^4*b + 3*A*a^3*b^2)*c^4 - (314*B*a^3*b^3 + 87*A*a^2*b^4)*c^3 + (225*B*a^2*b^5 + 31*A*a*b
^6)*c^2 - (58*B*a*b^7 + 3*A*b^8)*c)*x^4 + 4*(58*B*a^5*b + 27*A*a^4*b^2)*c^3 - (202*B*a^4*b^3 + 33*A*a^3*b^4)*c
^2 - 2*(5*B*a*b^8 + 4*(30*B*a^5 + 31*A*a^4*b)*c^4 - (346*B*a^4*b^2 + 119*A*a^3*b^3)*c^3 + (235*B*a^3*b^4 + 34*
A*a^2*b^5)*c^2 - (59*B*a^2*b^6 + 3*A*a*b^7)*c)*x^2 - 2*(3*B*a^2*b^6 + (3*B*b^6*c^2 - 30*(2*B*a^3 + A*a^2*b)*c^
5 + 10*(9*B*a^2*b^2 + A*a*b^3)*c^4 - (30*B*a*b^4 + A*b^5)*c^3)*x^8 + 2*(3*B*b^7*c - 30*(2*B*a^3*b + A*a^2*b^2)
*c^4 + 10*(9*B*a^2*b^3 + A*a*b^4)*c^3 - (30*B*a*b^5 + A*b^6)*c^2)*x^6 + (3*B*b^8 - 60*(2*B*a^4 + A*a^3*b)*c^4
+ 10*(12*B*a^3*b^2 - A*a^2*b^3)*c^3 + 2*(15*B*a^2*b^4 + 4*A*a*b^5)*c^2 - (24*B*a*b^6 + A*b^7)*c)*x^4 - 30*(2*B
*a^5 + A*a^4*b)*c^3 + 10*(9*B*a^4*b^2 + A*a^3*b^3)*c^2 + 2*(3*B*a*b^7 - 30*(2*B*a^4*b + A*a^3*b^2)*c^3 + 10*(9
*B*a^3*b^3 + A*a^2*b^4)*c^2 - (30*B*a^2*b^5 + A*a*b^6)*c)*x^2 - (30*B*a^3*b^4 + A*a^2*b^5)*c)*sqrt(-b^2 + 4*a*
c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (56*B*a^3*b^5 + 3*A*a^2*b^6)*c - (3*B*a^2*b^7 + 6
4*A*a^5*c^4 + (3*B*b^7*c^2 + 64*A*a^3*c^6 - 48*(4*B*a^3*b + A*a^2*b^2)*c^5 + 12*(12*B*a^2*b^3 + A*a*b^4)*c^4 -
(36*B*a*b^5 + A*b^6)*c^3)*x^8 + 2*(3*B*b^8*c + 64*A*a^3*b*c^5 - 48*(4*B*a^3*b^2 + A*a^2*b^3)*c^4 + 12*(12*B*a
^2*b^4 + A*a*b^5)*c^3 - (36*B*a*b^6 + A*b^7)*c^2)*x^6 + (3*B*b^9 + 128*A*a^4*c^5 - 32*(12*B*a^4*b + A*a^3*b^2)
*c^4 + 24*(4*B*a^3*b^3 - A*a^2*b^4)*c^3 + 2*(36*B*a^2*b^5 + 5*A*a*b^6)*c^2 - (30*B*a*b^7 + A*b^8)*c)*x^4 - 48*
(4*B*a^5*b + A*a^4*b^2)*c^3 + 12*(12*B*a^4*b^3 + A*a^3*b^4)*c^2 + 2*(3*B*a*b^8 + 64*A*a^4*b*c^4 - 48*(4*B*a^4*
b^2 + A*a^3*b^3)*c^3 + 12*(12*B*a^3*b^4 + A*a^2*b^5)*c^2 - (36*B*a^2*b^6 + A*a*b^7)*c)*x^2 - (36*B*a^3*b^5 + A
*a^2*b^6)*c)*log(c*x^4 + b*x^2 + a))/(a^2*b^6*c^4 - 12*a^3*b^4*c^5 + 48*a^4*b^2*c^6 - 64*a^5*c^7 + (b^6*c^6 -
12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*x^8 + 2*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*x
^6 + (b^8*c^4 - 10*a*b^6*c^5 + 24*a^2*b^4*c^6 + 32*a^3*b^2*c^7 - 128*a^4*c^8)*x^4 + 2*(a*b^7*c^4 - 12*a^2*b^5*
c^5 + 48*a^3*b^3*c^6 - 64*a^4*b*c^7)*x^2)]

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giac [A]  time = 5.88, size = 598, normalized size = 1.64 \[ \frac {{\left (3 \, B b^{6} - 30 \, B a b^{4} c - A b^{5} c + 90 \, B a^{2} b^{2} c^{2} + 10 \, A a b^{3} c^{2} - 60 \, B a^{3} c^{3} - 30 \, A a^{2} b c^{3}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {B x^{2}}{2 \, c^{3}} + \frac {9 \, B b^{5} c^{2} x^{8} - 72 \, B a b^{3} c^{3} x^{8} - 3 \, A b^{4} c^{3} x^{8} + 144 \, B a^{2} b c^{4} x^{8} + 24 \, A a b^{2} c^{4} x^{8} - 48 \, A a^{2} c^{5} x^{8} + 6 \, B b^{6} c x^{6} - 48 \, B a b^{4} c^{2} x^{6} + 2 \, A b^{5} c^{2} x^{6} + 84 \, B a^{2} b^{2} c^{3} x^{6} - 12 \, A a b^{3} c^{3} x^{6} + 72 \, B a^{3} c^{4} x^{6} + 4 \, A a^{2} b c^{4} x^{6} - B b^{7} x^{4} + 14 \, B a b^{5} c x^{4} + 3 \, A b^{6} c x^{4} - 82 \, B a^{2} b^{3} c^{2} x^{4} - 20 \, A a b^{4} c^{2} x^{4} + 204 \, B a^{3} b c^{3} x^{4} + 22 \, A a^{2} b^{2} c^{3} x^{4} - 32 \, A a^{3} c^{4} x^{4} - 2 \, B a b^{6} x^{2} + 8 \, B a^{2} b^{4} c x^{2} + 6 \, A a b^{5} c x^{2} + 4 \, B a^{3} b^{2} c^{2} x^{2} - 40 \, A a^{2} b^{3} c^{2} x^{2} + 56 \, B a^{4} c^{3} x^{2} + 28 \, A a^{3} b c^{3} x^{2} - B a^{2} b^{5} + 3 \, A a^{2} b^{4} c + 28 \, B a^{4} b c^{2} - 18 \, A a^{3} b^{2} c^{2}}{8 \, {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac {{\left (3 \, B b - A c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

1/2*(3*B*b^6 - 30*B*a*b^4*c - A*b^5*c + 90*B*a^2*b^2*c^2 + 10*A*a*b^3*c^2 - 60*B*a^3*c^3 - 30*A*a^2*b*c^3)*arc
tan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*sqrt(-b^2 + 4*a*c)) + 1/2*B*x^2/c^
3 + 1/8*(9*B*b^5*c^2*x^8 - 72*B*a*b^3*c^3*x^8 - 3*A*b^4*c^3*x^8 + 144*B*a^2*b*c^4*x^8 + 24*A*a*b^2*c^4*x^8 - 4
8*A*a^2*c^5*x^8 + 6*B*b^6*c*x^6 - 48*B*a*b^4*c^2*x^6 + 2*A*b^5*c^2*x^6 + 84*B*a^2*b^2*c^3*x^6 - 12*A*a*b^3*c^3
*x^6 + 72*B*a^3*c^4*x^6 + 4*A*a^2*b*c^4*x^6 - B*b^7*x^4 + 14*B*a*b^5*c*x^4 + 3*A*b^6*c*x^4 - 82*B*a^2*b^3*c^2*
x^4 - 20*A*a*b^4*c^2*x^4 + 204*B*a^3*b*c^3*x^4 + 22*A*a^2*b^2*c^3*x^4 - 32*A*a^3*c^4*x^4 - 2*B*a*b^6*x^2 + 8*B
*a^2*b^4*c*x^2 + 6*A*a*b^5*c*x^2 + 4*B*a^3*b^2*c^2*x^2 - 40*A*a^2*b^3*c^2*x^2 + 56*B*a^4*c^3*x^2 + 28*A*a^3*b*
c^3*x^2 - B*a^2*b^5 + 3*A*a^2*b^4*c + 28*B*a^4*b*c^2 - 18*A*a^3*b^2*c^2)/((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)
*(c*x^4 + b*x^2 + a)^2) - 1/4*(3*B*b - A*c)*log(c*x^4 + b*x^2 + a)/c^4

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maple [B]  time = 0.03, size = 2054, normalized size = 5.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x)

[Out]

1/2*B*x^2/c^3+6/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*B*a*b^3-1/2/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4
*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^5*A-2/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)
*A*a*b^2-12/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*B*a^2*b+1/c^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b
^2*c+b^4)*x^6*A*b^5-5/4/c^4/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*b^7+7/c/(c*x^4+b*x^2+a)^2*a^4/(
16*a^2*c^2-8*a*b^2*c+b^4)*x^2*B-21/4/c^2/(c*x^4+b*x^2+a)^2*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*A*b^2-29/2/c^2/(c*x^
4+b*x^2+a)^2*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*B*b-5/4/c^4/(c*x^4+b*x^2+a)^2*a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*B*b^5
+25/2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*A*a^2*b+3/2/c^4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^
(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^6*B-30/c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*
c*x^2+b)/(4*a*c-b^2)^(1/2))*a^3*B-3/2/c^3/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*B*b^6+3/4/c^3/(c*x^
4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*A*b^6+3/4/c^3/(c*x^4+b*x^2+a)^2*a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*A*b
^4+9/c^3/(c*x^4+b*x^2+a)^2*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*B*b^3+3/2/c^3/(c*x^4+b*x^2+a)^2*a/(16*a^2*c^2-8*a*b^
2*c+b^4)*x^2*A*b^5-11/c^2/(c*x^4+b*x^2+a)^2*a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^3+11/4/c/(c*x^4+b*x^2+a)^2/
(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*A*a^2*b^2-19/4/c^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*A*a*b^4-21/
2/c/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*a^3*b-41/4/c^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+
b^4)*x^4*B*a^2*b^3+31/2/c/(c*x^4+b*x^2+a)^2*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b-71/2/c^2/(c*x^4+b*x^2+a)^2*
a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*B*b^2-5/2/c^4/(c*x^4+b*x^2+a)^2*a/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*B*b^6-15/2
/c/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*A*a*b^3+12/c^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^6*B*a*b^4-15/c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*a^2*b+
5/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*a*b^3+45/c^2/(16*a^
2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*B*a^2*b^2-15/c^3/(16*a^2*c^2-8*a*
b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*B*a*b^4+19/c^3/(c*x^4+b*x^2+a)^2*a^2/(16*a^
2*c^2-8*a*b^2*c+b^4)*x^2*B*b^4+17/2/c^3/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*a*b^5-51/2/c/(c*x^4
+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*B*a^2*b^2+9/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*B*a^3+
8/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*A*a^3+6/c/(c*x^4+b*x^2+a)^2*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*
A+1/4/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*A*b^4+4/c/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*
A*a^2-3/4/c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*B*b^5

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(B*x^2+A)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 4.66, size = 4501, normalized size = 12.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^11*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x)

[Out]

((x^6*(18*B*a^3*c^3 - 3*B*b^6 + 2*A*b^5*c + 24*B*a*b^4*c - 15*A*a*b^3*c^2 + 25*A*a^2*b*c^3 - 51*B*a^2*b^2*c^2)
)/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a*(24*A*a^3*c^3 - 5*B*a*b^5 + 3*A*a*b^4*c + 36*B*a^2*b^3*c - 58*B*a^3*
b*c^2 - 21*A*a^2*b^2*c^2))/(4*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^2*(14*B*a^4*c^3 - 5*B*a*b^6 + 3*A*a*b^5*c
+ 31*A*a^3*b*c^3 + 38*B*a^2*b^4*c - 22*A*a^2*b^3*c^2 - 71*B*a^3*b^2*c^2))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)
) - (x^4*(5*B*b^7 - 32*A*a^3*c^4 - 3*A*b^6*c - 34*B*a*b^5*c + 19*A*a*b^4*c^2 + 42*B*a^3*b*c^3 - 11*A*a^2*b^2*c
^3 + 41*B*a^2*b^3*c^2))/(4*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(a^2*c^3 + c^5*x^8 + x^4*(2*a*c^4 + b^2*c^3) + 2
*b*c^4*x^6 + 2*a*b*c^3*x^2) + (B*x^2)/(2*c^3) + (log(((a*(A*c - 3*B*b)^2)/c^6 - (((8*a*(A*c - 3*B*b))/c^2 - (2
*(2*a + b*x^2)*(A*c - 3*B*b + c^4*(-(60*B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a
^2*b*c^3 - 90*B*a^2*b^2*c^2)^2/(c^8*(4*a*c - b^2)^5))^(1/2)))/c^2 + (2*x^2*(60*B*a^3*c^3 - 9*B*b^6 + 3*A*b^5*c
+ 78*B*a*b^4*c - 26*A*a*b^3*c^2 + 62*A*a^2*b*c^3 - 186*B*a^2*b^2*c^2))/(c^2*(4*a*c - b^2)^2))*(A*c - 3*B*b +
c^4*(-(60*B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b*c^3 - 90*B*a^2*b^2*c^2)^2
/(c^8*(4*a*c - b^2)^5))^(1/2)))/(4*c^4) + (x^2*(A*c - 3*B*b)*(30*B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 27*B*a*b^4*c
- 9*A*a*b^3*c^2 + 23*A*a^2*b*c^3 - 69*B*a^2*b^2*c^2))/(c^6*(4*a*c - b^2)^2))*((a*(A*c - 3*B*b)^2)/c^6 + (((2*(
2*a + b*x^2)*(3*B*b - A*c + c^4*(-(60*B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2
*b*c^3 - 90*B*a^2*b^2*c^2)^2/(c^8*(4*a*c - b^2)^5))^(1/2)))/c^2 + (8*a*(A*c - 3*B*b))/c^2 + (2*x^2*(60*B*a^3*c
^3 - 9*B*b^6 + 3*A*b^5*c + 78*B*a*b^4*c - 26*A*a*b^3*c^2 + 62*A*a^2*b*c^3 - 186*B*a^2*b^2*c^2))/(c^2*(4*a*c -
b^2)^2))*(3*B*b - A*c + c^4*(-(60*B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b*c
^3 - 90*B*a^2*b^2*c^2)^2/(c^8*(4*a*c - b^2)^5))^(1/2)))/(4*c^4) + (x^2*(A*c - 3*B*b)*(30*B*a^3*c^3 - 3*B*b^6 +
A*b^5*c + 27*B*a*b^4*c - 9*A*a*b^3*c^2 + 23*A*a^2*b*c^3 - 69*B*a^2*b^2*c^2))/(c^6*(4*a*c - b^2)^2)))*(6*B*b^1
1 + 2048*A*a^5*c^6 - 2*A*b^10*c - 120*B*a*b^9*c + 40*A*a*b^8*c^2 - 6144*B*a^5*b*c^5 - 320*A*a^2*b^6*c^3 + 1280
*A*a^3*b^4*c^4 - 2560*A*a^4*b^2*c^5 + 960*B*a^2*b^7*c^2 - 3840*B*a^3*b^5*c^3 + 7680*B*a^4*b^3*c^4))/(2*(4096*a
^5*c^9 - 4*b^10*c^4 + 80*a*b^8*c^5 - 640*a^2*b^6*c^6 + 2560*a^3*b^4*c^7 - 5120*a^4*b^2*c^8)) - (atan(((32*a^2*
c^8*(4*a*c - b^2)^5 + 2*b^4*c^6*(4*a*c - b^2)^5 - 16*a*b^2*c^7*(4*a*c - b^2)^5)*(x^2*(((((6*A*b^5*c^5 + 120*B*
a^3*c^7 - 18*B*b^6*c^4 - 52*A*a*b^3*c^6 + 124*A*a^2*b*c^7 + 156*B*a*b^4*c^5 - 372*B*a^2*b^2*c^6)/(16*a^2*c^8 +
b^4*c^6 - 8*a*b^2*c^7) - ((8*b^5*c^8 - 64*a*b^3*c^9 + 128*a^2*b*c^10)*(6*B*b^11 + 2048*A*a^5*c^6 - 2*A*b^10*c
- 120*B*a*b^9*c + 40*A*a*b^8*c^2 - 6144*B*a^5*b*c^5 - 320*A*a^2*b^6*c^3 + 1280*A*a^3*b^4*c^4 - 2560*A*a^4*b^2
*c^5 + 960*B*a^2*b^7*c^2 - 3840*B*a^3*b^5*c^3 + 7680*B*a^4*b^3*c^4))/(2*(16*a^2*c^8 + b^4*c^6 - 8*a*b^2*c^7)*(
4096*a^5*c^9 - 4*b^10*c^4 + 80*a*b^8*c^5 - 640*a^2*b^6*c^6 + 2560*a^3*b^4*c^7 - 5120*a^4*b^2*c^8)))*(60*B*a^3*
c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b*c^3 - 90*B*a^2*b^2*c^2))/(8*c^4*(4*a*c -
b^2)^(5/2)) - ((8*b^5*c^8 - 64*a*b^3*c^9 + 128*a^2*b*c^10)*(60*B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c -
10*A*a*b^3*c^2 + 30*A*a^2*b*c^3 - 90*B*a^2*b^2*c^2)*(6*B*b^11 + 2048*A*a^5*c^6 - 2*A*b^10*c - 120*B*a*b^9*c +
40*A*a*b^8*c^2 - 6144*B*a^5*b*c^5 - 320*A*a^2*b^6*c^3 + 1280*A*a^3*b^4*c^4 - 2560*A*a^4*b^2*c^5 + 960*B*a^2*b^
7*c^2 - 3840*B*a^3*b^5*c^3 + 7680*B*a^4*b^3*c^4))/(16*c^4*(4*a*c - b^2)^(5/2)*(16*a^2*c^8 + b^4*c^6 - 8*a*b^2*
c^7)*(4096*a^5*c^9 - 4*b^10*c^4 + 80*a*b^8*c^5 - 640*a^2*b^6*c^6 + 2560*a^3*b^4*c^7 - 5120*a^4*b^2*c^8)))/(a*(
4*a*c - b^2)^2) + (b*((((6*A*b^5*c^5 + 120*B*a^3*c^7 - 18*B*b^6*c^4 - 52*A*a*b^3*c^6 + 124*A*a^2*b*c^7 + 156*B
*a*b^4*c^5 - 372*B*a^2*b^2*c^6)/(16*a^2*c^8 + b^4*c^6 - 8*a*b^2*c^7) - ((8*b^5*c^8 - 64*a*b^3*c^9 + 128*a^2*b*
c^10)*(6*B*b^11 + 2048*A*a^5*c^6 - 2*A*b^10*c - 120*B*a*b^9*c + 40*A*a*b^8*c^2 - 6144*B*a^5*b*c^5 - 320*A*a^2*
b^6*c^3 + 1280*A*a^3*b^4*c^4 - 2560*A*a^4*b^2*c^5 + 960*B*a^2*b^7*c^2 - 3840*B*a^3*b^5*c^3 + 7680*B*a^4*b^3*c^
4))/(2*(16*a^2*c^8 + b^4*c^6 - 8*a*b^2*c^7)*(4096*a^5*c^9 - 4*b^10*c^4 + 80*a*b^8*c^5 - 640*a^2*b^6*c^6 + 2560
*a^3*b^4*c^7 - 5120*a^4*b^2*c^8)))*(6*B*b^11 + 2048*A*a^5*c^6 - 2*A*b^10*c - 120*B*a*b^9*c + 40*A*a*b^8*c^2 -
6144*B*a^5*b*c^5 - 320*A*a^2*b^6*c^3 + 1280*A*a^3*b^4*c^4 - 2560*A*a^4*b^2*c^5 + 960*B*a^2*b^7*c^2 - 3840*B*a^
3*b^5*c^3 + 7680*B*a^4*b^3*c^4))/(2*(4096*a^5*c^9 - 4*b^10*c^4 + 80*a*b^8*c^5 - 640*a^2*b^6*c^6 + 2560*a^3*b^4
*c^7 - 5120*a^4*b^2*c^8)) - (9*B^2*b^7 + A^2*b^5*c^2 - 6*A*B*b^6*c + 207*B^2*a^2*b^3*c^2 + 30*A*B*a^3*c^4 - 81
*B^2*a*b^5*c - 9*A^2*a*b^3*c^3 + 23*A^2*a^2*b*c^4 - 90*B^2*a^3*b*c^3 - 138*A*B*a^2*b^2*c^3 + 54*A*B*a*b^4*c^2)
/(16*a^2*c^8 + b^4*c^6 - 8*a*b^2*c^7) + (((b^5*c^8)/2 - 4*a*b^3*c^9 + 8*a^2*b*c^10)*(60*B*a^3*c^3 - 3*B*b^6 +
A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b*c^3 - 90*B*a^2*b^2*c^2)^2)/(c^8*(4*a*c - b^2)^5*(16*a^2*c
^8 + b^4*c^6 - 8*a*b^2*c^7))))/(2*a*(4*a*c - b^2)^(5/2))) + ((((8*A*a*c^5 - 24*B*a*b*c^4)/c^6 - (8*a*c^2*(6*B*
b^11 + 2048*A*a^5*c^6 - 2*A*b^10*c - 120*B*a*b^9*c + 40*A*a*b^8*c^2 - 6144*B*a^5*b*c^5 - 320*A*a^2*b^6*c^3 + 1
280*A*a^3*b^4*c^4 - 2560*A*a^4*b^2*c^5 + 960*B*a^2*b^7*c^2 - 3840*B*a^3*b^5*c^3 + 7680*B*a^4*b^3*c^4))/(4096*a
^5*c^9 - 4*b^10*c^4 + 80*a*b^8*c^5 - 640*a^2*b^6*c^6 + 2560*a^3*b^4*c^7 - 5120*a^4*b^2*c^8))*(60*B*a^3*c^3 - 3
*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b*c^3 - 90*B*a^2*b^2*c^2))/(8*c^4*(4*a*c - b^2)^(5
/2)) - (a*(60*B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b*c^3 - 90*B*a^2*b^2*c^
2)*(6*B*b^11 + 2048*A*a^5*c^6 - 2*A*b^10*c - 120*B*a*b^9*c + 40*A*a*b^8*c^2 - 6144*B*a^5*b*c^5 - 320*A*a^2*b^6
*c^3 + 1280*A*a^3*b^4*c^4 - 2560*A*a^4*b^2*c^5 + 960*B*a^2*b^7*c^2 - 3840*B*a^3*b^5*c^3 + 7680*B*a^4*b^3*c^4))
/(c^2*(4*a*c - b^2)^(5/2)*(4096*a^5*c^9 - 4*b^10*c^4 + 80*a*b^8*c^5 - 640*a^2*b^6*c^6 + 2560*a^3*b^4*c^7 - 512
0*a^4*b^2*c^8)))/(a*(4*a*c - b^2)^2) + (b*((((8*A*a*c^5 - 24*B*a*b*c^4)/c^6 - (8*a*c^2*(6*B*b^11 + 2048*A*a^5*
c^6 - 2*A*b^10*c - 120*B*a*b^9*c + 40*A*a*b^8*c^2 - 6144*B*a^5*b*c^5 - 320*A*a^2*b^6*c^3 + 1280*A*a^3*b^4*c^4
- 2560*A*a^4*b^2*c^5 + 960*B*a^2*b^7*c^2 - 3840*B*a^3*b^5*c^3 + 7680*B*a^4*b^3*c^4))/(4096*a^5*c^9 - 4*b^10*c^
4 + 80*a*b^8*c^5 - 640*a^2*b^6*c^6 + 2560*a^3*b^4*c^7 - 5120*a^4*b^2*c^8))*(6*B*b^11 + 2048*A*a^5*c^6 - 2*A*b^
10*c - 120*B*a*b^9*c + 40*A*a*b^8*c^2 - 6144*B*a^5*b*c^5 - 320*A*a^2*b^6*c^3 + 1280*A*a^3*b^4*c^4 - 2560*A*a^4
*b^2*c^5 + 960*B*a^2*b^7*c^2 - 3840*B*a^3*b^5*c^3 + 7680*B*a^4*b^3*c^4))/(2*(4096*a^5*c^9 - 4*b^10*c^4 + 80*a*
b^8*c^5 - 640*a^2*b^6*c^6 + 2560*a^3*b^4*c^7 - 5120*a^4*b^2*c^8)) - (A^2*a*c^2 + 9*B^2*a*b^2 - 6*A*B*a*b*c)/c^
6 + (a*(60*B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b*c^3 - 90*B*a^2*b^2*c^2)^
2)/(c^6*(4*a*c - b^2)^5)))/(2*a*(4*a*c - b^2)^(5/2))))/(9*B^2*b^12 + A^2*b^10*c^2 + 3600*B^2*a^6*c^6 - 6*A*B*b
^11*c + 160*A^2*a^2*b^6*c^4 - 600*A^2*a^3*b^4*c^5 + 900*A^2*a^4*b^2*c^6 + 1440*B^2*a^2*b^8*c^2 - 5760*B^2*a^3*
b^6*c^3 + 11700*B^2*a^4*b^4*c^4 - 10800*B^2*a^5*b^2*c^5 - 180*B^2*a*b^10*c - 20*A^2*a*b^8*c^3 - 960*A*B*a^2*b^
7*c^3 + 3720*A*B*a^3*b^5*c^4 - 6600*A*B*a^4*b^3*c^5 + 120*A*B*a*b^9*c^2 + 3600*A*B*a^5*b*c^6))*(60*B*a^3*c^3 -
3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b*c^3 - 90*B*a^2*b^2*c^2))/(2*c^4*(4*a*c - b^2)^
(5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(B*x**2+A)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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