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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.45459, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, 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 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]

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 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 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 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 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 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]

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*}

Mathematica [A]  time = 0.758272, size = 435, normalized size = 1.19 \[ \frac{\frac{-3 a^2 b^2 c^3 \left (13 A+34 B x^2\right )+2 a^2 b c^3 \left (25 A c x^2-39 a B\right )+4 a^3 c^4 \left (8 A+9 B x^2\right )+a b^4 c^2 \left (11 A+48 B x^2\right )+a b^3 c^2 \left (61 a B-30 A c x^2\right )+2 b^5 c \left (2 A c x^2-7 a B\right )-b^6 c \left (A+6 B x^2\right )+b^7 B}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{a^2 b c \left (-b c \left (4 A+9 B x^2\right )+5 A c^2 x^2+5 b^2 B\right )+a^3 c^2 \left (2 c \left (A+B x^2\right )-5 b B\right )+a b^3 \left (b c \left (A+6 B x^2\right )-5 A c^2 x^2+b^2 (-B)\right )+b^5 x^2 (A c-b B)}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{2 c \left (30 a^2 A b c^3-90 a^2 b^2 B c^2+60 a^3 B c^3-10 a A b^3 c^2+30 a b^4 B c+A b^5 c-3 b^6 B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+c (A c-3 b B) \log \left (a+b x^2+c x^4\right )+2 B c^2 x^2}{4 c^5} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.026, size = 2054, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}

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^3-12/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-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^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-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^2-2
9/2/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+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+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/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6
*B*b^6+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+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+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-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+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+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+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^3-51/2/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+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+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-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)*a
rctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*B*a*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
+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+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^4-11/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^3-71/2/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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [B]  time = 3.3471, size = 6745, normalized size = 18.48 \begin{align*} \text{result too large to display} \end{align*}

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

[Out]

Timed out

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Giac [A]  time = 38.2789, size = 807, normalized size = 2.21 \begin{align*} \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}} \end{align*}

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