3.59 \(\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{(a+b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=1177 \[ \text{result too large to display} \]

[Out]

-(x*(c^2*(a*b*f - b^2*(d + (a^2*j)/c^2) + 2*a*(c*d - a*h + (a^2*j)/c)) + (2*a*c^3*f - a*b^3*j - b*c*(c^2*d + a
*c*h - 3*a^2*j))*x^2))/(4*a*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (b*c^3*(c*e + a*i) - a*b^4*k + 4*a^2*b^
2*c*k - 2*a*c^2*(c^2*g + a^2*k) + (2*c^5*e + b^2*c^3*i - c^4*(b*g + 2*a*i) - b^5*k + 5*a*b^3*c*k - 5*a^2*b*c^2
*k)*x^2)/(4*c^4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(c*(a*b^3*f + 8*a^2*b*c*f + 4*a^2*(7*c^2*d + a*c*h -
9*a^2*j) + b^4*(3*d - (2*a^2*j)/c^2) - a*b^2*(25*c*d + 7*a*h - (11*a^2*j)/c)) + (a*b^2*c^2*f + 20*a^2*c^3*f +
b^3*(3*c^2*d + a^2*j) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*j))*x^2))/(8*a^2*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*
x^4)) + (b^3*c^2*i + 2*b*c^3*(3*c*e + a*i) + 11*a*b^4*k - (b^6*k)/c + 32*a^3*c^2*k - 3*b^2*(c^3*g + 13*a^2*c*k
) + 2*(6*c^5*e + b^2*c^3*i - c^4*(3*b*g - 2*a*i) + 2*b^5*k - 15*a*b^3*c*k + 25*a^2*b*c^2*k)*x^2)/(4*c^3*(b^2 -
4*a*c)^2*(a + b*x^2 + c*x^4)) + ((a*b^2*c^2*f + 20*a^2*c^3*f + b^3*(3*c^2*d + a^2*j) - 4*a*b*c*(6*c^2*d + 3*a
*c*h + 4*a^2*j) + (a*b^3*c^2*f - 52*a^2*b*c^3*f - 6*a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2
*j) + 8*a^2*c^2*(21*c^2*d + 3*a*c*h + 5*a^2*j))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^
2 - 4*a*c]]])/(8*Sqrt[2]*a^2*c^(3/2)*(b^2 - 4*a*c)^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^2*c^2*f + 20*a^2*c^3
*f + b^3*(3*c^2*d + a^2*j) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*j) - (a*b^3*c^2*f - 52*a^2*b*c^3*f - 6*a*b^2*c
*(5*c^2*d - 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2*j) + 8*a^2*c^2*(21*c^2*d + 3*a*c*h + 5*a^2*j))/Sqrt[b^2 -
4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*c^(3/2)*(b^2 - 4*a*c)^2*Sqrt[b
+ Sqrt[b^2 - 4*a*c]]) - ((12*c^5*e + 2*b^2*c^3*i - c^4*(6*b*g - 4*a*i) - b^5*k + 10*a*b^3*c*k - 30*a^2*b*c^2*
k)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(5/2)) + (k*Log[a + b*x^2 + c*x^4])/(4*c^3)

________________________________________________________________________________________

Rubi [A]  time = 7.92648, antiderivative size = 1179, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1673, 1678, 1166, 205, 1663, 1660, 634, 618, 206, 628} \[ -\frac{x \left (\left (-\left (\frac{j a^2}{c^2}+d\right ) b^2+a f b+2 a \left (\frac{j a^2}{c}-h a+c d\right )\right ) c^2+\left (-a j b^3-c \left (-3 j a^2+c h a+c^2 d\right ) b+2 a c^3 f\right ) x^2\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2}+\frac{\left (\left (\frac{j a^2}{c}+3 c d\right ) b^3+a c f b^2-4 a \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^2 f+\frac{\left (3 c^2 d-a^2 j\right ) b^4+a c^2 f b^3-6 a c \left (-3 j a^2-3 c h a+5 c^2 d\right ) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left (5 j a^2+3 c h a+21 c^2 d\right )}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\left (\frac{j a^2}{c}+3 c d\right ) b^3+a c f b^2-4 a \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^2 f-\frac{\left (3 c^2 d-a^2 j\right ) b^4+a c^2 f b^3-6 a c \left (-3 j a^2-3 c h a+5 c^2 d\right ) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left (5 j a^2+3 c h a+21 c^2 d\right )}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left (-k b^5+10 a c k b^3+2 c^3 i b^2-30 a^2 c^2 k b+12 c^5 e-c^4 (6 b g-4 a i)\right ) \tanh ^{-1}\left (\frac{2 c x^2+b}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{k \log \left (c x^4+b x^2+a\right )}{4 c^3}+\frac{x \left (\left (\left (j a^2+3 c^2 d\right ) b^3+a c^2 f b^2-4 a c \left (4 j a^2+3 c h a+6 c^2 d\right ) b+20 a^2 c^3 f\right ) x^2+c \left (\left (3 d-\frac{2 a^2 j}{c^2}\right ) b^4+a f b^3-a \left (-\frac{11 j a^2}{c}+7 h a+25 c d\right ) b^2+8 a^2 c f b+4 a^2 \left (-9 j a^2+c h a+7 c^2 d\right )\right )\right )}{8 a^2 c \left (b^2-4 a c\right )^2 \left (c x^4+b x^2+a\right )}+\frac{-\frac{k b^6}{c}+11 a k b^4+c^2 i b^3-3 \left (g c^3+13 a^2 k c\right ) b^2+2 c^3 (3 c e+a i) b+2 \left (2 k b^5-15 a c k b^3+c^3 i b^2+25 a^2 c^2 k b+6 c^5 e-c^4 (3 b g-2 a i)\right ) x^2+32 a^3 c^2 k}{4 c^3 \left (b^2-4 a c\right )^2 \left (c x^4+b x^2+a\right )}-\frac{-a k b^4+4 a^2 c k b^2+c^3 (c e+a i) b+\left (-k b^5+5 a c k b^3+c^3 i b^2-5 a^2 c^2 k b+2 c^5 e-c^4 (b g+2 a i)\right ) x^2-2 a c^2 \left (k a^2+c^2 g\right )}{4 c^4 \left (b^2-4 a c\right ) \left (c x^4+b x^2+a\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^8 + k*x^11)/(a + b*x^2 + c*x^4)^3,x]

[Out]

-(x*(c^2*(a*b*f - b^2*(d + (a^2*j)/c^2) + 2*a*(c*d - a*h + (a^2*j)/c)) + (2*a*c^3*f - a*b^3*j - b*c*(c^2*d + a
*c*h - 3*a^2*j))*x^2))/(4*a*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (b*c^3*(c*e + a*i) - a*b^4*k + 4*a^2*b^
2*c*k - 2*a*c^2*(c^2*g + a^2*k) + (2*c^5*e + b^2*c^3*i - c^4*(b*g + 2*a*i) - b^5*k + 5*a*b^3*c*k - 5*a^2*b*c^2
*k)*x^2)/(4*c^4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(c*(a*b^3*f + 8*a^2*b*c*f + 4*a^2*(7*c^2*d + a*c*h -
9*a^2*j) + b^4*(3*d - (2*a^2*j)/c^2) - a*b^2*(25*c*d + 7*a*h - (11*a^2*j)/c)) + (a*b^2*c^2*f + 20*a^2*c^3*f +
b^3*(3*c^2*d + a^2*j) - 4*a*b*c*(6*c^2*d + 3*a*c*h + 4*a^2*j))*x^2))/(8*a^2*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*
x^4)) + (b^3*c^2*i + 2*b*c^3*(3*c*e + a*i) + 11*a*b^4*k - (b^6*k)/c + 32*a^3*c^2*k - 3*b^2*(c^3*g + 13*a^2*c*k
) + 2*(6*c^5*e + b^2*c^3*i - c^4*(3*b*g - 2*a*i) + 2*b^5*k - 15*a*b^3*c*k + 25*a^2*b*c^2*k)*x^2)/(4*c^3*(b^2 -
4*a*c)^2*(a + b*x^2 + c*x^4)) + ((a*b^2*c*f + 20*a^2*c^2*f - 4*a*b*(6*c^2*d + 3*a*c*h + 4*a^2*j) + b^3*(3*c*d
+ (a^2*j)/c) + (a*b^3*c^2*f - 52*a^2*b*c^3*f - 6*a*b^2*c*(5*c^2*d - 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2*j
) + 8*a^2*c^2*(21*c^2*d + 3*a*c*h + 5*a^2*j))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[
b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*Sqrt[c]*(b^2 - 4*a*c)^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^2*c*f + 20*a^2*c^2
*f - 4*a*b*(6*c^2*d + 3*a*c*h + 4*a^2*j) + b^3*(3*c*d + (a^2*j)/c) - (a*b^3*c^2*f - 52*a^2*b*c^3*f - 6*a*b^2*c
*(5*c^2*d - 3*a*c*h - 3*a^2*j) + b^4*(3*c^2*d - a^2*j) + 8*a^2*c^2*(21*c^2*d + 3*a*c*h + 5*a^2*j))/(c*Sqrt[b^2
- 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*Sqrt[c]*(b^2 - 4*a*c)^2*Sq
rt[b + Sqrt[b^2 - 4*a*c]]) - ((12*c^5*e + 2*b^2*c^3*i - c^4*(6*b*g - 4*a*i) - b^5*k + 10*a*b^3*c*k - 30*a^2*b*
c^2*k)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(5/2)) + (k*Log[a + b*x^2 + c*x^4])/(4*c
^3)

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2
+ c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1166

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

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{d+e x+f x^2+g x^3+h x^4+59 x^5+j x^8+k x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx &=\int \frac{d+f x^2+h x^4+j x^8}{\left (a+b x^2+c x^4\right )^3} \, dx+\int \frac{x \left (e+g x^2+59 x^4+k x^{10}\right )}{\left (a+b x^2+c x^4\right )^3} \, dx\\ &=-\frac{x \left (c^2 \left (a b f-b^2 \left (d+\frac{a^2 j}{c^2}\right )+2 a \left (c d-a h+\frac{a^2 j}{c}\right )\right )+\left (2 a c^3 f-a b^3 j-b c \left (c^2 d+a c h-3 a^2 j\right )\right ) x^2\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x+59 x^2+k x^5}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )-\frac{\int \frac{-a b f-b^2 \left (3 d-\frac{a^2 j}{c^2}\right )+2 a \left (7 c d+a h-\frac{a^2 j}{c}\right )+\frac{\left (10 a c^3 f-a b^3 j-b c \left (5 c^2 d+5 a c h+a^2 j\right )\right ) x^2}{c^2}+4 a \left (4 a-\frac{b^2}{c}\right ) j x^4}{\left (a+b x^2+c x^4\right )^2} \, dx}{4 a \left (b^2-4 a c\right )}\\ &=-\frac{x \left (c^2 \left (a b f-b^2 \left (d+\frac{a^2 j}{c^2}\right )+2 a \left (c d-a h+\frac{a^2 j}{c}\right )\right )+\left (2 a c^3 f-a b^3 j-b c \left (c^2 d+a c h-3 a^2 j\right )\right ) x^2\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{b c^4 e+4 a^2 b^2 c k-2 a^3 c^2 k+a \left (59 b c^3-2 c^4 g-b^4 k\right )+\left (59 b^2 c^3-118 a c^4+2 c^5 e-b c^4 g-b^5 k+5 a b^3 c k-5 a^2 b c^2 k\right ) x^2}{4 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (c \left (a b^3 f+8 a^2 b c f+4 a^2 \left (7 c^2 d+a c h-9 a^2 j\right )+b^4 \left (3 d-\frac{2 a^2 j}{c^2}\right )-a b^2 \left (25 c d+7 a h-\frac{11 a^2 j}{c}\right )\right )+\left (a b^2 c^2 f+20 a^2 c^3 f+b^3 \left (3 c^2 d+a^2 j\right )-4 a b c \left (6 c^2 d+3 a c h+4 a^2 j\right )\right ) x^2\right )}{8 a^2 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\int \frac{3 b^4 d+a b^3 f-16 a^2 b c f+4 a^2 \left (21 c^2 d+3 a c h+5 a^2 j\right )-a b^2 \left (27 c d-3 a h-\frac{a^2 j}{c}\right )+\frac{\left (a b^2 c^2 f+20 a^2 c^3 f+b^3 \left (3 c^2 d+a^2 j\right )-4 a b c \left (6 c^2 d+3 a c h+4 a^2 j\right )\right ) x^2}{c}}{a+b x^2+c x^4} \, dx}{8 a^2 \left (b^2-4 a c\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{\frac{59 b^2}{c}+6 c e-3 b g-\frac{b^5 k}{c^4}+\frac{a^2 b k}{c^2}+a \left (118+\frac{3 b^3 k}{c^3}\right )-\frac{2 \left (b^4-5 a b^2 c+4 a^2 c^2\right ) k x}{c^3}+\frac{2 b \left (b^2-4 a c\right ) k x^2}{c^2}+2 \left (4 a-\frac{b^2}{c}\right ) k x^3}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=-\frac{x \left (c^2 \left (a b f-b^2 \left (d+\frac{a^2 j}{c^2}\right )+2 a \left (c d-a h+\frac{a^2 j}{c}\right )\right )+\left (2 a c^3 f-a b^3 j-b c \left (c^2 d+a c h-3 a^2 j\right )\right ) x^2\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{b c^4 e+4 a^2 b^2 c k-2 a^3 c^2 k+a \left (59 b c^3-2 c^4 g-b^4 k\right )+\left (59 b^2 c^3-118 a c^4+2 c^5 e-b c^4 g-b^5 k+5 a b^3 c k-5 a^2 b c^2 k\right ) x^2}{4 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (c \left (a b^3 f+8 a^2 b c f+4 a^2 \left (7 c^2 d+a c h-9 a^2 j\right )+b^4 \left (3 d-\frac{2 a^2 j}{c^2}\right )-a b^2 \left (25 c d+7 a h-\frac{11 a^2 j}{c}\right )\right )+\left (a b^2 c^2 f+20 a^2 c^3 f+b^3 \left (3 c^2 d+a^2 j\right )-4 a b c \left (6 c^2 d+3 a c h+4 a^2 j\right )\right ) x^2\right )}{8 a^2 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{59 b^3 c^2+2 b c^3 (59 a+3 c e)+11 a b^4 k-\frac{b^6 k}{c}+32 a^3 c^2 k-3 b^2 \left (c^3 g+13 a^2 c k\right )+2 \left (59 b^2 c^3+118 a c^4+6 c^5 e-3 b c^4 g+2 b^5 k-15 a b^3 c k+25 a^2 b c^2 k\right ) x^2}{4 c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{2 \left (59 b^2+118 a c+6 c^2 e-3 b c g+\frac{a b^3 k}{c^2}-\frac{7 a^2 b k}{c}\right )+\frac{2 \left (b^2-4 a c\right )^2 k x}{c^2}}{a+b x+c x^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )^2}+\frac{\left (a b^2 c f+20 a^2 c^2 f-4 a b \left (6 c^2 d+3 a c h+4 a^2 j\right )+b^3 \left (3 c d+\frac{a^2 j}{c}\right )-\frac{a b^3 c^2 f-52 a^2 b c^3 f-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 j\right )+b^4 \left (3 c^2 d-a^2 j\right )+8 a^2 c^2 \left (21 c^2 d+3 a c h+5 a^2 j\right )}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{16 a^2 \left (b^2-4 a c\right )^2}+\frac{\left (a b^2 c f+20 a^2 c^2 f-4 a b \left (6 c^2 d+3 a c h+4 a^2 j\right )+b^3 \left (3 c d+\frac{a^2 j}{c}\right )+\frac{a b^3 c^2 f-52 a^2 b c^3 f-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 j\right )+b^4 \left (3 c^2 d-a^2 j\right )+8 a^2 c^2 \left (21 c^2 d+3 a c h+5 a^2 j\right )}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{16 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac{x \left (c^2 \left (a b f-b^2 \left (d+\frac{a^2 j}{c^2}\right )+2 a \left (c d-a h+\frac{a^2 j}{c}\right )\right )+\left (2 a c^3 f-a b^3 j-b c \left (c^2 d+a c h-3 a^2 j\right )\right ) x^2\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{b c^4 e+4 a^2 b^2 c k-2 a^3 c^2 k+a \left (59 b c^3-2 c^4 g-b^4 k\right )+\left (59 b^2 c^3-118 a c^4+2 c^5 e-b c^4 g-b^5 k+5 a b^3 c k-5 a^2 b c^2 k\right ) x^2}{4 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (c \left (a b^3 f+8 a^2 b c f+4 a^2 \left (7 c^2 d+a c h-9 a^2 j\right )+b^4 \left (3 d-\frac{2 a^2 j}{c^2}\right )-a b^2 \left (25 c d+7 a h-\frac{11 a^2 j}{c}\right )\right )+\left (a b^2 c^2 f+20 a^2 c^3 f+b^3 \left (3 c^2 d+a^2 j\right )-4 a b c \left (6 c^2 d+3 a c h+4 a^2 j\right )\right ) x^2\right )}{8 a^2 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{59 b^3 c^2+2 b c^3 (59 a+3 c e)+11 a b^4 k-\frac{b^6 k}{c}+32 a^3 c^2 k-3 b^2 \left (c^3 g+13 a^2 c k\right )+2 \left (59 b^2 c^3+118 a c^4+6 c^5 e-3 b c^4 g+2 b^5 k-15 a b^3 c k+25 a^2 b c^2 k\right ) x^2}{4 c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (a b^2 c f+20 a^2 c^2 f-4 a b \left (6 c^2 d+3 a c h+4 a^2 j\right )+b^3 \left (3 c d+\frac{a^2 j}{c}\right )+\frac{a b^3 c^2 f-52 a^2 b c^3 f-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 j\right )+b^4 \left (3 c^2 d-a^2 j\right )+8 a^2 c^2 \left (21 c^2 d+3 a c h+5 a^2 j\right )}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (a b^2 c f+20 a^2 c^2 f-4 a b \left (6 c^2 d+3 a c h+4 a^2 j\right )+b^3 \left (3 c d+\frac{a^2 j}{c}\right )-\frac{a b^3 c^2 f-52 a^2 b c^3 f-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 j\right )+b^4 \left (3 c^2 d-a^2 j\right )+8 a^2 c^2 \left (21 c^2 d+3 a c h+5 a^2 j\right )}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{k \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3}+\frac{\left (118 b^2 c^3+4 c^4 (59 a+3 c e)-b^5 k+10 a b^3 c k-6 b \left (c^4 g+5 a^2 c^2 k\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac{x \left (c^2 \left (a b f-b^2 \left (d+\frac{a^2 j}{c^2}\right )+2 a \left (c d-a h+\frac{a^2 j}{c}\right )\right )+\left (2 a c^3 f-a b^3 j-b c \left (c^2 d+a c h-3 a^2 j\right )\right ) x^2\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{b c^4 e+4 a^2 b^2 c k-2 a^3 c^2 k+a \left (59 b c^3-2 c^4 g-b^4 k\right )+\left (59 b^2 c^3-118 a c^4+2 c^5 e-b c^4 g-b^5 k+5 a b^3 c k-5 a^2 b c^2 k\right ) x^2}{4 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (c \left (a b^3 f+8 a^2 b c f+4 a^2 \left (7 c^2 d+a c h-9 a^2 j\right )+b^4 \left (3 d-\frac{2 a^2 j}{c^2}\right )-a b^2 \left (25 c d+7 a h-\frac{11 a^2 j}{c}\right )\right )+\left (a b^2 c^2 f+20 a^2 c^3 f+b^3 \left (3 c^2 d+a^2 j\right )-4 a b c \left (6 c^2 d+3 a c h+4 a^2 j\right )\right ) x^2\right )}{8 a^2 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{59 b^3 c^2+2 b c^3 (59 a+3 c e)+11 a b^4 k-\frac{b^6 k}{c}+32 a^3 c^2 k-3 b^2 \left (c^3 g+13 a^2 c k\right )+2 \left (59 b^2 c^3+118 a c^4+6 c^5 e-3 b c^4 g+2 b^5 k-15 a b^3 c k+25 a^2 b c^2 k\right ) x^2}{4 c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (a b^2 c f+20 a^2 c^2 f-4 a b \left (6 c^2 d+3 a c h+4 a^2 j\right )+b^3 \left (3 c d+\frac{a^2 j}{c}\right )+\frac{a b^3 c^2 f-52 a^2 b c^3 f-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 j\right )+b^4 \left (3 c^2 d-a^2 j\right )+8 a^2 c^2 \left (21 c^2 d+3 a c h+5 a^2 j\right )}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (a b^2 c f+20 a^2 c^2 f-4 a b \left (6 c^2 d+3 a c h+4 a^2 j\right )+b^3 \left (3 c d+\frac{a^2 j}{c}\right )-\frac{a b^3 c^2 f-52 a^2 b c^3 f-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 j\right )+b^4 \left (3 c^2 d-a^2 j\right )+8 a^2 c^2 \left (21 c^2 d+3 a c h+5 a^2 j\right )}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{k \log \left (a+b x^2+c x^4\right )}{4 c^3}-\frac{\left (118 b^2 c^3+4 c^4 (59 a+3 c e)-b^5 k+10 a b^3 c k-6 b \left (c^4 g+5 a^2 c^2 k\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac{x \left (c^2 \left (a b f-b^2 \left (d+\frac{a^2 j}{c^2}\right )+2 a \left (c d-a h+\frac{a^2 j}{c}\right )\right )+\left (2 a c^3 f-a b^3 j-b c \left (c^2 d+a c h-3 a^2 j\right )\right ) x^2\right )}{4 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{b c^4 e+4 a^2 b^2 c k-2 a^3 c^2 k+a \left (59 b c^3-2 c^4 g-b^4 k\right )+\left (59 b^2 c^3-118 a c^4+2 c^5 e-b c^4 g-b^5 k+5 a b^3 c k-5 a^2 b c^2 k\right ) x^2}{4 c^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (c \left (a b^3 f+8 a^2 b c f+4 a^2 \left (7 c^2 d+a c h-9 a^2 j\right )+b^4 \left (3 d-\frac{2 a^2 j}{c^2}\right )-a b^2 \left (25 c d+7 a h-\frac{11 a^2 j}{c}\right )\right )+\left (a b^2 c^2 f+20 a^2 c^3 f+b^3 \left (3 c^2 d+a^2 j\right )-4 a b c \left (6 c^2 d+3 a c h+4 a^2 j\right )\right ) x^2\right )}{8 a^2 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{59 b^3 c^2+2 b c^3 (59 a+3 c e)+11 a b^4 k-\frac{b^6 k}{c}+32 a^3 c^2 k-3 b^2 \left (c^3 g+13 a^2 c k\right )+2 \left (59 b^2 c^3+118 a c^4+6 c^5 e-3 b c^4 g+2 b^5 k-15 a b^3 c k+25 a^2 b c^2 k\right ) x^2}{4 c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (a b^2 c f+20 a^2 c^2 f-4 a b \left (6 c^2 d+3 a c h+4 a^2 j\right )+b^3 \left (3 c d+\frac{a^2 j}{c}\right )+\frac{a b^3 c^2 f-52 a^2 b c^3 f-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 j\right )+b^4 \left (3 c^2 d-a^2 j\right )+8 a^2 c^2 \left (21 c^2 d+3 a c h+5 a^2 j\right )}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (a b^2 c f+20 a^2 c^2 f-4 a b \left (6 c^2 d+3 a c h+4 a^2 j\right )+b^3 \left (3 c d+\frac{a^2 j}{c}\right )-\frac{a b^3 c^2 f-52 a^2 b c^3 f-6 a b^2 c \left (5 c^2 d-3 a c h-3 a^2 j\right )+b^4 \left (3 c^2 d-a^2 j\right )+8 a^2 c^2 \left (21 c^2 d+3 a c h+5 a^2 j\right )}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left (118 b^2 c^3+236 a c^4+12 c^5 e-6 b c^4 g-b^5 k+10 a b^3 c k-30 a^2 b c^2 k\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{k \log \left (a+b x^2+c x^4\right )}{4 c^3}\\ \end{align*}

Mathematica [A]  time = 7.6312, size = 1649, normalized size = 1.4 \[ \frac{-a k x^2 b^5-a^2 k b^4-a c^2 j x^3 b^3+5 a^2 c k x^2 b^3+a c^3 i x^2 b^2+4 a^3 c k b^2-c^4 d x b^2-a^2 c^2 j x b^2-c^5 d x^3 b-a c^4 h x^3 b+3 a^2 c^3 j x^3 b-a c^4 g x^2 b-5 a^3 c^2 k x^2 b+a c^4 e b+a^2 c^3 i b+a c^4 f x b+2 a c^5 f x^3+2 a c^5 e x^2-2 a^2 c^4 i x^2-2 a^2 c^4 g-2 a^4 c^2 k+2 a c^5 d x-2 a^2 c^4 h x+2 a^3 c^3 j x}{4 a c^4 \left (4 a c-b^2\right ) \left (c x^4+b x^2+a\right )^2}+\frac{\left (40 c^2 j a^4+24 c^3 h a^3+18 b^2 c j a^3-16 b c \sqrt{b^2-4 a c} j a^3+168 c^4 d a^2-52 b c^3 f a^2+20 c^3 \sqrt{b^2-4 a c} f a^2+18 b^2 c^2 h a^2-12 b c^2 \sqrt{b^2-4 a c} h a^2-b^4 j a^2+b^3 \sqrt{b^2-4 a c} j a^2-30 b^2 c^3 d a-24 b c^3 \sqrt{b^2-4 a c} d a+b^3 c^2 f a+b^2 c^2 \sqrt{b^2-4 a c} f a+3 b^4 c^2 d+3 b^3 c^2 \sqrt{b^2-4 a c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (-40 c^2 j a^4-24 c^3 h a^3-18 b^2 c j a^3-16 b c \sqrt{b^2-4 a c} j a^3-168 c^4 d a^2+52 b c^3 f a^2+20 c^3 \sqrt{b^2-4 a c} f a^2-18 b^2 c^2 h a^2-12 b c^2 \sqrt{b^2-4 a c} h a^2+b^4 j a^2+b^3 \sqrt{b^2-4 a c} j a^2+30 b^2 c^3 d a-24 b c^3 \sqrt{b^2-4 a c} d a-b^3 c^2 f a+b^2 c^2 \sqrt{b^2-4 a c} f a-3 b^4 c^2 d+3 b^3 c^2 \sqrt{b^2-4 a c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left (-k b^5+\sqrt{b^2-4 a c} k b^4+10 a c k b^3+2 c^3 i b^2-8 a c \sqrt{b^2-4 a c} k b^2-6 c^4 g b-30 a^2 c^2 k b+12 c^5 e+4 a c^4 i+16 a^2 c^2 \sqrt{b^2-4 a c} k\right ) \log \left (-2 c x^2-b+\sqrt{b^2-4 a c}\right )}{4 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{\left (k b^5+\sqrt{b^2-4 a c} k b^4-10 a c k b^3-2 c^3 i b^2-8 a c \sqrt{b^2-4 a c} k b^2+6 c^4 g b+30 a^2 c^2 k b-12 c^5 e-4 a c^4 i+16 a^2 c^2 \sqrt{b^2-4 a c} k\right ) \log \left (2 c x^2+b+\sqrt{b^2-4 a c}\right )}{4 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a^2 k b^6+8 a^2 c k x^2 b^5+22 a^3 c k b^4+3 c^4 d x b^4-2 a^2 c^2 j x b^4+3 c^5 d x^3 b^3+a^2 c^3 j x^3 b^3-60 a^3 c^2 k x^2 b^3+2 a^2 c^3 i b^3+a c^4 f x b^3+a c^5 f x^3 b^2+4 a^2 c^4 i x^2 b^2-6 a^2 c^4 g b^2-78 a^4 c^2 k b^2-25 a c^5 d x b^2-7 a^2 c^4 h x b^2+11 a^3 c^3 j x b^2-24 a c^6 d x^3 b-12 a^2 c^5 h x^3 b-16 a^3 c^4 j x^3 b-12 a^2 c^5 g x^2 b+100 a^4 c^3 k x^2 b+12 a^2 c^5 e b+4 a^3 c^4 i b+8 a^2 c^5 f x b+20 a^2 c^6 f x^3+24 a^2 c^6 e x^2+8 a^3 c^5 i x^2+64 a^5 c^3 k+28 a^2 c^6 d x+4 a^3 c^5 h x-36 a^4 c^4 j x}{8 a^2 c^4 \left (4 a c-b^2\right )^2 \left (c x^4+b x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^8 + k*x^11)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(a*b*c^4*e - 2*a^2*c^4*g + a^2*b*c^3*i - a^2*b^4*k + 4*a^3*b^2*c*k - 2*a^4*c^2*k - b^2*c^4*d*x + 2*a*c^5*d*x +
a*b*c^4*f*x - 2*a^2*c^4*h*x - a^2*b^2*c^2*j*x + 2*a^3*c^3*j*x + 2*a*c^5*e*x^2 - a*b*c^4*g*x^2 + a*b^2*c^3*i*x
^2 - 2*a^2*c^4*i*x^2 - a*b^5*k*x^2 + 5*a^2*b^3*c*k*x^2 - 5*a^3*b*c^2*k*x^2 - b*c^5*d*x^3 + 2*a*c^5*f*x^3 - a*b
*c^4*h*x^3 - a*b^3*c^2*j*x^3 + 3*a^2*b*c^3*j*x^3)/(4*a*c^4*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (12*a^2*b*c
^5*e - 6*a^2*b^2*c^4*g + 2*a^2*b^3*c^3*i + 4*a^3*b*c^4*i - 2*a^2*b^6*k + 22*a^3*b^4*c*k - 78*a^4*b^2*c^2*k + 6
4*a^5*c^3*k + 3*b^4*c^4*d*x - 25*a*b^2*c^5*d*x + 28*a^2*c^6*d*x + a*b^3*c^4*f*x + 8*a^2*b*c^5*f*x - 7*a^2*b^2*
c^4*h*x + 4*a^3*c^5*h*x - 2*a^2*b^4*c^2*j*x + 11*a^3*b^2*c^3*j*x - 36*a^4*c^4*j*x + 24*a^2*c^6*e*x^2 - 12*a^2*
b*c^5*g*x^2 + 4*a^2*b^2*c^4*i*x^2 + 8*a^3*c^5*i*x^2 + 8*a^2*b^5*c*k*x^2 - 60*a^3*b^3*c^2*k*x^2 + 100*a^4*b*c^3
*k*x^2 + 3*b^3*c^5*d*x^3 - 24*a*b*c^6*d*x^3 + a*b^2*c^5*f*x^3 + 20*a^2*c^6*f*x^3 - 12*a^2*b*c^5*h*x^3 + a^2*b^
3*c^3*j*x^3 - 16*a^3*b*c^4*j*x^3)/(8*a^2*c^4*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*b^4*c^2*d - 30*a*b^2*
c^3*d + 168*a^2*c^4*d + 3*b^3*c^2*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c^3*Sqrt[b^2 - 4*a*c]*d + a*b^3*c^2*f - 52*a^2*
b*c^3*f + a*b^2*c^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c^3*Sqrt[b^2 - 4*a*c]*f + 18*a^2*b^2*c^2*h + 24*a^3*c^3*h - 1
2*a^2*b*c^2*Sqrt[b^2 - 4*a*c]*h - a^2*b^4*j + 18*a^3*b^2*c*j + 40*a^4*c^2*j + a^2*b^3*Sqrt[b^2 - 4*a*c]*j - 16
*a^3*b*c*Sqrt[b^2 - 4*a*c]*j)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*c^(3/2)*
(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((-3*b^4*c^2*d + 30*a*b^2*c^3*d - 168*a^2*c^4*d + 3*b^3*c^2
*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c^3*Sqrt[b^2 - 4*a*c]*d - a*b^3*c^2*f + 52*a^2*b*c^3*f + a*b^2*c^2*Sqrt[b^2 - 4*
a*c]*f + 20*a^2*c^3*Sqrt[b^2 - 4*a*c]*f - 18*a^2*b^2*c^2*h - 24*a^3*c^3*h - 12*a^2*b*c^2*Sqrt[b^2 - 4*a*c]*h +
a^2*b^4*j - 18*a^3*b^2*c*j - 40*a^4*c^2*j + a^2*b^3*Sqrt[b^2 - 4*a*c]*j - 16*a^3*b*c*Sqrt[b^2 - 4*a*c]*j)*Arc
Tan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt
[b^2 - 4*a*c]]) + ((12*c^5*e - 6*b*c^4*g + 2*b^2*c^3*i + 4*a*c^4*i - b^5*k + 10*a*b^3*c*k - 30*a^2*b*c^2*k + b
^4*Sqrt[b^2 - 4*a*c]*k - 8*a*b^2*c*Sqrt[b^2 - 4*a*c]*k + 16*a^2*c^2*Sqrt[b^2 - 4*a*c]*k)*Log[-b + Sqrt[b^2 - 4
*a*c] - 2*c*x^2])/(4*c^3*(b^2 - 4*a*c)^(5/2)) + ((-12*c^5*e + 6*b*c^4*g - 2*b^2*c^3*i - 4*a*c^4*i + b^5*k - 10
*a*b^3*c*k + 30*a^2*b*c^2*k + b^4*Sqrt[b^2 - 4*a*c]*k - 8*a*b^2*c*Sqrt[b^2 - 4*a*c]*k + 16*a^2*c^2*Sqrt[b^2 -
4*a*c]*k)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*c^3*(b^2 - 4*a*c)^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.089, size = 6130, normalized size = 5.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((k*x^11+j*x^8+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^3,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [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((k*x^11+j*x^8+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [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((k*x^11+j*x^8+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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((k*x**11+j*x**8+i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**3,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((k*x^11+j*x^8+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

Timed out