3.58 \(\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{(a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=645 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{b^2 c (c d-a h)-a b^3 j+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt{b^2-4 a c}}+\frac{a b^2 j}{c}+b (a h+c d)-2 a (3 a j+c f)\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{b^2 c (c d-a h)-a b^3 j+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt{b^2-4 a c}}+\frac{a b^2 j}{c}+b (a h+c d)-2 a (3 a j+c f)\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{x \left (x^2 \left (-a b^2 j+b c (a h+c d)-2 a c (c f-a j)\right )+c \left (-\frac{a b (a j+c f)}{c}-2 a (c d-a h)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{k \log \left (a+b x^2+c x^4\right )}{4 c^2} \]

[Out]

(x*(c*(b^2*d - 2*a*(c*d - a*h) - (a*b*(c*f + a*j))/c) + (b*c*(c*d + a*h) - a*b^2*j - 2*a*c*(c*f - a*j))*x^2))/
(2*a*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (b*c*(c*e + a*i) - a*b^2*k - 2*a*c*(c*g - a*k) + (2*c^3*e - c^2*(b
*g + 2*a*i) - b^3*k + b*c*(b*i + 3*a*k))*x^2)/(2*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b*(c*d + a*h) + (a
*b^2*j)/c - 2*a*(c*f + 3*a*j) + (b^2*c*(c*d - a*h) - 4*a*c^2*(3*c*d + a*h) - a*b^3*j + 4*a*b*c*(c*f + 2*a*j))/
(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*Sqrt[c]*(b^2 - 4*
a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b*(c*d + a*h) + (a*b^2*j)/c - 2*a*(c*f + 3*a*j) - (b^2*c*(c*d - a*h) - 4
*a*c^2*(3*c*d + a*h) - a*b^3*j + 4*a*b*c*(c*f + 2*a*j))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt
[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*Sqrt[c]*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((4*c^3*e - c^2*(2
*b*g - 4*a*i) + b^3*k - 6*a*b*c*k)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^2*(b^2 - 4*a*c)^(3/2)) + (k*
Log[a + b*x^2 + c*x^4])/(4*c^2)

________________________________________________________________________________________

Rubi [A]  time = 3.36662, antiderivative size = 645, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{b^2 c (c d-a h)-a b^3 j+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt{b^2-4 a c}}+\frac{a b^2 j}{c}+b (a h+c d)-2 a (3 a j+c f)\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{b^2 c (c d-a h)-a b^3 j+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt{b^2-4 a c}}+\frac{a b^2 j}{c}+b (a h+c d)-2 a (3 a j+c f)\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{x \left (x^2 \left (-a b^2 j+b c (a h+c d)-2 a c (c f-a j)\right )+c \left (-\frac{a b (a j+c f)}{c}-2 a (c d-a h)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{k \log \left (a+b x^2+c x^4\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^6 + k*x^7)/(a + b*x^2 + c*x^4)^2,x]

[Out]

(x*(c*(b^2*d - 2*a*(c*d - a*h) - (a*b*(c*f + a*j))/c) + (b*c*(c*d + a*h) - a*b^2*j - 2*a*c*(c*f - a*j))*x^2))/
(2*a*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (b*c*(c*e + a*i) - a*b^2*k - 2*a*c*(c*g - a*k) + (2*c^3*e - c^2*(b
*g + 2*a*i) - b^3*k + b*c*(b*i + 3*a*k))*x^2)/(2*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b*(c*d + a*h) + (a
*b^2*j)/c - 2*a*(c*f + 3*a*j) + (b^2*c*(c*d - a*h) - 4*a*c^2*(3*c*d + a*h) - a*b^3*j + 4*a*b*c*(c*f + 2*a*j))/
(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*Sqrt[c]*(b^2 - 4*
a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b*(c*d + a*h) + (a*b^2*j)/c - 2*a*(c*f + 3*a*j) - (b^2*c*(c*d - a*h) - 4
*a*c^2*(3*c*d + a*h) - a*b^3*j + 4*a*b*c*(c*f + 2*a*j))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt
[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*Sqrt[c]*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((4*c^3*e - c^2*(2
*b*g - 4*a*i) + b^3*k - 6*a*b*c*k)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^2*(b^2 - 4*a*c)^(3/2)) + (k*
Log[a + b*x^2 + c*x^4])/(4*c^2)

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+58 x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx &=\int \frac{d+f x^2+h x^4+j x^6}{\left (a+b x^2+c x^4\right )^2} \, dx+\int \frac{x \left (e+g x^2+58 x^4+k x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{x \left (c \left (b^2 d-2 a (c d-a h)-\frac{a b (c f+a j)}{c}\right )+\left (b c (c d+a h)-a b^2 j-2 a c (c f-a j)\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x+58 x^2+k x^3}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )-\frac{\int \frac{-b^2 d+2 a (3 c d+a h)-\frac{a b (c f+a j)}{c}+\left (-b (c d+a h)-\frac{a b^2 j}{c}+2 a (c f+3 a j)\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{x \left (c \left (b^2 d-2 a (c d-a h)-\frac{a b (c f+a j)}{c}\right )+\left (b c (c d+a h)-a b^2 j-2 a c (c f-a j)\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b c^2 e+2 a^2 c k+a \left (58 b c-2 c^2 g-b^2 k\right )+\left (58 b^2 c-2 c^2 (58 a-c e)-b^3 k-b c (c g-3 a k)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{2 c e-b g+a \left (116-\frac{b k}{c}\right )+\left (4 a-\frac{b^2}{c}\right ) k x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}+\frac{\left (b (c d+a h)+\frac{a b^2 j}{c}-2 a (c f+3 a j)-\frac{b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{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}{4 a \left (b^2-4 a c\right )}+\frac{\left (b (c d+a h)+\frac{a b^2 j}{c}-2 a (c f+3 a j)+\frac{b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{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}{4 a \left (b^2-4 a c\right )}\\ &=\frac{x \left (c \left (b^2 d-2 a (c d-a h)-\frac{a b (c f+a j)}{c}\right )+\left (b c (c d+a h)-a b^2 j-2 a c (c f-a j)\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b c^2 e+2 a^2 c k+a \left (58 b c-2 c^2 g-b^2 k\right )+\left (58 b^2 c-2 c^2 (58 a-c e)-b^3 k-b c (c g-3 a k)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (b (c d+a h)+\frac{a b^2 j}{c}-2 a (c f+3 a j)+\frac{b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{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 )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (b (c d+a h)+\frac{a b^2 j}{c}-2 a (c f+3 a j)-\frac{b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{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 )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \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^2}-\frac{\left (4 c^3 e-2 b c^2 g+b^3 k+2 a c (116 c-3 b k)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2 \left (b^2-4 a c\right )}\\ &=\frac{x \left (c \left (b^2 d-2 a (c d-a h)-\frac{a b (c f+a j)}{c}\right )+\left (b c (c d+a h)-a b^2 j-2 a c (c f-a j)\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b c^2 e+2 a^2 c k+a \left (58 b c-2 c^2 g-b^2 k\right )+\left (58 b^2 c-2 c^2 (58 a-c e)-b^3 k-b c (c g-3 a k)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (b (c d+a h)+\frac{a b^2 j}{c}-2 a (c f+3 a j)+\frac{b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{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 )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (b (c d+a h)+\frac{a b^2 j}{c}-2 a (c f+3 a j)-\frac{b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{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 )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{k \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{\left (4 c^3 e-2 b c^2 g+b^3 k+2 a c (116 c-3 b k)\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^2 \left (b^2-4 a c\right )}\\ &=\frac{x \left (c \left (b^2 d-2 a (c d-a h)-\frac{a b (c f+a j)}{c}\right )+\left (b c (c d+a h)-a b^2 j-2 a c (c f-a j)\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b c^2 e+2 a^2 c k+a \left (58 b c-2 c^2 g-b^2 k\right )+\left (58 b^2 c-2 c^2 (58 a-c e)-b^3 k-b c (c g-3 a k)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (b (c d+a h)+\frac{a b^2 j}{c}-2 a (c f+3 a j)+\frac{b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{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 )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (b (c d+a h)+\frac{a b^2 j}{c}-2 a (c f+3 a j)-\frac{b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{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 )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left (4 c^3 e-2 b c^2 g+b^3 k+2 a c (116 c-3 b k)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{k \log \left (a+b x^2+c x^4\right )}{4 c^2}\\ \end{align*}

Mathematica [A]  time = 6.14003, size = 775, normalized size = 1.2 \[ \frac{\frac{2 \left (a^2 \left (b^2 (-k)+b c (i+x (j+3 k x))-2 c^2 (g+x (h+x (i+j x)))\right )+2 a^3 c k+a \left (b^2 c x^2 (i+j x)+b^3 (-k) x^2+b c^2 (e+x (f-x (g+h x)))+2 c^3 x (d+x (e+f x))\right )-b c^2 d x \left (b+c x^2\right )\right )}{a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}-\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-b c \left (8 a^2 j+c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}+4 a c f\right )+2 a c \left (c f \sqrt{b^2-4 a c}+3 a j \sqrt{b^2-4 a c}+2 a c h+6 c^2 d\right )-b^2 \left (a j \sqrt{b^2-4 a c}-a c h+c^2 d\right )+a b^3 j\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (b c \left (-8 a^2 j+c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}-4 a c f\right )+2 a c \left (-c f \sqrt{b^2-4 a c}-3 a j \sqrt{b^2-4 a c}+2 a c h+6 c^2 d\right )+b^2 \left (a j \sqrt{b^2-4 a c}+a c h+c^2 (-d)\right )+a b^3 j\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right ) \left (a c \left (6 b k-4 k \sqrt{b^2-4 a c}\right )+b^2 k \left (\sqrt{b^2-4 a c}-b\right )+2 c^2 (b g-2 a i)-4 c^3 e\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (-2 a c k \left (2 \sqrt{b^2-4 a c}+3 b\right )+b^2 k \left (\sqrt{b^2-4 a c}+b\right )+c^2 (4 a i-2 b g)+4 c^3 e\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^6 + k*x^7)/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*(2*a^3*c*k - b*c^2*d*x*(b + c*x^2) + a*(-(b^3*k*x^2) + b^2*c*x^2*(i + j*x) + 2*c^3*x*(d + x*(e + f*x)) + b
*c^2*(e + x*(f - x*(g + h*x)))) + a^2*(-(b^2*k) + b*c*(i + x*(j + 3*k*x)) - 2*c^2*(g + x*(h + x*(i + j*x))))))
/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*Sqrt[c]*(a*b^3*j - b*c*(c*Sqrt[b^2 - 4*a*c]*d + 4*a*c*f + a
*Sqrt[b^2 - 4*a*c]*h + 8*a^2*j) - b^2*(c^2*d - a*c*h + a*Sqrt[b^2 - 4*a*c]*j) + 2*a*c*(6*c^2*d + c*Sqrt[b^2 -
4*a*c]*f + 2*a*c*h + 3*a*Sqrt[b^2 - 4*a*c]*j))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^
2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(a*b^3*j + b*c*(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c
*f + a*Sqrt[b^2 - 4*a*c]*h - 8*a^2*j) + 2*a*c*(6*c^2*d - c*Sqrt[b^2 - 4*a*c]*f + 2*a*c*h - 3*a*Sqrt[b^2 - 4*a*
c]*j) + b^2*(-(c^2*d) + a*c*h + a*Sqrt[b^2 - 4*a*c]*j))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]
])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((-4*c^3*e + 2*c^2*(b*g - 2*a*i) + b^2*(-b + Sqrt[b^2
- 4*a*c])*k + a*c*(6*b*k - 4*Sqrt[b^2 - 4*a*c]*k))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2)
+ ((4*c^3*e + c^2*(-2*b*g + 4*a*i) + b^2*(b + Sqrt[b^2 - 4*a*c])*k - 2*a*c*(3*b + 2*Sqrt[b^2 - 4*a*c])*k)*Log
[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*c^2)

________________________________________________________________________________________

Maple [B]  time = 0.061, size = 3107, normalized size = 4.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((k*x^7+j*x^6+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x)

[Out]

1/4*c/(4*a*c-b^2)^2/a*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(
1/2))*b^3*d-1/4*c/(4*a*c-b^2)^2/a*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^
(1/2)-b)*c)^(1/2))*b^3*d-c/(4*a*c-b^2)^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*
c+b^2)^(1/2)-b)*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b*f-c/(4*a*c-b^2)^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
an(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b*f-1/4*c/(4*a*c-b^2)^2/a*2^(1/2)/((b+(-4*
a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b^2*d-1/4*c/(
4*a*c-b^2)^2/a*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*
(-4*a*c+b^2)^(1/2)*b^2*d+a/(4*a*c-b^2)^2*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4
*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*h-a/(4*a*c-b^2)^2*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc
tan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*h+a/(4*a*c-b^2)^2*c*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/
2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*(-4*a*c+b^2)^(1/2)*h+a/(4*a*c-b^2)^2*c*2^(1/2)/(((-4*
a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b*h+(-1/2/a*(2*a^2*c*j-a*b^2*
j+a*b*c*h-2*a*c^2*f+b*c^2*d)/(4*a*c-b^2)/c*x^3+1/2*(3*a*b*c*k-2*a*c^2*i-b^3*k+b^2*c*i-b*c^2*g+2*c^3*e)/(4*a*c-
b^2)/c^2*x^2+1/2*(a^2*b*j-2*a^2*c*h+a*b*c*f+2*a*c^2*d-b^2*c*d)/a/c/(4*a*c-b^2)*x+1/2*(2*a^2*c*k-a*b^2*k+a*b*c*
i-2*a*c^2*g+b*c^2*e)/(4*a*c-b^2)/c^2)/(c*x^4+b*x^2+a)+3*c^2/(4*a*c-b^2)^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(
1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*d-c^2/(4*a*c-b^2)^2*2^(1/2)/((b+(
-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*d+3*c^2/(4*a*c-b^2)^2*2^(1/
2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*(-4*a*c+b^2)^(1/2)*d
+c^2/(4*a*c-b^2)^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/
2))*b*d-2*a/(4*a*c-b^2)^2/c*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*b^2*k-2*a/(4*a*c-b^2)^2/c*ln(-2*c*x^2+(-4*a*c+b^2
)^(1/2)-b)*b^2*k-1/4/(4*a*c-b^2)^2/c^2*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(-4*a*c+b^2)^(1/2)*b^3*k+1/4/(4*a*c-b
^2)^2/c^2*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(-4*a*c+b^2)^(1/2)*b^3*k+1/4/(4*a*c-b^2)^2/c^2*ln(2*c*x^2+(-4*a*c+b
^2)^(1/2)+b)*b^4*k+1/4/(4*a*c-b^2)^2/c^2*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*b^4*k+1/2/(4*a*c-b^2)^2*ln(-2*c*x^2
+(-4*a*c+b^2)^(1/2)-b)*(-4*a*c+b^2)^(1/2)*b*g-1/2/(4*a*c-b^2)^2*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(-4*a*c+b^2)^
(1/2)*b*g+1/4/(4*a*c-b^2)^2/c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/
2))*c)^(1/2))*b^4*j-1/4/(4*a*c-b^2)^2/c*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c
+b^2)^(1/2)-b)*c)^(1/2))*b^4*j+3/2*a/(4*a*c-b^2)^2/c*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(-4*a*c+b^2)^(1/2)*b*k-
6*a^2/(4*a*c-b^2)^2*c*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^
(1/2))*j+5/2*a/(4*a*c-b^2)^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)
-b)*c)^(1/2))*b^2*j-5/2*a/(4*a*c-b^2)^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*
c+b^2)^(1/2))*c)^(1/2))*b^2*j-3/2*a/(4*a*c-b^2)^2/c*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(-4*a*c+b^2)^(1/2)*b*k+6*
a^2/(4*a*c-b^2)^2*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/
2))*j+2*c^2/(4*a*c-b^2)^2*a*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2))*f-1/2*c/(4*a*c-b^2)^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)
^(1/2))*c)^(1/2))*b^2*f-2*c^2/(4*a*c-b^2)^2*a*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((
-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*f+1/2*c/(4*a*c-b^2)^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1
/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^2*f+1/4/(4*a*c-b^2)^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh
(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b^2*h+1/4/(4*a*c-b^2)^2*2^(1/2)/((b+(-4*a*c+
b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b^2*h+c/(4*a*c-b^
2)^2*(-4*a*c+b^2)^(1/2)*e*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)-c/(4*a*c-b^2)^2*(-4*a*c+b^2)^(1/2)*e*ln(-2*c*x^2+(-
4*a*c+b^2)^(1/2)-b)+1/4/(4*a*c-b^2)^2/c*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c
+b^2)^(1/2)-b)*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b^3*j+1/4/(4*a*c-b^2)^2/c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)
*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b^3*j-2*a/(4*a*c-b^2)^2*2^(1/2)/((b+(
-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b*j-2*a/(4
*a*c-b^2)^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*(-4
*a*c+b^2)^(1/2)*b*j+a/(4*a*c-b^2)^2*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(-4*a*c+b^2)^(1/2)*i-a/(4*a*c-b^2)^2*ln(-
2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(-4*a*c+b^2)^(1/2)*i-1/4/(4*a*c-b^2)^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*
arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^3*h+1/4/(4*a*c-b^2)^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*
c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*h+4*a^2/(4*a*c-b^2)^2*ln(2*c*x^2+(-4*a*c+b^2
)^(1/2)+b)*k+4*a^2/(4*a*c-b^2)^2*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*k

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a b c^{2} e - 2 \, a^{2} c^{2} g + a^{2} b c i -{\left (b c^{3} d - 2 \, a c^{3} f + a b c^{2} h -{\left (a b^{2} c - 2 \, a^{2} c^{2}\right )} j\right )} x^{3} +{\left (2 \, a c^{3} e - a b c^{2} g +{\left (a b^{2} c - 2 \, a^{2} c^{2}\right )} i -{\left (a b^{3} - 3 \, a^{2} b c\right )} k\right )} x^{2} -{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} k +{\left (a b c^{2} f - 2 \, a^{2} c^{2} h + a^{2} b c j -{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d\right )} x}{2 \,{\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3} +{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} x^{4} +{\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} x^{2}\right )}} - \frac{-\int \frac{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} k x^{3} + a b c f - 2 \, a^{2} c h + a^{2} b j +{\left (b c^{2} d - 2 \, a c^{2} f + a b c h +{\left (a b^{2} - 6 \, a^{2} c\right )} j\right )} x^{2} +{\left (b^{2} c - 6 \, a c^{2}\right )} d - 2 \,{\left (2 \, a c^{2} e - a b c g + 2 \, a^{2} c i - a^{2} b k\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((k*x^7+j*x^6+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*(a*b*c^2*e - 2*a^2*c^2*g + a^2*b*c*i - (b*c^3*d - 2*a*c^3*f + a*b*c^2*h - (a*b^2*c - 2*a^2*c^2)*j)*x^3 +
(2*a*c^3*e - a*b*c^2*g + (a*b^2*c - 2*a^2*c^2)*i - (a*b^3 - 3*a^2*b*c)*k)*x^2 - (a^2*b^2 - 2*a^3*c)*k + (a*b*c
^2*f - 2*a^2*c^2*h + a^2*b*c*j - (b^2*c^2 - 2*a*c^3)*d)*x)/(a^2*b^2*c^2 - 4*a^3*c^3 + (a*b^2*c^3 - 4*a^2*c^4)*
x^4 + (a*b^3*c^2 - 4*a^2*b*c^3)*x^2) - 1/2*integrate(-(2*(a*b^2 - 4*a^2*c)*k*x^3 + a*b*c*f - 2*a^2*c*h + a^2*b
*j + (b*c^2*d - 2*a*c^2*f + a*b*c*h + (a*b^2 - 6*a^2*c)*j)*x^2 + (b^2*c - 6*a*c^2)*d - 2*(2*a*c^2*e - a*b*c*g
+ 2*a^2*c*i - a^2*b*k)*x)/(c*x^4 + b*x^2 + a), x)/(a*b^2*c - 4*a^2*c^2)

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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^7+j*x^6+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

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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((k*x**7+j*x**6+i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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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^7+j*x^6+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

Timed out