3.41 \(\int \frac{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{(a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=770 \[ -\frac{x \left (x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )+a b^2 c k-a b^3 m+2 a c^2 (c f-a k)\right )+b^2 \left (-\left (a^2 m+c^2 d\right )\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-\frac{-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )+a b^3 c k-3 a b^4 m-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )+a b^2 c k-3 a b^3 m-2 a c^2 (3 a k+c f)\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (\frac{-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )+a b^3 c k-3 a b^4 m-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )+a b^2 c k-3 a b^3 m-2 a c^2 (3 a k+c f)\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{l \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{m x}{c^2} \]

[Out]

(m*x)/c^2 - (b*c*(c*e + a*j) - a*b^2*l - 2*a*c*(c*g - a*l) + (2*c^3*e - c^2*(b*g + 2*a*j) - b^3*l + b*c*(b*j +
3*a*l))*x^2)/(2*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (x*(a*b*c*(c*f + a*k) - b^2*(c^2*d + a^2*m) + 2*a*c*
(c^2*d - a*c*h + a^2*m) + (a*b^2*c*k + 2*a*c^2*(c*f - a*k) - a*b^3*m - b*c*(c^2*d + a*c*h - 3*a^2*m))*x^2))/(2
*a*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((a*b^2*c*k - 2*a*c^2*(c*f + 3*a*k) - 3*a*b^3*m + b*c*(c^2*d + a*c
*h + 13*a^2*m) - (a*b^3*c*k - 4*a*b*c^2*(c*f + 2*a*k) - 3*a*b^4*m - b^2*c*(c^2*d - a*c*h - 19*a^2*m) + 4*a*c^2
*(3*c^2*d + a*c*h - 5*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*S
qrt[2]*a*c^(5/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^2*c*k - 2*a*c^2*(c*f + 3*a*k) - 3*a*b^3*m
+ b*c*(c^2*d + a*c*h + 13*a^2*m) + (a*b^3*c*k - 4*a*b*c^2*(c*f + 2*a*k) - 3*a*b^4*m - b^2*c*(c^2*d - a*c*h - 1
9*a^2*m) + 4*a*c^2*(3*c^2*d + a*c*h - 5*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^
2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(5/2)*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((4*c^3*e - c^2*(2*b*g - 4*a*j
) + b^3*l - 6*a*b*c*l)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^2*(b^2 - 4*a*c)^(3/2)) + (l*Log[a + b*x^
2 + c*x^4])/(4*c^2)

________________________________________________________________________________________

Rubi [A]  time = 7.83472, antiderivative size = 770, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1673, 1678, 1676, 1166, 205, 1663, 1660, 634, 618, 206, 628} \[ -\frac{x \left (x^2 \left (-b c \left (-3 a^2 m+a c h+c^2 d\right )+a b^2 c k-a b^3 m+2 a c^2 (c f-a k)\right )+b^2 \left (-\left (a^2 m+c^2 d\right )\right )+2 a c \left (a^2 m-a c h+c^2 d\right )+a b c (a k+c f)\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (-\frac{-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )+a b^3 c k-3 a b^4 m-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )+a b^2 c k-3 a b^3 m-2 a c^2 (3 a k+c f)\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (\frac{-b^2 c \left (-19 a^2 m-a c h+c^2 d\right )+4 a c^2 \left (-5 a^2 m+a c h+3 c^2 d\right )+a b^3 c k-3 a b^4 m-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left (13 a^2 m+a c h+c^2 d\right )+a b^2 c k-3 a b^3 m-2 a c^2 (3 a k+c f)\right )}{2 \sqrt{2} a c^{5/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{x^2 \left (-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right )-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{l \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{m x}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4)^2,x]

[Out]

(m*x)/c^2 - (b*c*(c*e + a*j) - a*b^2*l - 2*a*c*(c*g - a*l) + (2*c^3*e - c^2*(b*g + 2*a*j) - b^3*l + b*c*(b*j +
3*a*l))*x^2)/(2*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (x*(a*b*c*(c*f + a*k) - b^2*(c^2*d + a^2*m) + 2*a*c*
(c^2*d - a*c*h + a^2*m) + (a*b^2*c*k + 2*a*c^2*(c*f - a*k) - a*b^3*m - b*c*(c^2*d + a*c*h - 3*a^2*m))*x^2))/(2
*a*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((a*b^2*c*k - 2*a*c^2*(c*f + 3*a*k) - 3*a*b^3*m + b*c*(c^2*d + a*c
*h + 13*a^2*m) - (a*b^3*c*k - 4*a*b*c^2*(c*f + 2*a*k) - 3*a*b^4*m - b^2*c*(c^2*d - a*c*h - 19*a^2*m) + 4*a*c^2
*(3*c^2*d + a*c*h - 5*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*S
qrt[2]*a*c^(5/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^2*c*k - 2*a*c^2*(c*f + 3*a*k) - 3*a*b^3*m
+ b*c*(c^2*d + a*c*h + 13*a^2*m) + (a*b^3*c*k - 4*a*b*c^2*(c*f + 2*a*k) - 3*a*b^4*m - b^2*c*(c^2*d - a*c*h - 1
9*a^2*m) + 4*a*c^2*(3*c^2*d + a*c*h - 5*a^2*m))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^
2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(5/2)*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((4*c^3*e - c^2*(2*b*g - 4*a*j
) + b^3*l - 6*a*b*c*l)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^2*(b^2 - 4*a*c)^(3/2)) + (l*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 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 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+j x^5+k x^6+l x^7+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx &=\int \frac{x \left (e+g x^2+j x^4+l x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx+\int \frac{d+f x^2+h x^4+k x^6+m x^8}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=-\frac{x \left (a b c (c f+a k)-b^2 \left (c^2 d+a^2 m\right )+2 a c \left (c^2 d-a c h+a^2 m\right )+\left (a b^2 c k+2 a c^2 (c f-a k)-a b^3 m-b c \left (c^2 d+a c h-3 a^2 m\right )\right ) x^2\right )}{2 a c^2 \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+j x^2+l x^3}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )-\frac{\int \frac{-\frac{a b c (c f+a k)+b^2 \left (c^2 d-a^2 m\right )-2 a c \left (3 c^2 d+a c h-a^2 m\right )}{c^2}-\frac{\left (a b^2 c k-2 a c^2 (c f+3 a k)-a b^3 m+b c \left (c^2 d+a c h+5 a^2 m\right )\right ) x^2}{c^2}+2 a \left (4 a-\frac{b^2}{c}\right ) m x^4}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac{b c (c e+a j)-a b^2 l-2 a c (c g-a l)+\left (2 c^3 e-c^2 (b g+2 a j)-b^3 l+b c (b j+3 a l)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{x \left (a b c (c f+a k)-b^2 \left (c^2 d+a^2 m\right )+2 a c \left (c^2 d-a c h+a^2 m\right )+\left (a b^2 c k+2 a c^2 (c f-a k)-a b^3 m-b c \left (c^2 d+a c h-3 a^2 m\right )\right ) x^2\right )}{2 a 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+2 a j-\frac{a b l}{c}+\left (4 a-\frac{b^2}{c}\right ) l x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}-\frac{\int \left (-\frac{2 a \left (b^2-4 a c\right ) m}{c^2}-\frac{a b c (c f+a k)-2 a c \left (3 c^2 d+a c h-5 a^2 m\right )+b^2 \left (c^2 d-3 a^2 m\right )+\left (a b^2 c k-2 a c^2 (c f+3 a k)-3 a b^3 m+b c \left (c^2 d+a c h+13 a^2 m\right )\right ) x^2}{c^2 \left (a+b x^2+c x^4\right )}\right ) \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{m x}{c^2}-\frac{b c (c e+a j)-a b^2 l-2 a c (c g-a l)+\left (2 c^3 e-c^2 (b g+2 a j)-b^3 l+b c (b j+3 a l)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{x \left (a b c (c f+a k)-b^2 \left (c^2 d+a^2 m\right )+2 a c \left (c^2 d-a c h+a^2 m\right )+\left (a b^2 c k+2 a c^2 (c f-a k)-a b^3 m-b c \left (c^2 d+a c h-3 a^2 m\right )\right ) x^2\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\int \frac{a b c (c f+a k)-2 a c \left (3 c^2 d+a c h-5 a^2 m\right )+b^2 \left (c^2 d-3 a^2 m\right )+\left (a b^2 c k-2 a c^2 (c f+3 a k)-3 a b^3 m+b c \left (c^2 d+a c h+13 a^2 m\right )\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a c^2 \left (b^2-4 a c\right )}+\frac{l \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-c^2 (2 b g-4 a j)+b^3 l-6 a b c l\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{m x}{c^2}-\frac{b c (c e+a j)-a b^2 l-2 a c (c g-a l)+\left (2 c^3 e-c^2 (b g+2 a j)-b^3 l+b c (b j+3 a l)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{x \left (a b c (c f+a k)-b^2 \left (c^2 d+a^2 m\right )+2 a c \left (c^2 d-a c h+a^2 m\right )+\left (a b^2 c k+2 a c^2 (c f-a k)-a b^3 m-b c \left (c^2 d+a c h-3 a^2 m\right )\right ) x^2\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{l \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{\left (4 c^3 e-c^2 (2 b g-4 a j)+b^3 l-6 a b c l\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{\left (a b^2 c k-2 a c^2 (c f+3 a k)-3 a b^3 m+b c \left (c^2 d+a c h+13 a^2 m\right )-\frac{a b^3 c k-4 a b c^2 (c f+2 a k)-3 a b^4 m-b^2 c \left (c^2 d-a c h-19 a^2 m\right )+4 a c^2 \left (3 c^2 d+a c h-5 a^2 m\right )}{\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 c^2 \left (b^2-4 a c\right )}+\frac{\left (a b^2 c k-2 a c^2 (c f+3 a k)-3 a b^3 m+b c \left (c^2 d+a c h+13 a^2 m\right )+\frac{a b^3 c k-4 a b c^2 (c f+2 a k)-3 a b^4 m-b^2 c \left (c^2 d-a c h-19 a^2 m\right )+4 a c^2 \left (3 c^2 d+a c h-5 a^2 m\right )}{\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 c^2 \left (b^2-4 a c\right )}\\ &=\frac{m x}{c^2}-\frac{b c (c e+a j)-a b^2 l-2 a c (c g-a l)+\left (2 c^3 e-c^2 (b g+2 a j)-b^3 l+b c (b j+3 a l)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{x \left (a b c (c f+a k)-b^2 \left (c^2 d+a^2 m\right )+2 a c \left (c^2 d-a c h+a^2 m\right )+\left (a b^2 c k+2 a c^2 (c f-a k)-a b^3 m-b c \left (c^2 d+a c h-3 a^2 m\right )\right ) x^2\right )}{2 a c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (a b^2 c k-2 a c^2 (c f+3 a k)-3 a b^3 m+b c \left (c^2 d+a c h+13 a^2 m\right )-\frac{a b^3 c k-4 a b c^2 (c f+2 a k)-3 a b^4 m-b^2 c \left (c^2 d-a c h-19 a^2 m\right )+4 a c^2 \left (3 c^2 d+a c h-5 a^2 m\right )}{\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 c^{5/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (a b^2 c k-2 a c^2 (c f+3 a k)-3 a b^3 m+b c \left (c^2 d+a c h+13 a^2 m\right )+\frac{a b^3 c k-4 a b c^2 (c f+2 a k)-3 a b^4 m-b^2 c \left (c^2 d-a c h-19 a^2 m\right )+4 a c^2 \left (3 c^2 d+a c h-5 a^2 m\right )}{\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 c^{5/2} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left (4 c^3 e-c^2 (2 b g-4 a j)+b^3 l-6 a b c l\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{l \log \left (a+b x^2+c x^4\right )}{4 c^2}\\ \end{align*}

Mathematica [A]  time = 6.56622, size = 935, normalized size = 1.21 \[ \frac{4 \sqrt{c} m x+\frac{2 \sqrt{c} \left (2 c (l+m x) a^3-\left ((l+m x) b^2-c (j+x (k+3 x (l+m x))) b+2 c^2 (g+x (h+x (j+k x)))\right ) a^2+\left (-x^2 (l+m x) b^3+c x^2 (j+k x) b^2+c^2 (e+x (f-x (g+h x))) b+2 c^3 x (d+x (e+f x))\right ) a-b c^2 d x \left (c x^2+b\right )\right )}{a \left (4 a c-b^2\right ) \left (c x^4+b x^2+a\right )}-\frac{\sqrt{2} \left (-3 a m b^4+a \left (c k+3 \sqrt{b^2-4 a c} m\right ) b^3+c \left (-d c^2+a h c+a \left (19 a m-\sqrt{b^2-4 a c} k\right )\right ) b^2-c \left (13 \sqrt{b^2-4 a c} m a^2+c \left (\sqrt{b^2-4 a c} h+8 a k\right ) a+c^2 \left (\sqrt{b^2-4 a c} d+4 a f\right )\right ) b+2 a c^2 \left (-10 m a^2+2 c h a+3 \sqrt{b^2-4 a c} k a+6 c^2 d+c \sqrt{b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (3 a m b^4+a \left (3 \sqrt{b^2-4 a c} m-c k\right ) b^3+c \left (d c^2-a h c-a \left (\sqrt{b^2-4 a c} k+19 a m\right )\right ) b^2-c \left (13 \sqrt{b^2-4 a c} m a^2+c \left (\sqrt{b^2-4 a c} h-8 a k\right ) a+c^2 \left (\sqrt{b^2-4 a c} d-4 a f\right )\right ) b+2 a c^2 \left (10 m a^2-2 c h a+3 \sqrt{b^2-4 a c} k a-6 c^2 d+c \sqrt{b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (-4 e c^3+2 (b g-2 a j) c^2+a \left (6 b l-4 \sqrt{b^2-4 a c} l\right ) c+b^2 \left (\sqrt{b^2-4 a c}-b\right ) l\right ) \log \left (-2 c x^2-b+\sqrt{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\sqrt{c} \left (4 e c^3+(4 a j-2 b g) c^2-2 a \left (3 b+2 \sqrt{b^2-4 a c}\right ) l c+b^2 \left (b+\sqrt{b^2-4 a c}\right ) l\right ) \log \left (2 c x^2+b+\sqrt{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 c^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4)^2,x]

[Out]

(4*Sqrt[c]*m*x + (2*Sqrt[c]*(2*a^3*c*(l + m*x) - b*c^2*d*x*(b + c*x^2) + a*(b^2*c*x^2*(j + k*x) - b^3*x^2*(l +
m*x) + 2*c^3*x*(d + x*(e + f*x)) + b*c^2*(e + x*(f - x*(g + h*x)))) - a^2*(b^2*(l + m*x) + 2*c^2*(g + x*(h +
x*(j + k*x))) - b*c*(j + x*(k + 3*x*(l + m*x))))))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*(-3*a*b^4
*m + 2*a*c^2*(6*c^2*d + c*Sqrt[b^2 - 4*a*c]*f + 2*a*c*h + 3*a*Sqrt[b^2 - 4*a*c]*k - 10*a^2*m) + a*b^3*(c*k + 3
*Sqrt[b^2 - 4*a*c]*m) - b*c*(c^2*(Sqrt[b^2 - 4*a*c]*d + 4*a*f) + a*c*(Sqrt[b^2 - 4*a*c]*h + 8*a*k) + 13*a^2*Sq
rt[b^2 - 4*a*c]*m) + b^2*c*(-(c^2*d) + a*c*h + a*(-(Sqrt[b^2 - 4*a*c]*k) + 19*a*m)))*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]*(3*a*b^4*m + 2*
a*c^2*(-6*c^2*d + c*Sqrt[b^2 - 4*a*c]*f - 2*a*c*h + 3*a*Sqrt[b^2 - 4*a*c]*k + 10*a^2*m) + a*b^3*(-(c*k) + 3*Sq
rt[b^2 - 4*a*c]*m) - b*c*(c^2*(Sqrt[b^2 - 4*a*c]*d - 4*a*f) + a*c*(Sqrt[b^2 - 4*a*c]*h - 8*a*k) + 13*a^2*Sqrt[
b^2 - 4*a*c]*m) + b^2*c*(c^2*d - a*c*h - a*(Sqrt[b^2 - 4*a*c]*k + 19*a*m)))*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[c]*(-4*c^3*e + 2*c^2*(b*g -
2*a*j) + b^2*(-b + Sqrt[b^2 - 4*a*c])*l + a*c*(6*b*l - 4*Sqrt[b^2 - 4*a*c]*l))*Log[-b + Sqrt[b^2 - 4*a*c] - 2
*c*x^2])/(b^2 - 4*a*c)^(3/2) + (Sqrt[c]*(4*c^3*e + c^2*(-2*b*g + 4*a*j) + b^2*(b + Sqrt[b^2 - 4*a*c])*l - 2*a*
c*(3*b + 2*Sqrt[b^2 - 4*a*c])*l)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*c^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.07, size = 4570, normalized size = 5.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((m*x^8+l*x^7+k*x^6+j*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+19/4/c*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^2*m+19/4/c
*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^2*m-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)*arctan(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+m*x/c^2+3*c^2/(4*a*c-b^2)^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*a
rctan(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-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)*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)*j+c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*f+c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*d+1/c/(c*
x^4+b*x^2+a)/(4*a*c-b^2)*a^2*l+c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*e-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b*g-1/(
c*x^4+b*x^2+a)*a/(4*a*c-b^2)*x*h-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3*b*h+1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b*f
-1/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*x^3*k-1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a*j+1/4/c^2/(4*a*c-b^2)^2*ln(2*c*x^2+
(-4*a*c+b^2)^(1/2)+b)*b^4*l+1/4/c^2/(4*a*c-b^2)^2*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*b^4*l+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/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^4*k+1/4/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^4*k+3/2/c/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*x^3*b*m-1/2*c/(c*x^4+b*x^2+a)/a/(4
*a*c-b^2)*x^3*b*d+3/2/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a*b*l+1/2/c/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*x*b*k-1/2/c^
2/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*x*b^2*m+13*a^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*m+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*k-5*a^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)*m-13*a^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*m-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*k-5*a^
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)*m-3/4/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^5*m-3/2/c*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)*b*l
+6*c*a^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))*k+3/2/c*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)*b*l-6*c*a^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))*k+3/4/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^5*m+
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/(c*x^4+b*x^2+a)/(4*a*c-b^2)*a*g+1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b*e+4*a^2/(4*a*c-b^2)^2*ln(-2*
c*x^2+(-4*a*c+b^2)^(1/2)-b)*l+4*a^2/(4*a*c-b^2)^2*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*l-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*k
-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*k-3/4/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)*b^4*m-25/4/c*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^3*m+25/4/c*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^3*m+1/4/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^
3*k-3/4/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)*b^4*m+1/4/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))*(-4*a*c+b^2)^(1/2)*b^3*k-1/2/(c*x^4+b*x^2+a)/a/(4*a*c-b^2)*x*b^2*d+1/c
/(c*x^4+b*x^2+a)*a^2/(4*a*c-b^2)*x*m+1/2/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*a*b*j+1/2/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)
*x^3*b^2*k-2/c*a/(4*a*c-b^2)^2*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*b^2*l-2/c*a/(4*a*c-b^2)^2*ln(2*c*x^2+(-4*a*c+
b^2)^(1/2)+b)*b^2*l+1/4/c^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^3*l-1/4/c^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^3*l-1/2/c^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^3
*b^3*m-1/2/c^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*a*b^2*l-1/2/c^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b^3*l+1/2/c/(c*x^4+
b*x^2+a)/(4*a*c-b^2)*x^2*b^2*j-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

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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 j -{\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 )} k +{\left (a b^{3} - 3 \, a^{2} b c\right )} m\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 )} j -{\left (a b^{3} - 3 \, a^{2} b c\right )} l\right )} x^{2} -{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} l +{\left (a b c^{2} f - 2 \, a^{2} c^{2} h + a^{2} b c k -{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d -{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} m\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{m x}{c^{2}} - \frac{-\int \frac{a b c^{2} f - 2 \, a^{2} c^{2} h + a^{2} b c k + 2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} l x^{3} +{\left (b c^{3} d - 2 \, a c^{3} f + a b c^{2} h +{\left (a b^{2} c - 6 \, a^{2} c^{2}\right )} k -{\left (3 \, a b^{3} - 13 \, a^{2} b c\right )} m\right )} x^{2} +{\left (b^{2} c^{2} - 6 \, a c^{3}\right )} d -{\left (3 \, a^{2} b^{2} - 10 \, a^{3} c\right )} m - 2 \,{\left (2 \, a c^{3} e - a b c^{2} g + 2 \, a^{2} c^{2} j - a^{2} b c l\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((m*x^8+l*x^7+k*x^6+j*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*j - (b*c^3*d - 2*a*c^3*f + a*b*c^2*h - (a*b^2*c - 2*a^2*c^2)*k + (a*b^
3 - 3*a^2*b*c)*m)*x^3 + (2*a*c^3*e - a*b*c^2*g + (a*b^2*c - 2*a^2*c^2)*j - (a*b^3 - 3*a^2*b*c)*l)*x^2 - (a^2*b
^2 - 2*a^3*c)*l + (a*b*c^2*f - 2*a^2*c^2*h + a^2*b*c*k - (b^2*c^2 - 2*a*c^3)*d - (a^2*b^2 - 2*a^3*c)*m)*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) + m*x/c^2 - 1/2*integrate
(-(a*b*c^2*f - 2*a^2*c^2*h + a^2*b*c*k + 2*(a*b^2*c - 4*a^2*c^2)*l*x^3 + (b*c^3*d - 2*a*c^3*f + a*b*c^2*h + (a
*b^2*c - 6*a^2*c^2)*k - (3*a*b^3 - 13*a^2*b*c)*m)*x^2 + (b^2*c^2 - 6*a*c^3)*d - (3*a^2*b^2 - 10*a^3*c)*m - 2*(
2*a*c^3*e - a*b*c^2*g + 2*a^2*c^2*j - a^2*b*c*l)*x)/(c*x^4 + b*x^2 + a), x)/(a*b^2*c^2 - 4*a^2*c^3)

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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((m*x^8+l*x^7+k*x^6+j*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((m*x**8+l*x**7+k*x**6+j*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((m*x^8+l*x^7+k*x^6+j*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