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

Optimal. Leaf size=728 \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )+8 a^2 b c f+4 a^2 c (a h+7 c d)-a b^2 (7 a h+25 c d)+a b^3 f+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)-6 a b^2 (5 c d-3 a h)+a b^3 f+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)-6 a b^2 (5 c d-3 a h)+a b^3 f+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x^2 \left (-\left (-2 a c i+b^2 i-b c g+2 c^2 e\right )\right )-b (a i+c e)+2 a c g}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a c i+b^2 i-3 b c g+6 c^2 e\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\left (b+2 c x^2\right ) \left (2 a i+\frac{b^2 i}{c}-3 b g+6 c e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]

[Out]

(x*(b^2*d - a*b*f - 2*a*(c*d - a*h) + (b*c*d - 2*a*c*f + a*b*h)*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2
) + (2*a*c*g - b*(c*e + a*i) - (2*c^2*e - b*c*g + b^2*i - 2*a*c*i)*x^2)/(4*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)
^2) + ((6*c*e - 3*b*g + 2*a*i + (b^2*i)/c)*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (x*(3*b^4*
d + a*b^3*f + 8*a^2*b*c*f + 4*a^2*c*(7*c*d + a*h) - a*b^2*(25*c*d + 7*a*h) + c*(3*b^3*d + a*b^2*f + 20*a^2*c*f
- 12*a*b*(2*c*d + a*h))*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^3*d + a*b^2*f + 20*
a^2*c*f - 12*a*b*(2*c*d + a*h) + (3*b^4*d + a*b^3*f - 52*a^2*b*c*f - 6*a*b^2*(5*c*d - 3*a*h) + 24*a^2*c*(7*c*d
+ a*h))/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*(b^2 - 4*a
*c)^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(3*b^3*d + a*b^2*f + 20*a^2*c*f - 12*a*b*(2*c*d + a*h) - (3*b^4*
d + a*b^3*f - 52*a^2*b*c*f - 6*a*b^2*(5*c*d - 3*a*h) + 24*a^2*c*(7*c*d + a*h))/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*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - ((6
*c^2*e - 3*b*c*g + b^2*i + 2*a*c*i)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 2.73276, antiderivative size = 728, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {1673, 1678, 1178, 1166, 205, 1663, 1660, 12, 614, 618, 206} \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )+8 a^2 b c f+4 a^2 c (a h+7 c d)-a b^2 (7 a h+25 c d)+a b^3 f+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)-6 a b^2 (5 c d-3 a h)+a b^3 f+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)-6 a b^2 (5 c d-3 a h)+a b^3 f+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x^2 \left (-\left (-2 a c i+b^2 i-b c g+2 c^2 e\right )\right )-b (a i+c e)+2 a c g}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a c i+b^2 i-3 b c g+6 c^2 e\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\left (b+2 c x^2\right ) \left (2 a i+\frac{b^2 i}{c}-3 b g+6 c e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b^2*d - a*b*f - 2*a*(c*d - a*h) + (b*c*d - 2*a*c*f + a*b*h)*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2
) + (2*a*c*g - b*(c*e + a*i) - (2*c^2*e - b*c*g + b^2*i - 2*a*c*i)*x^2)/(4*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)
^2) + ((6*c*e - 3*b*g + 2*a*i + (b^2*i)/c)*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (x*(3*b^4*
d + a*b^3*f + 8*a^2*b*c*f + 4*a^2*c*(7*c*d + a*h) - a*b^2*(25*c*d + 7*a*h) + c*(3*b^3*d + a*b^2*f + 20*a^2*c*f
- 12*a*b*(2*c*d + a*h))*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^3*d + a*b^2*f + 20*
a^2*c*f - 12*a*b*(2*c*d + a*h) + (3*b^4*d + a*b^3*f - 52*a^2*b*c*f - 6*a*b^2*(5*c*d - 3*a*h) + 24*a^2*c*(7*c*d
+ a*h))/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*(b^2 - 4*a
*c)^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(3*b^3*d + a*b^2*f + 20*a^2*c*f - 12*a*b*(2*c*d + a*h) - (3*b^4*
d + a*b^3*f - 52*a^2*b*c*f - 6*a*b^2*(5*c*d - 3*a*h) + 24*a^2*c*(7*c*d + a*h))/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*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - ((6
*c^2*e - 3*b*c*g + b^2*i + 2*a*c*i)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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])

Rubi steps

\begin{align*} \int \frac{d+e x+f x^2+g x^3+h x^4+56 x^5}{\left (a+b x^2+c x^4\right )^3} \, dx &=\int \frac{x \left (e+g x^2+56 x^4\right )}{\left (a+b x^2+c x^4\right )^3} \, dx+\int \frac{d+f x^2+h x^4}{\left (a+b x^2+c x^4\right )^3} \, dx\\ &=\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{4 a \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+56 x^2}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )-\frac{\int \frac{-3 b^2 d-a b f+2 a (7 c d+a h)-5 (b c d-2 a c f+a b h) x^2}{\left (a+b x^2+c x^4\right )^2} \, dx}{4 a \left (b^2-4 a c\right )}\\ &=-\frac{56 a b+b c e-2 a c g+\left (56 b^2-2 c (56 a-c e)-b c g\right ) x^2}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (3 b^4 d+a b^3 f+8 a^2 b c f+4 a^2 c (7 c d+a h)-a b^2 (25 c d+7 a h)+c \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)\right ) x^2\right )}{8 a^2 \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-3 a b^2 (9 c d-a h)+12 a^2 c (7 c d+a h)+c \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)\right ) x^2}{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{112 a+\frac{56 b^2}{c}+6 c e-3 b g}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=-\frac{56 a b+b c e-2 a c g+\left (56 b^2-2 c (56 a-c e)-b c g\right ) x^2}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (3 b^4 d+a b^3 f+8 a^2 b c f+4 a^2 c (7 c d+a h)-a b^2 (25 c d+7 a h)+c \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (112 a+\frac{56 b^2}{c}+6 c e-3 b g\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}+\frac{\left (c \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)-\frac{3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\sqrt{b^2-4 a c}}\right )\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 (c \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)+\frac{3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\sqrt{b^2-4 a c}}\right )\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{56 a b+b c e-2 a c g+\left (56 b^2-2 c (56 a-c e)-b c g\right ) x^2}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\left (112 a+\frac{56 b^2}{c}+6 c e-3 b g\right ) \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x \left (3 b^4 d+a b^3 f+8 a^2 b c f+4 a^2 c (7 c d+a h)-a b^2 (25 c d+7 a h)+c \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)+\frac{3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\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 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)-\frac{3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\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 \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left (56 b^2+112 a c+6 c^2 e-3 b c g\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )^2}\\ &=-\frac{56 a b+b c e-2 a c g+\left (56 b^2-2 c (56 a-c e)-b c g\right ) x^2}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\left (112 a+\frac{56 b^2}{c}+6 c e-3 b g\right ) \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x \left (3 b^4 d+a b^3 f+8 a^2 b c f+4 a^2 c (7 c d+a h)-a b^2 (25 c d+7 a h)+c \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)+\frac{3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\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 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)-\frac{3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\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 \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left (56 b^2+112 a c+6 c^2 e-3 b c g\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{56 a b+b c e-2 a c g+\left (56 b^2-2 c (56 a-c e)-b c g\right ) x^2}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\left (112 a+\frac{56 b^2}{c}+6 c e-3 b g\right ) \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x \left (3 b^4 d+a b^3 f+8 a^2 b c f+4 a^2 c (7 c d+a h)-a b^2 (25 c d+7 a h)+c \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)+\frac{3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\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 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (3 b^3 d+a b^2 f+20 a^2 c f-12 a b (2 c d+a h)-\frac{3 b^4 d+a b^3 f-52 a^2 b c f-6 a b^2 (5 c d-3 a h)+24 a^2 c (7 c d+a h)}{\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 \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left (56 b^2+112 a c+6 c^2 e-3 b c g\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 6.85054, size = 980, normalized size = 1.35 \[ \frac{-b c^2 d x^3+2 a c^2 f x^3-a b c h x^3+2 a c^2 e x^2-a b c g x^2+a b^2 i x^2-2 a^2 c i x^2+2 a c^2 d x-b^2 c d x+a b c f x-2 a^2 c h x+a b c e-2 a^2 c g+a^2 b i}{4 a c \left (4 a c-b^2\right ) \left (c x^4+b x^2+a\right )^2}+\frac{\sqrt{c} \left (3 d b^4+3 \sqrt{b^2-4 a c} d b^3+a f b^3-30 a c d b^2+a \sqrt{b^2-4 a c} f b^2+18 a^2 h b^2-24 a c \sqrt{b^2-4 a c} d b-52 a^2 c f b-12 a^2 \sqrt{b^2-4 a c} h b+168 a^2 c^2 d+20 a^2 c \sqrt{b^2-4 a c} f+24 a^3 c h\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (-3 d b^4+3 \sqrt{b^2-4 a c} d b^3-a f b^3+30 a c d b^2+a \sqrt{b^2-4 a c} f b^2-18 a^2 h b^2-24 a c \sqrt{b^2-4 a c} d b+52 a^2 c f b-12 a^2 \sqrt{b^2-4 a c} h b-168 a^2 c^2 d+20 a^2 c \sqrt{b^2-4 a c} f-24 a^3 c h\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left (i b^2-3 c g b+6 c^2 e+2 a c i\right ) \log \left (-2 c x^2-b+\sqrt{b^2-4 a c}\right )}{2 \left (b^2-4 a c\right )^{5/2}}+\frac{\left (-i b^2+3 c g b-6 c^2 e-2 a c i\right ) \log \left (2 c x^2+b+\sqrt{b^2-4 a c}\right )}{2 \left (b^2-4 a c\right )^{5/2}}+\frac{3 c d x b^4+3 c^2 d x^3 b^3+2 a^2 i b^3+a c f x b^3+a c^2 f x^3 b^2+4 a^2 c i x^2 b^2-6 a^2 c g b^2-25 a c^2 d x b^2-7 a^2 c h x b^2-24 a c^3 d x^3 b-12 a^2 c^2 h x^3 b-12 a^2 c^2 g x^2 b+12 a^2 c^2 e b+4 a^3 c i b+8 a^2 c^2 f x b+20 a^2 c^3 f x^3+24 a^2 c^3 e x^2+8 a^3 c^2 i x^2+28 a^2 c^3 d x+4 a^3 c^2 h x}{8 a^2 c \left (4 a c-b^2\right )^2 \left (c x^4+b x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*c*e - 2*a^2*c*g + a^2*b*i - b^2*c*d*x + 2*a*c^2*d*x + a*b*c*f*x - 2*a^2*c*h*x + 2*a*c^2*e*x^2 - a*b*c*g*x
^2 + a*b^2*i*x^2 - 2*a^2*c*i*x^2 - b*c^2*d*x^3 + 2*a*c^2*f*x^3 - a*b*c*h*x^3)/(4*a*c*(-b^2 + 4*a*c)*(a + b*x^2
+ c*x^4)^2) + (12*a^2*b*c^2*e - 6*a^2*b^2*c*g + 2*a^2*b^3*i + 4*a^3*b*c*i + 3*b^4*c*d*x - 25*a*b^2*c^2*d*x +
28*a^2*c^3*d*x + a*b^3*c*f*x + 8*a^2*b*c^2*f*x - 7*a^2*b^2*c*h*x + 4*a^3*c^2*h*x + 24*a^2*c^3*e*x^2 - 12*a^2*b
*c^2*g*x^2 + 4*a^2*b^2*c*i*x^2 + 8*a^3*c^2*i*x^2 + 3*b^3*c^2*d*x^3 - 24*a*b*c^3*d*x^3 + a*b^2*c^2*f*x^3 + 20*a
^2*c^3*f*x^3 - 12*a^2*b*c^2*h*x^3)/(8*a^2*c*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^4*d - 30*a*b
^2*c*d + 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4*a*c]*d + a*b^3*f - 52*a^2*b*c*f + a
*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 - 4*a*c]*f + 18*a^2*b^2*h + 24*a^3*c*h - 12*a^2*b*Sqrt[b^2 - 4*a*
c]*h)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqr
t[b^2 - 4*a*c]]) + (Sqrt[c]*(-3*b^4*d + 30*a*b^2*c*d - 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sq
rt[b^2 - 4*a*c]*d - a*b^3*f + 52*a^2*b*c*f + a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 - 4*a*c]*f - 18*a^2
*b^2*h - 24*a^3*c*h - 12*a^2*b*Sqrt[b^2 - 4*a*c]*h)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(
8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((6*c^2*e - 3*b*c*g + b^2*i + 2*a*c*i)*Log[-b
+ Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2)) + ((-6*c^2*e + 3*b*c*g - b^2*i - 2*a*c*i)*Log[b + Sqr
t[b^2 - 4*a*c] + 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.046, size = 3824, normalized size = 5.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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]

(-1/8*c^2*(12*a^2*b*h-20*a^2*c*f-a*b^2*f+24*a*b*c*d-3*b^3*d)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7+1/2*c*(2*a*c*i
+b^2*i-3*b*c*g+6*c^2*e)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/8/a^2*c*(4*a^3*c*h-19*a^2*b^2*h+28*a^2*b*c*f+28*a^2*c
^2*d+2*a*b^3*f-49*a*b^2*c*d+6*b^4*d)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+3/4*b*(2*a*c*i+b^2*i-3*b*c*g+6*c^2*e)/(16*
a^2*c^2-8*a*b^2*c+b^4)*x^4-1/8*(16*a^3*b*c*h-36*a^3*c^2*f+5*a^2*b^3*h-5*a^2*b^2*c*f+4*a^2*b*c^2*d-a*b^4*f+20*a
*b^3*c*d-3*b^5*d)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*(2*a^2*c*i-5*a*b^2*i+5*a*b*c*g-10*a*c^2*e+b^3*g-2*b^2
*c*e)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-1/8*(12*a^3*c*h+3*a^2*b^2*h-16*a^2*b*c*f-44*a^2*c^2*d+a*b^3*f+37*a*b^2*c*
d-5*b^4*d)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x+1/4*(6*a^2*b*i-8*a^2*c*g-a*b^2*g+10*a*b*c*e-b^3*e)/(16*a^2*c^2-8*a*b
^2*c+b^4))/(c*x^4+b*x^2+a)^2-12*a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b^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*h+9/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c
-4*b^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*d+12
*a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b^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*h-9/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b^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*d+6*a/(16*a^2*c^2-8*a*b^2*c+b^4)*
c^2/(16*a*c-4*b^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)*h-1/4/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^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*f-13/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b
^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-13/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b^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*f+9/2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4
*b^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+9/2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctan
h(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/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(1
6*a*c-4*b^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*f+3/4/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^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*d-3/4/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^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*d+6*a/(16*a^2*c^2-8*a*b
^2*c+b^4)*c^2/(16*a*c-4*b^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)*h+1/4/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^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*f+1/4/a/(16*a^2*c^2
-8*a*b^2*c+b^4)*c/(16*a*c-4*b^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*f+3/4/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^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*d+3/4/a^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^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*d-15/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b^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*d-15/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b^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*d+2/(16*a^2*c^2-8*a*b^2*c+b^4)/(16*a*c-4*
b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(-4*a*c+b^2)^(1/2)*b^2*i-2/(16*a^2*c^2-8*a*b^2*c+b^4)/(16*a*c-4*b^2)*ln(
-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(-4*a*c+b^2)^(1/2)*b^2*i+6/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^2)*ln(-2*c*
x^2+(-4*a*c+b^2)^(1/2)-b)*(-4*a*c+b^2)^(1/2)*b*g-6/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^2)*ln(2*c*x^2+(-4*
a*c+b^2)^(1/2)+b)*(-4*a*c+b^2)^(1/2)*b*g+4*a/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^2)*ln(2*c*x^2+(-4*a*c+b^
2)^(1/2)+b)*(-4*a*c+b^2)^(1/2)*i-4*a/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^2)*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2
)-b)*(-4*a*c+b^2)^(1/2)*i+42/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(16*a*c-4*b^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-4/(16*a^2*c^2-8*a*b^2*c+b^4)*c
^2/(16*a*c-4*b^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-24/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(16*a*c-4*b^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+4/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b^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+24/(16*a^2*c^2-8*a*b^2*c+b
^4)*c^3/(16*a*c-4*b^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+42/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(16*a*c-4*b^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+3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^
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-3/(16*a
^2*c^2-8*a*b^2*c+b^4)*c/(16*a*c-4*b^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+20*a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(16*a*c-4*b^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))*f-20*a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(16*a*c
-4*b^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))*f-12/(1
6*a^2*c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b^2)*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(-4*a*c+b^2)^(1/2)*e+12/(16*a^2*
c^2-8*a*b^2*c+b^4)*c^2/(16*a*c-4*b^2)*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(-4*a*c+b^2)^(1/2)*e

________________________________________________________________________________________

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((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((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((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 [B]  time = 29.7546, size = 16458, normalized size = 22.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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]

1/64*(32*a^2*b^7 - 192*a^3*b^5*c + 128*a^2*b^6*c - 256*a^3*b^4*c^2 + 128*a^2*b^5*c^2 + 1024*a^5*b*c^3 - 1024*a
^4*b^2*c^3 + 256*a^3*b^3*c^3 - 16*a^2*b^6 + 96*a^3*b^4*c - 128*a^2*b^5*c + 256*a^3*b^3*c^2 - 192*a^2*b^4*c^2 -
512*a^5*c^3 + 1024*a^4*b*c^3 - 384*a^3*b^2*c^3 + 32*a^2*b^4*c - 64*a^3*b^2*c^2 + 96*a^2*b^3*c^2 - 256*a^4*c^3
+ 192*a^3*b*c^3 - 16*a^2*b^2*c^2 - 32*a^3*c^3 + 3*(2*b^8 - 36*a*b^6*c + 8*b^7*c + 304*a^2*b^4*c^2 - 112*a*b^5
*c^2 + 8*b^6*c^2 - 1216*a^3*b^2*c^3 + 768*a^2*b^3*c^3 - 80*a*b^4*c^3 + 1792*a^4*c^4 - 1792*a^3*b*c^4 + 448*a^2
*b^2*c^4 + b^7 - 16*a*b^5*c + 80*a^2*b^3*c^2 + 8*a*b^4*c^2 - 4*b^5*c^2 - 128*a^3*b*c^3 - 256*a^2*b^2*c^3 + 48*
a*b^3*c^3 + 896*a^3*c^4 - 448*a^2*b*c^4 - 2*b^5*c + 24*a*b^3*c^2 - 2*b^4*c^2 - 64*a^2*b*c^3 + 12*a*b^2*c^3 + 1
12*a^2*c^4 + b^3*c^2 - 8*a*b*c^3)*sqrt(2*b*c - c)*d + (2*a*b^7 - 120*a^2*b^5*c + 8*a*b^6*c + 864*a^3*b^3*c^2 -
448*a^2*b^4*c^2 + 8*a*b^5*c^2 - 1664*a^4*b*c^3 + 1664*a^3*b^2*c^3 - 416*a^2*b^3*c^3 + a*b^6 + 12*a^2*b^4*c -
144*a^3*b^2*c^2 + 288*a^2*b^3*c^2 - 4*a*b^4*c^2 + 320*a^4*c^3 - 1152*a^3*b*c^3 + 496*a^2*b^2*c^3 - 2*a*b^4*c -
32*a^2*b^2*c^2 - 2*a*b^3*c^2 + 160*a^3*c^3 - 184*a^2*b*c^3 + a*b^2*c^2 + 20*a^2*c^3)*sqrt(2*b*c - c)*f + 12*(
3*a^2*b^6 - 20*a^3*b^4*c + 12*a^2*b^5*c + 16*a^4*b^2*c^2 - 32*a^3*b^3*c^2 + 12*a^2*b^4*c^2 + 64*a^5*c^3 - 64*a
^4*b*c^3 + 16*a^3*b^2*c^3 - a^2*b^5 + 8*a^3*b^3*c - 10*a^2*b^4*c - 16*a^4*b*c^2 + 32*a^3*b^2*c^2 - 16*a^2*b^3*
c^2 + 32*a^4*c^3 - 16*a^3*b*c^3 + 2*a^2*b^3*c - 8*a^3*b*c^2 + 7*a^2*b^2*c^2 + 4*a^3*c^3 - a^2*b*c^2)*sqrt(2*b*
c - c)*h + 48*(2*a^2*b^6*c*i - 16*a^3*b^4*c^2*i + 8*a^2*b^5*c^2*i + 32*a^4*b^2*c^3*i - 32*a^3*b^3*c^3*i + 8*a^
2*b^4*c^3*i - a^2*b^5*c*i + 8*a^3*b^3*c^2*i - 8*a^2*b^4*c^2*i - 16*a^4*b*c^3*i + 32*a^3*b^2*c^3*i - 12*a^2*b^3
*c^3*i + 2*a^2*b^3*c^2*i - 8*a^3*b*c^3*i + 6*a^2*b^2*c^3*i - a^2*b*c^3*i)*g - 96*(2*a^2*b^5*c^2*i - 16*a^3*b^3
*c^3*i + 8*a^2*b^4*c^3*i + 32*a^4*b*c^4*i - 32*a^3*b^2*c^4*i + 8*a^2*b^3*c^4*i - a^2*b^4*c^2*i + 8*a^3*b^2*c^3
*i - 8*a^2*b^3*c^3*i - 16*a^4*c^4*i + 32*a^3*b*c^4*i - 12*a^2*b^2*c^4*i + 2*a^2*b^2*c^3*i - 8*a^3*c^4*i + 6*a^
2*b*c^4*i - a^2*c^4*i)*e)*log(x + 1/4*sqrt(-(8*a^2*b^5*i - 64*a^3*b^3*c*i + 128*a^4*b*c^2*i + sqrt(-64*(a^2*b^
5 - 8*a^3*b^3*c + 16*a^4*b*c^2)^2 + 256*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a
^4*c^3)))/(a^2*b^4*c*i - 8*a^3*b^2*c^2*i + 16*a^4*c^3*i)))/(a^2*b^10*i - 20*a^3*b^8*c*i + 4*a^2*b^9*c*i + 160*
a^4*b^6*c^2*i - 64*a^3*b^7*c^2*i + 4*a^2*b^8*c^2*i - 640*a^5*b^4*c^3*i + 384*a^4*b^5*c^3*i - 48*a^3*b^6*c^3*i
+ 1280*a^6*b^2*c^4*i - 1024*a^5*b^3*c^4*i + 192*a^4*b^4*c^4*i - 1024*a^7*c^5*i + 1024*a^6*b*c^5*i - 256*a^5*b^
2*c^5*i - 2*a^2*b^8*c*i + 32*a^3*b^6*c^2*i - 4*a^2*b^7*c^2*i - 192*a^4*b^4*c^3*i + 48*a^3*b^5*c^3*i + 512*a^5*
b^2*c^4*i - 192*a^4*b^3*c^4*i - 512*a^6*c^5*i + 256*a^5*b*c^5*i + a^2*b^6*c^2*i - 12*a^3*b^4*c^3*i + 48*a^4*b^
2*c^4*i - 64*a^5*c^5*i + sqrt(-b^18 + 36*a*b^16*c - 8*b^17*c - 576*a^2*b^14*c^2 + 256*a*b^15*c^2 - 24*b^16*c^2
+ 5376*a^3*b^12*c^3 - 3584*a^2*b^13*c^3 + 672*a*b^14*c^3 - 32*b^15*c^3 - 32256*a^4*b^10*c^4 + 28672*a^3*b^11*
c^4 - 8064*a^2*b^12*c^4 + 768*a*b^13*c^4 - 16*b^14*c^4 + 129024*a^5*b^8*c^5 - 143360*a^4*b^9*c^5 + 53760*a^3*b
^10*c^5 - 7680*a^2*b^11*c^5 + 320*a*b^12*c^5 - 344064*a^6*b^6*c^6 + 458752*a^5*b^7*c^6 - 215040*a^4*b^8*c^6 +
40960*a^3*b^9*c^6 - 2560*a^2*b^10*c^6 + 589824*a^7*b^4*c^7 - 917504*a^6*b^5*c^7 - 1125899906064384*a^5*b^6*c^7
- 122880*a^4*b^7*c^7 + 10240*a^3*b^8*c^7 - 589824*a^8*b^2*c^8 + 1048576*a^7*b^3*c^8 - 688128*a^6*b^4*c^8 + 19
6608*a^5*b^5*c^8 - 20480*a^4*b^6*c^8 + 262144*a^9*c^9 - 524288*a^8*b*c^9 + 393216*a^7*b^2*c^9 - 131072*a^6*b^3
*c^9 + 16384*a^5*b^4*c^9 + 4*b^16*c - 128*a*b^14*c^2 + 24*b^15*c^2 + 1792*a^2*b^12*c^3 - 672*a*b^13*c^3 + 48*b
^14*c^3 - 14336*a^3*b^10*c^4 + 8064*a^2*b^11*c^4 - 1152*a*b^12*c^4 + 32*b^13*c^4 + 71680*a^4*b^8*c^5 - 53760*a
^3*b^9*c^5 + 11520*a^2*b^10*c^5 - 640*a*b^11*c^5 - 229376*a^5*b^6*c^6 + 215040*a^4*b^7*c^6 - 61440*a^3*b^8*c^6
+ 5120*a^2*b^9*c^6 + 458752*a^6*b^4*c^7 - 516096*a^5*b^5*c^7 + 184320*a^4*b^6*c^7 - 20480*a^3*b^7*c^7 - 52428
8*a^7*b^2*c^8 + 688128*a^6*b^3*c^8 - 294912*a^5*b^4*c^8 + 40960*a^4*b^5*c^8 + 262144*a^8*c^9 - 393216*a^7*b*c^
9 + 196608*a^6*b^2*c^9 - 32768*a^5*b^3*c^9 - 6*b^14*c^2 + 168*a*b^12*c^3 - 24*b^13*c^3 - 2016*a^2*b^10*c^4 + 5
76*a*b^11*c^4 - 24*b^12*c^4 + 13440*a^3*b^8*c^5 - 5760*a^2*b^9*c^5 + 480*a*b^10*c^5 - 53760*a^4*b^6*c^6 + 3072
0*a^3*b^7*c^6 - 3840*a^2*b^8*c^6 + 129024*a^5*b^4*c^7 - 92160*a^4*b^5*c^7 + 15360*a^3*b^6*c^7 - 172032*a^6*b^2
*c^8 + 147456*a^5*b^3*c^8 - 30720*a^4*b^4*c^8 + 98304*a^7*c^9 - 98304*a^6*b*c^9 + 24576*a^5*b^2*c^9 + 4*b^12*c
^3 - 96*a*b^10*c^4 + 8*b^11*c^4 + 960*a^2*b^8*c^5 - 160*a*b^9*c^5 - 5120*a^3*b^6*c^6 + 1280*a^2*b^7*c^6 + 1536
0*a^4*b^4*c^7 - 5120*a^3*b^5*c^7 - 24576*a^5*b^2*c^8 + 10240*a^4*b^3*c^8 + 16384*a^6*c^9 - 8192*a^5*b*c^9 - b^
10*c^4 + 20*a*b^8*c^5 - 160*a^2*b^6*c^6 + 640*a^3*b^4*c^7 - 1280*a^4*b^2*c^8 + 1024*a^5*c^9)*a^2*b) + 1/64*(32
*a^2*b^7 - 192*a^3*b^5*c + 128*a^2*b^6*c - 256*a^3*b^4*c^2 + 128*a^2*b^5*c^2 + 1024*a^5*b*c^3 - 1024*a^4*b^2*c
^3 + 256*a^3*b^3*c^3 - 16*a^2*b^6 + 96*a^3*b^4*c - 128*a^2*b^5*c + 256*a^3*b^3*c^2 - 192*a^2*b^4*c^2 - 512*a^5
*c^3 + 1024*a^4*b*c^3 - 384*a^3*b^2*c^3 + 32*a^2*b^4*c - 64*a^3*b^2*c^2 + 96*a^2*b^3*c^2 - 256*a^4*c^3 + 192*a
^3*b*c^3 - 16*a^2*b^2*c^2 - 32*a^3*c^3 - 3*(2*b^8 - 36*a*b^6*c + 8*b^7*c + 304*a^2*b^4*c^2 - 112*a*b^5*c^2 + 8
*b^6*c^2 - 1216*a^3*b^2*c^3 + 768*a^2*b^3*c^3 - 80*a*b^4*c^3 + 1792*a^4*c^4 - 1792*a^3*b*c^4 + 448*a^2*b^2*c^4
+ b^7 - 16*a*b^5*c + 80*a^2*b^3*c^2 + 8*a*b^4*c^2 - 4*b^5*c^2 - 128*a^3*b*c^3 - 256*a^2*b^2*c^3 + 48*a*b^3*c^
3 + 896*a^3*c^4 - 448*a^2*b*c^4 - 2*b^5*c + 24*a*b^3*c^2 - 2*b^4*c^2 - 64*a^2*b*c^3 + 12*a*b^2*c^3 + 112*a^2*c
^4 + b^3*c^2 - 8*a*b*c^3)*sqrt(2*b*c - c)*d - (2*a*b^7 - 120*a^2*b^5*c + 8*a*b^6*c + 864*a^3*b^3*c^2 - 448*a^2
*b^4*c^2 + 8*a*b^5*c^2 - 1664*a^4*b*c^3 + 1664*a^3*b^2*c^3 - 416*a^2*b^3*c^3 + a*b^6 + 12*a^2*b^4*c - 144*a^3*
b^2*c^2 + 288*a^2*b^3*c^2 - 4*a*b^4*c^2 + 320*a^4*c^3 - 1152*a^3*b*c^3 + 496*a^2*b^2*c^3 - 2*a*b^4*c - 32*a^2*
b^2*c^2 - 2*a*b^3*c^2 + 160*a^3*c^3 - 184*a^2*b*c^3 + a*b^2*c^2 + 20*a^2*c^3)*sqrt(2*b*c - c)*f - 12*(3*a^2*b^
6 - 20*a^3*b^4*c + 12*a^2*b^5*c + 16*a^4*b^2*c^2 - 32*a^3*b^3*c^2 + 12*a^2*b^4*c^2 + 64*a^5*c^3 - 64*a^4*b*c^3
+ 16*a^3*b^2*c^3 - a^2*b^5 + 8*a^3*b^3*c - 10*a^2*b^4*c - 16*a^4*b*c^2 + 32*a^3*b^2*c^2 - 16*a^2*b^3*c^2 + 32
*a^4*c^3 - 16*a^3*b*c^3 + 2*a^2*b^3*c - 8*a^3*b*c^2 + 7*a^2*b^2*c^2 + 4*a^3*c^3 - a^2*b*c^2)*sqrt(2*b*c - c)*h
+ 48*(2*a^2*b^6*c*i - 16*a^3*b^4*c^2*i + 8*a^2*b^5*c^2*i + 32*a^4*b^2*c^3*i - 32*a^3*b^3*c^3*i + 8*a^2*b^4*c^
3*i - a^2*b^5*c*i + 8*a^3*b^3*c^2*i - 8*a^2*b^4*c^2*i - 16*a^4*b*c^3*i + 32*a^3*b^2*c^3*i - 12*a^2*b^3*c^3*i +
2*a^2*b^3*c^2*i - 8*a^3*b*c^3*i + 6*a^2*b^2*c^3*i - a^2*b*c^3*i)*g - 96*(2*a^2*b^5*c^2*i - 16*a^3*b^3*c^3*i +
8*a^2*b^4*c^3*i + 32*a^4*b*c^4*i - 32*a^3*b^2*c^4*i + 8*a^2*b^3*c^4*i - a^2*b^4*c^2*i + 8*a^3*b^2*c^3*i - 8*a
^2*b^3*c^3*i - 16*a^4*c^4*i + 32*a^3*b*c^4*i - 12*a^2*b^2*c^4*i + 2*a^2*b^2*c^3*i - 8*a^3*c^4*i + 6*a^2*b*c^4*
i - a^2*c^4*i)*e)*log(x - 1/4*sqrt(-(8*a^2*b^5*i - 64*a^3*b^3*c*i + 128*a^4*b*c^2*i + sqrt(-64*(a^2*b^5 - 8*a^
3*b^3*c + 16*a^4*b*c^2)^2 + 256*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3))
)/(a^2*b^4*c*i - 8*a^3*b^2*c^2*i + 16*a^4*c^3*i)))/(a^2*b^10*i - 20*a^3*b^8*c*i + 4*a^2*b^9*c*i + 160*a^4*b^6*
c^2*i - 64*a^3*b^7*c^2*i + 4*a^2*b^8*c^2*i - 640*a^5*b^4*c^3*i + 384*a^4*b^5*c^3*i - 48*a^3*b^6*c^3*i + 1280*a
^6*b^2*c^4*i - 1024*a^5*b^3*c^4*i + 192*a^4*b^4*c^4*i - 1024*a^7*c^5*i + 1024*a^6*b*c^5*i - 256*a^5*b^2*c^5*i
- 2*a^2*b^8*c*i + 32*a^3*b^6*c^2*i - 4*a^2*b^7*c^2*i - 192*a^4*b^4*c^3*i + 48*a^3*b^5*c^3*i + 512*a^5*b^2*c^4*
i - 192*a^4*b^3*c^4*i - 512*a^6*c^5*i + 256*a^5*b*c^5*i + a^2*b^6*c^2*i - 12*a^3*b^4*c^3*i + 48*a^4*b^2*c^4*i
- 64*a^5*c^5*i + sqrt(-b^18 + 36*a*b^16*c - 8*b^17*c - 576*a^2*b^14*c^2 + 256*a*b^15*c^2 - 24*b^16*c^2 + 5376*
a^3*b^12*c^3 - 3584*a^2*b^13*c^3 + 672*a*b^14*c^3 - 32*b^15*c^3 - 32256*a^4*b^10*c^4 + 28672*a^3*b^11*c^4 - 80
64*a^2*b^12*c^4 + 768*a*b^13*c^4 - 16*b^14*c^4 + 129024*a^5*b^8*c^5 - 143360*a^4*b^9*c^5 + 53760*a^3*b^10*c^5
- 7680*a^2*b^11*c^5 + 320*a*b^12*c^5 - 344064*a^6*b^6*c^6 + 458752*a^5*b^7*c^6 - 215040*a^4*b^8*c^6 + 40960*a^
3*b^9*c^6 - 2560*a^2*b^10*c^6 + 589824*a^7*b^4*c^7 - 917504*a^6*b^5*c^7 - 1125899906064384*a^5*b^6*c^7 - 12288
0*a^4*b^7*c^7 + 10240*a^3*b^8*c^7 - 589824*a^8*b^2*c^8 + 1048576*a^7*b^3*c^8 - 688128*a^6*b^4*c^8 + 196608*a^5
*b^5*c^8 - 20480*a^4*b^6*c^8 + 262144*a^9*c^9 - 524288*a^8*b*c^9 + 393216*a^7*b^2*c^9 - 131072*a^6*b^3*c^9 + 1
6384*a^5*b^4*c^9 + 4*b^16*c - 128*a*b^14*c^2 + 24*b^15*c^2 + 1792*a^2*b^12*c^3 - 672*a*b^13*c^3 + 48*b^14*c^3
- 14336*a^3*b^10*c^4 + 8064*a^2*b^11*c^4 - 1152*a*b^12*c^4 + 32*b^13*c^4 + 71680*a^4*b^8*c^5 - 53760*a^3*b^9*c
^5 + 11520*a^2*b^10*c^5 - 640*a*b^11*c^5 - 229376*a^5*b^6*c^6 + 215040*a^4*b^7*c^6 - 61440*a^3*b^8*c^6 + 5120*
a^2*b^9*c^6 + 458752*a^6*b^4*c^7 - 516096*a^5*b^5*c^7 + 184320*a^4*b^6*c^7 - 20480*a^3*b^7*c^7 - 524288*a^7*b^
2*c^8 + 688128*a^6*b^3*c^8 - 294912*a^5*b^4*c^8 + 40960*a^4*b^5*c^8 + 262144*a^8*c^9 - 393216*a^7*b*c^9 + 1966
08*a^6*b^2*c^9 - 32768*a^5*b^3*c^9 - 6*b^14*c^2 + 168*a*b^12*c^3 - 24*b^13*c^3 - 2016*a^2*b^10*c^4 + 576*a*b^1
1*c^4 - 24*b^12*c^4 + 13440*a^3*b^8*c^5 - 5760*a^2*b^9*c^5 + 480*a*b^10*c^5 - 53760*a^4*b^6*c^6 + 30720*a^3*b^
7*c^6 - 3840*a^2*b^8*c^6 + 129024*a^5*b^4*c^7 - 92160*a^4*b^5*c^7 + 15360*a^3*b^6*c^7 - 172032*a^6*b^2*c^8 + 1
47456*a^5*b^3*c^8 - 30720*a^4*b^4*c^8 + 98304*a^7*c^9 - 98304*a^6*b*c^9 + 24576*a^5*b^2*c^9 + 4*b^12*c^3 - 96*
a*b^10*c^4 + 8*b^11*c^4 + 960*a^2*b^8*c^5 - 160*a*b^9*c^5 - 5120*a^3*b^6*c^6 + 1280*a^2*b^7*c^6 + 15360*a^4*b^
4*c^7 - 5120*a^3*b^5*c^7 - 24576*a^5*b^2*c^8 + 10240*a^4*b^3*c^8 + 16384*a^6*c^9 - 8192*a^5*b*c^9 - b^10*c^4 +
20*a*b^8*c^5 - 160*a^2*b^6*c^6 + 640*a^3*b^4*c^7 - 1280*a^4*b^2*c^8 + 1024*a^5*c^9)*a^2*b) + 1/64*(32*a^2*b^7
- 192*a^3*b^5*c + 128*a^2*b^6*c - 256*a^3*b^4*c^2 + 128*a^2*b^5*c^2 + 1024*a^5*b*c^3 - 1024*a^4*b^2*c^3 + 256
*a^3*b^3*c^3 + 16*a^2*b^6 - 96*a^3*b^4*c + 128*a^2*b^5*c - 256*a^3*b^3*c^2 + 192*a^2*b^4*c^2 + 512*a^5*c^3 - 1
024*a^4*b*c^3 + 384*a^3*b^2*c^3 + 32*a^2*b^4*c - 64*a^3*b^2*c^2 + 96*a^2*b^3*c^2 - 256*a^4*c^3 + 192*a^3*b*c^3
+ 16*a^2*b^2*c^2 + 32*a^3*c^3 - 3*(2*b^8 - 36*a*b^6*c + 8*b^7*c + 304*a^2*b^4*c^2 - 112*a*b^5*c^2 + 8*b^6*c^2
- 1216*a^3*b^2*c^3 + 768*a^2*b^3*c^3 - 80*a*b^4*c^3 + 1792*a^4*c^4 - 1792*a^3*b*c^4 + 448*a^2*b^2*c^4 - b^7 +
16*a*b^5*c - 80*a^2*b^3*c^2 - 8*a*b^4*c^2 + 4*b^5*c^2 + 128*a^3*b*c^3 + 256*a^2*b^2*c^3 - 48*a*b^3*c^3 - 896*
a^3*c^4 + 448*a^2*b*c^4 - 2*b^5*c + 24*a*b^3*c^2 - 2*b^4*c^2 - 64*a^2*b*c^3 + 12*a*b^2*c^3 + 112*a^2*c^4 - b^3
*c^2 + 8*a*b*c^3)*sqrt(2*b*c + c)*d - (2*a*b^7 - 120*a^2*b^5*c + 8*a*b^6*c + 864*a^3*b^3*c^2 - 448*a^2*b^4*c^2
+ 8*a*b^5*c^2 - 1664*a^4*b*c^3 + 1664*a^3*b^2*c^3 - 416*a^2*b^3*c^3 - a*b^6 - 12*a^2*b^4*c + 144*a^3*b^2*c^2
- 288*a^2*b^3*c^2 + 4*a*b^4*c^2 - 320*a^4*c^3 + 1152*a^3*b*c^3 - 496*a^2*b^2*c^3 - 2*a*b^4*c - 32*a^2*b^2*c^2
- 2*a*b^3*c^2 + 160*a^3*c^3 - 184*a^2*b*c^3 - a*b^2*c^2 - 20*a^2*c^3)*sqrt(2*b*c + c)*f - 12*(3*a^2*b^6 - 20*a
^3*b^4*c + 12*a^2*b^5*c + 16*a^4*b^2*c^2 - 32*a^3*b^3*c^2 + 12*a^2*b^4*c^2 + 64*a^5*c^3 - 64*a^4*b*c^3 + 16*a^
3*b^2*c^3 + a^2*b^5 - 8*a^3*b^3*c + 10*a^2*b^4*c + 16*a^4*b*c^2 - 32*a^3*b^2*c^2 + 16*a^2*b^3*c^2 - 32*a^4*c^3
+ 16*a^3*b*c^3 + 2*a^2*b^3*c - 8*a^3*b*c^2 + 7*a^2*b^2*c^2 + 4*a^3*c^3 + a^2*b*c^2)*sqrt(2*b*c + c)*h + 48*(2
*a^2*b^6*c*i - 16*a^3*b^4*c^2*i + 8*a^2*b^5*c^2*i + 32*a^4*b^2*c^3*i - 32*a^3*b^3*c^3*i + 8*a^2*b^4*c^3*i + a^
2*b^5*c*i - 8*a^3*b^3*c^2*i + 8*a^2*b^4*c^2*i + 16*a^4*b*c^3*i - 32*a^3*b^2*c^3*i + 12*a^2*b^3*c^3*i + 2*a^2*b
^3*c^2*i - 8*a^3*b*c^3*i + 6*a^2*b^2*c^3*i + a^2*b*c^3*i)*g - 96*(2*a^2*b^5*c^2*i - 16*a^3*b^3*c^3*i + 8*a^2*b
^4*c^3*i + 32*a^4*b*c^4*i - 32*a^3*b^2*c^4*i + 8*a^2*b^3*c^4*i + a^2*b^4*c^2*i - 8*a^3*b^2*c^3*i + 8*a^2*b^3*c
^3*i + 16*a^4*c^4*i - 32*a^3*b*c^4*i + 12*a^2*b^2*c^4*i + 2*a^2*b^2*c^3*i - 8*a^3*c^4*i + 6*a^2*b*c^4*i + a^2*
c^4*i)*e)*log(x + 1/4*sqrt(-(8*a^2*b^5*i - 64*a^3*b^3*c*i + 128*a^4*b*c^2*i - sqrt(-64*(a^2*b^5 - 8*a^3*b^3*c
+ 16*a^4*b*c^2)^2 + 256*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)))/(a^2*b
^4*c*i - 8*a^3*b^2*c^2*i + 16*a^4*c^3*i)))/(a^2*b^10*i - 20*a^3*b^8*c*i + 4*a^2*b^9*c*i + 160*a^4*b^6*c^2*i -
64*a^3*b^7*c^2*i + 4*a^2*b^8*c^2*i - 640*a^5*b^4*c^3*i + 384*a^4*b^5*c^3*i - 48*a^3*b^6*c^3*i + 1280*a^6*b^2*c
^4*i - 1024*a^5*b^3*c^4*i + 192*a^4*b^4*c^4*i - 1024*a^7*c^5*i + 1024*a^6*b*c^5*i - 256*a^5*b^2*c^5*i + 2*a^2*
b^8*c*i - 32*a^3*b^6*c^2*i + 4*a^2*b^7*c^2*i + 192*a^4*b^4*c^3*i - 48*a^3*b^5*c^3*i - 512*a^5*b^2*c^4*i + 192*
a^4*b^3*c^4*i + 512*a^6*c^5*i - 256*a^5*b*c^5*i + a^2*b^6*c^2*i - 12*a^3*b^4*c^3*i + 48*a^4*b^2*c^4*i - 64*a^5
*c^5*i - sqrt(-b^18 + 36*a*b^16*c - 8*b^17*c - 576*a^2*b^14*c^2 + 256*a*b^15*c^2 - 24*b^16*c^2 + 5376*a^3*b^12
*c^3 - 3584*a^2*b^13*c^3 + 672*a*b^14*c^3 - 32*b^15*c^3 - 32256*a^4*b^10*c^4 + 28672*a^3*b^11*c^4 - 8064*a^2*b
^12*c^4 + 768*a*b^13*c^4 - 16*b^14*c^4 + 129024*a^5*b^8*c^5 - 143360*a^4*b^9*c^5 + 53760*a^3*b^10*c^5 - 7680*a
^2*b^11*c^5 + 320*a*b^12*c^5 - 344064*a^6*b^6*c^6 + 458752*a^5*b^7*c^6 - 215040*a^4*b^8*c^6 + 40960*a^3*b^9*c^
6 - 2560*a^2*b^10*c^6 + 589824*a^7*b^4*c^7 - 917504*a^6*b^5*c^7 - 1125899906064384*a^5*b^6*c^7 - 122880*a^4*b^
7*c^7 + 10240*a^3*b^8*c^7 - 589824*a^8*b^2*c^8 + 1048576*a^7*b^3*c^8 - 688128*a^6*b^4*c^8 + 196608*a^5*b^5*c^8
- 20480*a^4*b^6*c^8 + 262144*a^9*c^9 - 524288*a^8*b*c^9 + 393216*a^7*b^2*c^9 - 131072*a^6*b^3*c^9 + 16384*a^5
*b^4*c^9 - 4*b^16*c + 128*a*b^14*c^2 - 24*b^15*c^2 - 1792*a^2*b^12*c^3 + 672*a*b^13*c^3 - 48*b^14*c^3 + 14336*
a^3*b^10*c^4 - 8064*a^2*b^11*c^4 + 1152*a*b^12*c^4 - 32*b^13*c^4 - 71680*a^4*b^8*c^5 + 53760*a^3*b^9*c^5 - 115
20*a^2*b^10*c^5 + 640*a*b^11*c^5 + 229376*a^5*b^6*c^6 - 215040*a^4*b^7*c^6 + 61440*a^3*b^8*c^6 - 5120*a^2*b^9*
c^6 - 458752*a^6*b^4*c^7 + 516096*a^5*b^5*c^7 - 184320*a^4*b^6*c^7 + 20480*a^3*b^7*c^7 + 524288*a^7*b^2*c^8 -
688128*a^6*b^3*c^8 + 294912*a^5*b^4*c^8 - 40960*a^4*b^5*c^8 - 262144*a^8*c^9 + 393216*a^7*b*c^9 - 196608*a^6*b
^2*c^9 + 32768*a^5*b^3*c^9 - 2251799813160960*a^6*b*c^10 - 6*b^14*c^2 + 168*a*b^12*c^3 - 24*b^13*c^3 - 2016*a^
2*b^10*c^4 + 576*a*b^11*c^4 - 24*b^12*c^4 + 13440*a^3*b^8*c^5 - 5760*a^2*b^9*c^5 + 480*a*b^10*c^5 - 53760*a^4*
b^6*c^6 + 30720*a^3*b^7*c^6 - 3840*a^2*b^8*c^6 + 129024*a^5*b^4*c^7 - 92160*a^4*b^5*c^7 + 15360*a^3*b^6*c^7 -
172032*a^6*b^2*c^8 + 147456*a^5*b^3*c^8 - 30720*a^4*b^4*c^8 + 98304*a^7*c^9 - 98304*a^6*b*c^9 + 24576*a^5*b^2*
c^9 - 4*b^12*c^3 + 96*a*b^10*c^4 - 8*b^11*c^4 - 960*a^2*b^8*c^5 + 160*a*b^9*c^5 + 5120*a^3*b^6*c^6 - 1280*a^2*
b^7*c^6 - 15360*a^4*b^4*c^7 + 5120*a^3*b^5*c^7 + 24576*a^5*b^2*c^8 - 10240*a^4*b^3*c^8 - 16384*a^6*c^9 + 8192*
a^5*b*c^9 - b^10*c^4 + 20*a*b^8*c^5 - 160*a^2*b^6*c^6 + 640*a^3*b^4*c^7 - 1280*a^4*b^2*c^8 + 1024*a^5*c^9)*a^2
*b) + 1/64*(32*a^2*b^7 - 192*a^3*b^5*c + 128*a^2*b^6*c - 256*a^3*b^4*c^2 + 128*a^2*b^5*c^2 + 1024*a^5*b*c^3 -
1024*a^4*b^2*c^3 + 256*a^3*b^3*c^3 + 16*a^2*b^6 - 96*a^3*b^4*c + 128*a^2*b^5*c - 256*a^3*b^3*c^2 + 192*a^2*b^4
*c^2 + 512*a^5*c^3 - 1024*a^4*b*c^3 + 384*a^3*b^2*c^3 + 32*a^2*b^4*c - 64*a^3*b^2*c^2 + 96*a^2*b^3*c^2 - 256*a
^4*c^3 + 192*a^3*b*c^3 + 16*a^2*b^2*c^2 + 32*a^3*c^3 + 3*(2*b^8 - 36*a*b^6*c + 8*b^7*c + 304*a^2*b^4*c^2 - 112
*a*b^5*c^2 + 8*b^6*c^2 - 1216*a^3*b^2*c^3 + 768*a^2*b^3*c^3 - 80*a*b^4*c^3 + 1792*a^4*c^4 - 1792*a^3*b*c^4 + 4
48*a^2*b^2*c^4 - b^7 + 16*a*b^5*c - 80*a^2*b^3*c^2 - 8*a*b^4*c^2 + 4*b^5*c^2 + 128*a^3*b*c^3 + 256*a^2*b^2*c^3
- 48*a*b^3*c^3 - 896*a^3*c^4 + 448*a^2*b*c^4 - 2*b^5*c + 24*a*b^3*c^2 - 2*b^4*c^2 - 64*a^2*b*c^3 + 12*a*b^2*c
^3 + 112*a^2*c^4 - b^3*c^2 + 8*a*b*c^3)*sqrt(2*b*c + c)*d + (2*a*b^7 - 120*a^2*b^5*c + 8*a*b^6*c + 864*a^3*b^3
*c^2 - 448*a^2*b^4*c^2 + 8*a*b^5*c^2 - 1664*a^4*b*c^3 + 1664*a^3*b^2*c^3 - 416*a^2*b^3*c^3 - a*b^6 - 12*a^2*b^
4*c + 144*a^3*b^2*c^2 - 288*a^2*b^3*c^2 + 4*a*b^4*c^2 - 320*a^4*c^3 + 1152*a^3*b*c^3 - 496*a^2*b^2*c^3 - 2*a*b
^4*c - 32*a^2*b^2*c^2 - 2*a*b^3*c^2 + 160*a^3*c^3 - 184*a^2*b*c^3 - a*b^2*c^2 - 20*a^2*c^3)*sqrt(2*b*c + c)*f
+ 12*(3*a^2*b^6 - 20*a^3*b^4*c + 12*a^2*b^5*c + 16*a^4*b^2*c^2 - 32*a^3*b^3*c^2 + 12*a^2*b^4*c^2 + 64*a^5*c^3
- 64*a^4*b*c^3 + 16*a^3*b^2*c^3 + a^2*b^5 - 8*a^3*b^3*c + 10*a^2*b^4*c + 16*a^4*b*c^2 - 32*a^3*b^2*c^2 + 16*a^
2*b^3*c^2 - 32*a^4*c^3 + 16*a^3*b*c^3 + 2*a^2*b^3*c - 8*a^3*b*c^2 + 7*a^2*b^2*c^2 + 4*a^3*c^3 + a^2*b*c^2)*sqr
t(2*b*c + c)*h + 48*(2*a^2*b^6*c*i - 16*a^3*b^4*c^2*i + 8*a^2*b^5*c^2*i + 32*a^4*b^2*c^3*i - 32*a^3*b^3*c^3*i
+ 8*a^2*b^4*c^3*i + a^2*b^5*c*i - 8*a^3*b^3*c^2*i + 8*a^2*b^4*c^2*i + 16*a^4*b*c^3*i - 32*a^3*b^2*c^3*i + 12*a
^2*b^3*c^3*i + 2*a^2*b^3*c^2*i - 8*a^3*b*c^3*i + 6*a^2*b^2*c^3*i + a^2*b*c^3*i)*g - 96*(2*a^2*b^5*c^2*i - 16*a
^3*b^3*c^3*i + 8*a^2*b^4*c^3*i + 32*a^4*b*c^4*i - 32*a^3*b^2*c^4*i + 8*a^2*b^3*c^4*i + a^2*b^4*c^2*i - 8*a^3*b
^2*c^3*i + 8*a^2*b^3*c^3*i + 16*a^4*c^4*i - 32*a^3*b*c^4*i + 12*a^2*b^2*c^4*i + 2*a^2*b^2*c^3*i - 8*a^3*c^4*i
+ 6*a^2*b*c^4*i + a^2*c^4*i)*e)*log(x - 1/4*sqrt(-(8*a^2*b^5*i - 64*a^3*b^3*c*i + 128*a^4*b*c^2*i - sqrt(-64*(
a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)^2 + 256*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*(a^2*b^4*c - 8*a^3*b^2*c^2
+ 16*a^4*c^3)))/(a^2*b^4*c*i - 8*a^3*b^2*c^2*i + 16*a^4*c^3*i)))/(a^2*b^10*i - 20*a^3*b^8*c*i + 4*a^2*b^9*c*i
+ 160*a^4*b^6*c^2*i - 64*a^3*b^7*c^2*i + 4*a^2*b^8*c^2*i - 640*a^5*b^4*c^3*i + 384*a^4*b^5*c^3*i - 48*a^3*b^6*
c^3*i + 1280*a^6*b^2*c^4*i - 1024*a^5*b^3*c^4*i + 192*a^4*b^4*c^4*i - 1024*a^7*c^5*i + 1024*a^6*b*c^5*i - 256*
a^5*b^2*c^5*i + 2*a^2*b^8*c*i - 32*a^3*b^6*c^2*i + 4*a^2*b^7*c^2*i + 192*a^4*b^4*c^3*i - 48*a^3*b^5*c^3*i - 51
2*a^5*b^2*c^4*i + 192*a^4*b^3*c^4*i + 512*a^6*c^5*i - 256*a^5*b*c^5*i + a^2*b^6*c^2*i - 12*a^3*b^4*c^3*i + 48*
a^4*b^2*c^4*i - 64*a^5*c^5*i - sqrt(-b^18 + 36*a*b^16*c - 8*b^17*c - 576*a^2*b^14*c^2 + 256*a*b^15*c^2 - 24*b^
16*c^2 + 5376*a^3*b^12*c^3 - 3584*a^2*b^13*c^3 + 672*a*b^14*c^3 - 32*b^15*c^3 - 32256*a^4*b^10*c^4 + 28672*a^3
*b^11*c^4 - 8064*a^2*b^12*c^4 + 768*a*b^13*c^4 - 16*b^14*c^4 + 129024*a^5*b^8*c^5 - 143360*a^4*b^9*c^5 + 53760
*a^3*b^10*c^5 - 7680*a^2*b^11*c^5 + 320*a*b^12*c^5 - 344064*a^6*b^6*c^6 + 458752*a^5*b^7*c^6 - 215040*a^4*b^8*
c^6 + 40960*a^3*b^9*c^6 - 2560*a^2*b^10*c^6 + 589824*a^7*b^4*c^7 - 917504*a^6*b^5*c^7 - 1125899906064384*a^5*b
^6*c^7 - 122880*a^4*b^7*c^7 + 10240*a^3*b^8*c^7 - 589824*a^8*b^2*c^8 + 1048576*a^7*b^3*c^8 - 688128*a^6*b^4*c^
8 + 196608*a^5*b^5*c^8 - 20480*a^4*b^6*c^8 + 262144*a^9*c^9 - 524288*a^8*b*c^9 + 393216*a^7*b^2*c^9 - 131072*a
^6*b^3*c^9 + 16384*a^5*b^4*c^9 - 4*b^16*c + 128*a*b^14*c^2 - 24*b^15*c^2 - 1792*a^2*b^12*c^3 + 672*a*b^13*c^3
- 48*b^14*c^3 + 14336*a^3*b^10*c^4 - 8064*a^2*b^11*c^4 + 1152*a*b^12*c^4 - 32*b^13*c^4 - 71680*a^4*b^8*c^5 + 5
3760*a^3*b^9*c^5 - 11520*a^2*b^10*c^5 + 640*a*b^11*c^5 + 229376*a^5*b^6*c^6 - 215040*a^4*b^7*c^6 + 61440*a^3*b
^8*c^6 - 5120*a^2*b^9*c^6 - 458752*a^6*b^4*c^7 + 516096*a^5*b^5*c^7 - 184320*a^4*b^6*c^7 + 20480*a^3*b^7*c^7 +
524288*a^7*b^2*c^8 - 688128*a^6*b^3*c^8 + 294912*a^5*b^4*c^8 - 40960*a^4*b^5*c^8 - 262144*a^8*c^9 + 393216*a^
7*b*c^9 - 196608*a^6*b^2*c^9 + 32768*a^5*b^3*c^9 - 2251799813160960*a^6*b*c^10 - 6*b^14*c^2 + 168*a*b^12*c^3 -
24*b^13*c^3 - 2016*a^2*b^10*c^4 + 576*a*b^11*c^4 - 24*b^12*c^4 + 13440*a^3*b^8*c^5 - 5760*a^2*b^9*c^5 + 480*a
*b^10*c^5 - 53760*a^4*b^6*c^6 + 30720*a^3*b^7*c^6 - 3840*a^2*b^8*c^6 + 129024*a^5*b^4*c^7 - 92160*a^4*b^5*c^7
+ 15360*a^3*b^6*c^7 - 172032*a^6*b^2*c^8 + 147456*a^5*b^3*c^8 - 30720*a^4*b^4*c^8 + 98304*a^7*c^9 - 98304*a^6*
b*c^9 + 24576*a^5*b^2*c^9 - 4*b^12*c^3 + 96*a*b^10*c^4 - 8*b^11*c^4 - 960*a^2*b^8*c^5 + 160*a*b^9*c^5 + 5120*a
^3*b^6*c^6 - 1280*a^2*b^7*c^6 - 15360*a^4*b^4*c^7 + 5120*a^3*b^5*c^7 + 24576*a^5*b^2*c^8 - 10240*a^4*b^3*c^8 -
16384*a^6*c^9 + 8192*a^5*b*c^9 - b^10*c^4 + 20*a*b^8*c^5 - 160*a^2*b^6*c^6 + 640*a^3*b^4*c^7 - 1280*a^4*b^2*c
^8 + 1024*a^5*c^9)*a^2*b) + 1/8*(3*b^3*c^2*d*i*x^7 - 24*a*b*c^3*d*i*x^7 + a*b^2*c^2*f*i*x^7 + 20*a^2*c^3*f*i*x
^7 - 12*a^2*b*c^2*h*i*x^7 - 12*a^2*b*c^2*g*i*x^6 + 24*a^2*c^3*i*x^6*e + 6*b^4*c*d*i*x^5 - 49*a*b^2*c^2*d*i*x^5
+ 28*a^2*c^3*d*i*x^5 + 2*a*b^3*c*f*i*x^5 + 28*a^2*b*c^2*f*i*x^5 - 19*a^2*b^2*c*h*i*x^5 + 4*a^3*c^2*h*i*x^5 -
18*a^2*b^2*c*g*i*x^4 - 4*a^2*b^2*c*x^6 - 8*a^3*c^2*x^6 + 36*a^2*b*c^2*i*x^4*e + 3*b^5*d*i*x^3 - 20*a*b^3*c*d*i
*x^3 - 4*a^2*b*c^2*d*i*x^3 + a*b^4*f*i*x^3 + 5*a^2*b^2*c*f*i*x^3 + 36*a^3*c^2*f*i*x^3 - 5*a^2*b^3*h*i*x^3 - 16
*a^3*b*c*h*i*x^3 - 4*a^2*b^3*g*i*x^2 - 20*a^3*b*c*g*i*x^2 - 6*a^2*b^3*x^4 - 12*a^3*b*c*x^4 + 8*a^2*b^2*c*i*x^2
*e + 40*a^3*c^2*i*x^2*e + 5*a*b^4*d*i*x - 37*a^2*b^2*c*d*i*x + 44*a^3*c^2*d*i*x - a^2*b^3*f*i*x + 16*a^3*b*c*f
*i*x - 3*a^3*b^2*h*i*x - 12*a^4*c*h*i*x - 2*a^3*b^2*g*i - 16*a^4*c*g*i - 20*a^3*b^2*x^2 + 8*a^4*c*x^2 - 2*a^2*
b^3*i*e + 20*a^3*b*c*i*e - 12*a^4*b)/((a^2*b^4*i - 8*a^3*b^2*c*i + 16*a^4*c^2*i)*(c*x^4 + b*x^2 + a)^2)