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

Optimal. Leaf size=1668 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 4.00758, antiderivative size = 1668, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 16, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.291, Rules used = {1790, 1789, 1422, 200, 31, 634, 617, 204, 628, 1758, 1510, 292, 1745, 1657, 618, 206} \[ \text{result too large to display} \]

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^3 + c*x^6),x]

[Out]

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

Rule 1790

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[x^j*Sum[Coeff[Pq, x, j + k*n]*x^(k*n), {k, 0, (q - j)/n + 1}]*(a + b*x^n + c*x^(2*n))^p, {j, 0, n - 1}], x
]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !PolyQ[P
q, x^n]

Rule 1789

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Int[ExpandToSum[Pq - Pqq*x^q - (Pqq*(a*(q - 2*n + 1)*x^(q - 2*n) + b*(q + n*(p - 1) + 1)*x^
(q - n)))/(c*(q + 2*n*p + 1)), x]*(a + b*x^n + c*x^(2*n))^p, x] + Simp[(Pqq*x^(q - 2*n + 1)*(a + b*x^n + c*x^(
2*n))^(p + 1))/(c*(q + 2*n*p + 1)), x]] /; GeQ[q, 2*n] && NeQ[q + 2*n*p + 1, 0] && (IntegerQ[2*p] || (EqQ[n, 1
] && IntegerQ[4*p]) || IntegerQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^
n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rule 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1758

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> With[{q = Expon[P
q, x]}, With[{Pqq = Coeff[Pq, x, q]}, Int[(d*x)^m*ExpandToSum[Pq - Pqq*x^q - (Pqq*(a*(m + q - 2*n + 1)*x^(q -
2*n) + b*(m + q + n*(p - 1) + 1)*x^(q - n)))/(c*(m + q + 2*n*p + 1)), x]*(a + b*x^n + c*x^(2*n))^p, x] + Simp[
(Pqq*(d*x)^(m + q - 2*n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(c*d^(q - 2*n + 1)*(m + q + 2*n*p + 1)), x]] /;
GeQ[q, 2*n] && NeQ[m + q + 2*n*p + 1, 0] && (IntegerQ[2*p] || (EqQ[n, 1] && IntegerQ[4*p]) || IntegerQ[p + (q
+ 1)/(2*n)])] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ
[n, 0]

Rule 1510

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[e/
2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2
, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 1745

Int[(Pq_)*(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[Subs
tFor[x^n, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ
[Pq, x^n] && EqQ[Simplify[m - n + 1], 0]

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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+j x^5+k x^6+l x^7+m x^8}{a+b x^3+c x^6} \, dx &=\int \left (\frac{d+g x^3+k x^6}{a+b x^3+c x^6}+\frac{x \left (e+h x^3+l x^6\right )}{a+b x^3+c x^6}+\frac{x^2 \left (f+j x^3+m x^6\right )}{a+b x^3+c x^6}\right ) \, dx\\ &=\int \frac{d+g x^3+k x^6}{a+b x^3+c x^6} \, dx+\int \frac{x \left (e+h x^3+l x^6\right )}{a+b x^3+c x^6} \, dx+\int \frac{x^2 \left (f+j x^3+m x^6\right )}{a+b x^3+c x^6} \, dx\\ &=\frac{k x}{c}+\frac{l x^2}{2 c}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{f+j x+m x^2}{a+b x+c x^2} \, dx,x,x^3\right )+\int \frac{d-\frac{a k}{c}+\left (g-\frac{b k}{c}\right ) x^3}{a+b x^3+c x^6} \, dx+\int \frac{x \left (e-\frac{a l}{c}+\left (h-\frac{b l}{c}\right ) x^3\right )}{a+b x^3+c x^6} \, dx\\ &=\frac{k x}{c}+\frac{l x^2}{2 c}+\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{m}{c}+\frac{c f-a m+(c j-b m) x}{c \left (a+b x+c x^2\right )}\right ) \, dx,x,x^3\right )+\frac{1}{2} \left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{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^3} \, dx+\frac{1}{2} \left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{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^3} \, dx+\frac{1}{2} \left (h-\frac{b l}{c}-\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \int \frac{x}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx+\frac{1}{2} \left (h-\frac{b l}{c}+\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \int \frac{x}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx\\ &=\frac{k x}{c}+\frac{l x^2}{2 c}+\frac{m x^3}{3 c}+\frac{\operatorname{Subst}\left (\int \frac{c f-a m+(c j-b m) x}{a+b x+c x^2} \, dx,x,x^3\right )}{3 c}+\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \int \frac{2^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}-\sqrt [3]{c} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \int \frac{2^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}-\sqrt [3]{c} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \int \frac{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (-h+\frac{b l}{c}+\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (-h+\frac{b l}{c}-\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \int \frac{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}\\ &=\frac{k x}{c}+\frac{l x^2}{2 c}+\frac{m x^3}{3 c}+\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}-\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6 \sqrt [3]{2} \sqrt [3]{c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2\ 2^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}-\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6 \sqrt [3]{2} \sqrt [3]{c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2\ 2^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}+\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}+\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{(c j-b m) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^2}+\frac{\left (2 c^2 f-b c j+b^2 m-2 a c m\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^2}\\ &=\frac{k x}{c}+\frac{l x^2}{2 c}+\frac{m x^3}{3 c}+\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}-\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{(c j-b m) \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}\right )}{2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}\right )}{2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (2 c^2 f-b c j+b^2 m-2 a c m\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^3\right )}{3 c^2}\\ &=\frac{k x}{c}+\frac{l x^2}{2 c}+\frac{m x^3}{3 c}-\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}-\frac{\left (2 c^2 f-b c j+b^2 m-2 a c m\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^2 \sqrt{b^2-4 a c}}+\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}-\frac{\left (g-\frac{b k}{c}+\frac{2 c^2 d+b^2 k-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (h-\frac{b l}{c}+\frac{2 c^2 e+b^2 l-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right ) \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (g-\frac{b k}{c}-\frac{2 c^2 d-b c g+b^2 k-2 a c k}{c \sqrt{b^2-4 a c}}\right ) \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (h-\frac{b l}{c}-\frac{2 c^2 e-b c h+b^2 l-2 a c l}{c \sqrt{b^2-4 a c}}\right ) \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{(c j-b m) \log \left (a+b x^3+c x^6\right )}{6 c^2}\\ \end{align*}

Mathematica [C]  time = 2.01828, size = 223, normalized size = 0.13 \[ \frac{-2 \text{RootSum}\left [\text{$\#$1}^3 b+\text{$\#$1}^6 c+a\& ,\frac{\text{$\#$1}^2 a m \log (x-\text{$\#$1})+\text{$\#$1}^3 b k \log (x-\text{$\#$1})+\text{$\#$1}^4 b l \log (x-\text{$\#$1})+\text{$\#$1}^5 b m \log (x-\text{$\#$1})-\text{$\#$1}^2 c f \log (x-\text{$\#$1})-\text{$\#$1}^3 c g \log (x-\text{$\#$1})-\text{$\#$1}^4 c h \log (x-\text{$\#$1})+\text{$\#$1}^5 (-c) j \log (x-\text{$\#$1})+a k \log (x-\text{$\#$1})+\text{$\#$1} a l \log (x-\text{$\#$1})-c d \log (x-\text{$\#$1})-\text{$\#$1} c e \log (x-\text{$\#$1})}{\text{$\#$1}^2 b+2 \text{$\#$1}^5 c}\& \right ]+6 k x+3 l x^2+2 m x^3}{6 c} \]

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^3 + c*x^6),x]

[Out]

(6*k*x + 3*l*x^2 + 2*m*x^3 - 2*RootSum[a + b*#1^3 + c*#1^6 & , (-(c*d*Log[x - #1]) + a*k*Log[x - #1] - c*e*Log
[x - #1]*#1 + a*l*Log[x - #1]*#1 - c*f*Log[x - #1]*#1^2 + a*m*Log[x - #1]*#1^2 - c*g*Log[x - #1]*#1^3 + b*k*Lo
g[x - #1]*#1^3 - c*h*Log[x - #1]*#1^4 + b*l*Log[x - #1]*#1^4 - c*j*Log[x - #1]*#1^5 + b*m*Log[x - #1]*#1^5)/(b
*#1^2 + 2*c*#1^5) & ])/(6*c)

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Maple [C]  time = 0.022, size = 134, normalized size = 0.1 \begin{align*}{\frac{m{x}^{3}}{3\,c}}+{\frac{l{x}^{2}}{2\,c}}+{\frac{kx}{c}}+{\frac{1}{3\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{ \left ( \left ( -bm+cj \right ){{\it \_R}}^{5}+ \left ( -bl+ch \right ){{\it \_R}}^{4}+ \left ( -bk+cg \right ){{\it \_R}}^{3}+ \left ( -am+cf \right ){{\it \_R}}^{2}+ \left ( -al+ce \right ){\it \_R}-ak+cd \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \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^6+b*x^3+a),x)

[Out]

1/3*m*x^3/c+1/2*l*x^2/c+k*x/c+1/3/c*sum(((-b*m+c*j)*_R^5+(-b*l+c*h)*_R^4+(-b*k+c*g)*_R^3+(-a*m+c*f)*_R^2+(-a*l
+c*e)*_R-a*k+c*d)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_Z^6*c+_Z^3*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, m x^{3} + 3 \, l x^{2} + 6 \, k x}{6 \, c} - \frac{-\int \frac{{\left (c j - b m\right )} x^{5} +{\left (c h - b l\right )} x^{4} +{\left (c g - b k\right )} x^{3} +{\left (c f - a m\right )} x^{2} + c d - a k +{\left (c e - a l\right )} x}{c x^{6} + b x^{3} + a}\,{d x}}{c} \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^6+b*x^3+a),x, algorithm="maxima")

[Out]

1/6*(2*m*x^3 + 3*l*x^2 + 6*k*x)/c - integrate(-((c*j - b*m)*x^5 + (c*h - b*l)*x^4 + (c*g - b*k)*x^3 + (c*f - a
*m)*x^2 + c*d - a*k + (c*e - a*l)*x)/(c*x^6 + b*x^3 + a), x)/c

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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^6+b*x^3+a),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**6+b*x**3+a),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^6+b*x^3+a),x, algorithm="giac")

[Out]

Timed out