3.413 \(\int \frac{1}{(d+e x)^2 (a+c x^4)^3} \, dx\)

Optimal. Leaf size=1830 \[ \text{result too large to display} \]

[Out]

-(e^11/((c*d^4 + a*e^4)^3*(d + e*x))) + (c*x*(7*d^2*(c*d^4 - 3*a*e^4) - 12*d*e*(c*d^4 - a*e^4)*x + 5*e^2*(3*c*
d^4 - a*e^4)*x^2))/(32*a^2*(c*d^4 + a*e^4)^2*(a + c*x^4)) + (c*(4*a*d^3*e^3 + x*(d^2*(c*d^4 - 3*a*e^4) - 2*d*e
*(c*d^4 - a*e^4)*x + e^2*(3*c*d^4 - a*e^4)*x^2)))/(8*a*(c*d^4 + a*e^4)^2*(a + c*x^4)^2) + (c*e^4*(8*a*d^3*e^3
+ x*(d^2*(5*c*d^4 - 3*a*e^4) - 2*d*e*(3*c*d^4 - a*e^4)*x + e^2*(7*c*d^4 - a*e^4)*x^2)))/(4*a*(c*d^4 + a*e^4)^3
*(a + c*x^4)) - (Sqrt[c]*d*e^9*(5*c*d^4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*(c*d^4 + a*e^4)^4) -
(Sqrt[c]*d*e^5*(3*c*d^4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/2)*(c*d^4 + a*e^4)^3) - (3*Sqrt[c]*d*e
*(c*d^4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(8*a^(5/2)*(c*d^4 + a*e^4)^2) - (c^(1/4)*(21*Sqrt[c]*d^2*(c*d^
4 - 3*a*e^4) + 5*Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*
(c*d^4 + a*e^4)^2) - (c^(1/4)*e^4*(3*Sqrt[c]*d^2*(5*c*d^4 - 3*a*e^4) + Sqrt[a]*e^2*(7*c*d^4 - a*e^4))*ArcTan[1
- (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) - (c^(1/4)*e^8*(3*Sqrt[c]*d^2*(3*c*d^4
- a*e^4) + Sqrt[a]*e^2*(11*c*d^4 - a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4
+ a*e^4)^4) + (c^(1/4)*(21*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) + 5*Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*ArcTan[1 + (Sqrt[2
]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^2) + (c^(1/4)*e^4*(3*Sqrt[c]*d^2*(5*c*d^4 - 3*a*e^
4) + Sqrt[a]*e^2*(7*c*d^4 - a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4
)^3) + (c^(1/4)*e^8*(3*Sqrt[c]*d^2*(3*c*d^4 - a*e^4) + Sqrt[a]*e^2*(11*c*d^4 - a*e^4))*ArcTan[1 + (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (12*c*d^3*e^11*Log[d + e*x])/(c*d^4 + a*e^4)^4 - (c^
(1/4)*e^8*(9*c^(3/2)*d^6 - 11*Sqrt[a]*c*d^4*e^2 - 3*a*Sqrt[c]*d^2*e^4 + a^(3/2)*e^6)*Log[Sqrt[a] - Sqrt[2]*a^(
1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) - (c^(1/4)*(21*Sqrt[c]*d^2*(c*d^4 - 3*a*e
^4) - 5*Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^
(11/4)*(c*d^4 + a*e^4)^2) - (c^(1/4)*e^4*(3*Sqrt[c]*d^2*(5*c*d^4 - 3*a*e^4) - Sqrt[a]*e^2*(7*c*d^4 - a*e^4))*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) + (c^(1/4)*e^8*(
9*c^(3/2)*d^6 - 11*Sqrt[a]*c*d^4*e^2 - 3*a*Sqrt[c]*d^2*e^4 + a^(3/2)*e^6)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4
)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (c^(1/4)*(21*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) - 5*Sqr
t[a]*e^2*(3*c*d^4 - a*e^4))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*(c*d
^4 + a*e^4)^2) + (c^(1/4)*e^4*(3*Sqrt[c]*d^2*(5*c*d^4 - 3*a*e^4) - Sqrt[a]*e^2*(7*c*d^4 - a*e^4))*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) - (3*c*d^3*e^11*Log[a + c*x
^4])/(c*d^4 + a*e^4)^4

________________________________________________________________________________________

Rubi [A]  time = 2.78142, antiderivative size = 1830, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 15, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.882, Rules used = {6742, 1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 260} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(a + c*x^4)^3),x]

[Out]

-(e^11/((c*d^4 + a*e^4)^3*(d + e*x))) + (c*x*(7*d^2*(c*d^4 - 3*a*e^4) - 12*d*e*(c*d^4 - a*e^4)*x + 5*e^2*(3*c*
d^4 - a*e^4)*x^2))/(32*a^2*(c*d^4 + a*e^4)^2*(a + c*x^4)) + (c*(4*a*d^3*e^3 + x*(d^2*(c*d^4 - 3*a*e^4) - 2*d*e
*(c*d^4 - a*e^4)*x + e^2*(3*c*d^4 - a*e^4)*x^2)))/(8*a*(c*d^4 + a*e^4)^2*(a + c*x^4)^2) + (c*e^4*(8*a*d^3*e^3
+ x*(d^2*(5*c*d^4 - 3*a*e^4) - 2*d*e*(3*c*d^4 - a*e^4)*x + e^2*(7*c*d^4 - a*e^4)*x^2)))/(4*a*(c*d^4 + a*e^4)^3
*(a + c*x^4)) - (Sqrt[c]*d*e^9*(5*c*d^4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*(c*d^4 + a*e^4)^4) -
(Sqrt[c]*d*e^5*(3*c*d^4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/2)*(c*d^4 + a*e^4)^3) - (3*Sqrt[c]*d*e
*(c*d^4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(8*a^(5/2)*(c*d^4 + a*e^4)^2) - (c^(1/4)*(21*Sqrt[c]*d^2*(c*d^
4 - 3*a*e^4) + 5*Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*
(c*d^4 + a*e^4)^2) - (c^(1/4)*e^4*(3*Sqrt[c]*d^2*(5*c*d^4 - 3*a*e^4) + Sqrt[a]*e^2*(7*c*d^4 - a*e^4))*ArcTan[1
- (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) - (c^(1/4)*e^8*(3*Sqrt[c]*d^2*(3*c*d^4
- a*e^4) + Sqrt[a]*e^2*(11*c*d^4 - a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4
+ a*e^4)^4) + (c^(1/4)*(21*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) + 5*Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*ArcTan[1 + (Sqrt[2
]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^2) + (c^(1/4)*e^4*(3*Sqrt[c]*d^2*(5*c*d^4 - 3*a*e^
4) + Sqrt[a]*e^2*(7*c*d^4 - a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4
)^3) + (c^(1/4)*e^8*(3*Sqrt[c]*d^2*(3*c*d^4 - a*e^4) + Sqrt[a]*e^2*(11*c*d^4 - a*e^4))*ArcTan[1 + (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (12*c*d^3*e^11*Log[d + e*x])/(c*d^4 + a*e^4)^4 - (c^
(1/4)*e^8*(9*c^(3/2)*d^6 - 11*Sqrt[a]*c*d^4*e^2 - 3*a*Sqrt[c]*d^2*e^4 + a^(3/2)*e^6)*Log[Sqrt[a] - Sqrt[2]*a^(
1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) - (c^(1/4)*(21*Sqrt[c]*d^2*(c*d^4 - 3*a*e
^4) - 5*Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^
(11/4)*(c*d^4 + a*e^4)^2) - (c^(1/4)*e^4*(3*Sqrt[c]*d^2*(5*c*d^4 - 3*a*e^4) - Sqrt[a]*e^2*(7*c*d^4 - a*e^4))*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) + (c^(1/4)*e^8*(
9*c^(3/2)*d^6 - 11*Sqrt[a]*c*d^4*e^2 - 3*a*Sqrt[c]*d^2*e^4 + a^(3/2)*e^6)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4
)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (c^(1/4)*(21*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) - 5*Sqr
t[a]*e^2*(3*c*d^4 - a*e^4))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*(c*d
^4 + a*e^4)^2) + (c^(1/4)*e^4*(3*Sqrt[c]*d^2*(5*c*d^4 - 3*a*e^4) - Sqrt[a]*e^2*(7*c*d^4 - a*e^4))*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) - (3*c*d^3*e^11*Log[a + c*x
^4])/(c*d^4 + a*e^4)^4

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 1854

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[((a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*(a + b*x^n)^(p + 1))/(a*b*n*(p + 1)), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1855

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Di
st[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b},
x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 1168

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

Rule 1162

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

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 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 260

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

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^2 \left (a+c x^4\right )^3} \, dx &=\int \left (\frac{e^{12}}{\left (c d^4+a e^4\right )^3 (d+e x)^2}+\frac{12 c d^3 e^{12}}{\left (c d^4+a e^4\right )^4 (d+e x)}+\frac{c \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2-4 c d^3 e^3 x^3\right )}{\left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^3}+\frac{c e^4 \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2-8 c d^3 e^3 x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )^2}+\frac{c e^8 \left (3 d^2 \left (3 c d^4-a e^4\right )-2 d e \left (5 c d^4-a e^4\right ) x+e^2 \left (11 c d^4-a e^4\right ) x^2-12 c d^3 e^3 x^3\right )}{\left (c d^4+a e^4\right )^4 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac{e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{12 c d^3 e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^8\right ) \int \frac{3 d^2 \left (3 c d^4-a e^4\right )-2 d e \left (5 c d^4-a e^4\right ) x+e^2 \left (11 c d^4-a e^4\right ) x^2-12 c d^3 e^3 x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^4\right ) \int \frac{d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2-8 c d^3 e^3 x^3}{\left (a+c x^4\right )^2} \, dx}{\left (c d^4+a e^4\right )^3}+\frac{c \int \frac{d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2-4 c d^3 e^3 x^3}{\left (a+c x^4\right )^3} \, dx}{\left (c d^4+a e^4\right )^2}\\ &=-\frac{e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac{c e^4 \left (8 a d^3 e^3+x \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac{12 c d^3 e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^8\right ) \int \left (\frac{x \left (-2 d e \left (5 c d^4-a e^4\right )-12 c d^3 e^3 x^2\right )}{a+c x^4}+\frac{3 d^2 \left (3 c d^4-a e^4\right )+e^2 \left (11 c d^4-a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^4}-\frac{\left (c e^4\right ) \int \frac{-3 d^2 \left (5 c d^4-3 a e^4\right )+4 d e \left (3 c d^4-a e^4\right ) x-e^2 \left (7 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^3}-\frac{c \int \frac{-7 d^2 \left (c d^4-3 a e^4\right )+12 d e \left (c d^4-a e^4\right ) x-5 e^2 \left (3 c d^4-a e^4\right ) x^2}{\left (a+c x^4\right )^2} \, dx}{8 a \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c x \left (7 d^2 \left (c d^4-3 a e^4\right )-12 d e \left (c d^4-a e^4\right ) x+5 e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac{c e^4 \left (8 a d^3 e^3+x \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac{12 c d^3 e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^8\right ) \int \frac{x \left (-2 d e \left (5 c d^4-a e^4\right )-12 c d^3 e^3 x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^8\right ) \int \frac{3 d^2 \left (3 c d^4-a e^4\right )+e^2 \left (11 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}-\frac{\left (c e^4\right ) \int \left (\frac{4 d e \left (3 c d^4-a e^4\right ) x}{a+c x^4}+\frac{-3 d^2 \left (5 c d^4-3 a e^4\right )-e^2 \left (7 c d^4-a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{4 a \left (c d^4+a e^4\right )^3}+\frac{c \int \frac{21 d^2 \left (c d^4-3 a e^4\right )-24 d e \left (c d^4-a e^4\right ) x+5 e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{32 a^2 \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c x \left (7 d^2 \left (c d^4-3 a e^4\right )-12 d e \left (c d^4-a e^4\right ) x+5 e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac{c e^4 \left (8 a d^3 e^3+x \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac{12 c d^3 e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^8\right ) \operatorname{Subst}\left (\int \frac{-2 d e \left (5 c d^4-a e^4\right )-12 c d^3 e^3 x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^4}-\frac{\left (c e^4\right ) \int \frac{-3 d^2 \left (5 c d^4-3 a e^4\right )-e^2 \left (7 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^3}-\frac{\left (c d e^5 \left (3 c d^4-a e^4\right )\right ) \int \frac{x}{a+c x^4} \, dx}{a \left (c d^4+a e^4\right )^3}+\frac{c \int \left (-\frac{24 d e \left (c d^4-a e^4\right ) x}{a+c x^4}+\frac{21 d^2 \left (c d^4-3 a e^4\right )+5 e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{32 a^2 \left (c d^4+a e^4\right )^2}-\frac{\left (e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^4}+\frac{\left (e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^4}\\ &=-\frac{e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c x \left (7 d^2 \left (c d^4-3 a e^4\right )-12 d e \left (c d^4-a e^4\right ) x+5 e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac{c e^4 \left (8 a d^3 e^3+x \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac{12 c d^3 e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^4}-\frac{\left (6 c^2 d^3 e^{11}\right ) \operatorname{Subst}\left (\int \frac{x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^4}-\frac{\left (c d e^9 \left (5 c d^4-a e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^4}-\frac{\left (c d e^5 \left (3 c d^4-a e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^4+a e^4\right )^3}+\frac{c \int \frac{21 d^2 \left (c d^4-3 a e^4\right )+5 e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{32 a^2 \left (c d^4+a e^4\right )^2}-\frac{\left (3 c d e \left (c d^4-a e^4\right )\right ) \int \frac{x}{a+c x^4} \, dx}{4 a^2 \left (c d^4+a e^4\right )^2}-\frac{\left (e^4 \left (7 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^3}+\frac{\left (e^4 \left (7 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^3}+\frac{\left (\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{\left (\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{\left (e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^4}+\frac{\left (e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^4}\\ &=-\frac{e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c x \left (7 d^2 \left (c d^4-3 a e^4\right )-12 d e \left (c d^4-a e^4\right ) x+5 e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac{c e^4 \left (8 a d^3 e^3+x \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac{\sqrt{c} d e^9 \left (5 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt{c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac{12 c d^3 e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}-\frac{3 c d^3 e^{11} \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}-\frac{\left (3 c d e \left (c d^4-a e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{8 a^2 \left (c d^4+a e^4\right )^2}+\frac{\left (15 c d^4 e^2-5 a e^6+\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{64 a^2 \left (c d^4+a e^4\right )^2}+\frac{\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac{\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac{\left (e^4 \left (7 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )^3}+\frac{\left (e^4 \left (7 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )^3}+\frac{\left (\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}-\frac{\left (\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{\left (\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}-5 e^2 \left (3 c d^4-a e^4\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{64 a^2 \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c x \left (7 d^2 \left (c d^4-3 a e^4\right )-12 d e \left (c d^4-a e^4\right ) x+5 e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac{c e^4 \left (8 a d^3 e^3+x \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac{\sqrt{c} d e^9 \left (5 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt{c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac{3 \sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{12 c d^3 e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}-\frac{3 c d^3 e^{11} \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}+\frac{\left (15 c d^4 e^2-5 a e^6+\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 \left (c d^4+a e^4\right )^2}+\frac{\left (15 c d^4 e^2-5 a e^6+\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 \left (c d^4+a e^4\right )^2}+\frac{\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac{\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac{\left (\sqrt [4]{c} \left (\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}-5 e^2 \left (3 c d^4-a e^4\right )\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}-\frac{\left (\sqrt [4]{c} \left (\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}-5 e^2 \left (3 c d^4-a e^4\right )\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c x \left (7 d^2 \left (c d^4-3 a e^4\right )-12 d e \left (c d^4-a e^4\right ) x+5 e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac{c e^4 \left (8 a d^3 e^3+x \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac{\sqrt{c} d e^9 \left (5 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt{c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac{3 \sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{12 c d^3 e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt [4]{c} \left (\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}-5 e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{\sqrt [4]{c} \left (\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}-5 e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}-\frac{3 c d^3 e^{11} \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}+\frac{\left (\sqrt [4]{c} \left (15 c d^4 e^2-5 a e^6+\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}-\frac{\left (\sqrt [4]{c} \left (15 c d^4 e^2-5 a e^6+\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}\\ &=-\frac{e^{11}}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c x \left (7 d^2 \left (c d^4-3 a e^4\right )-12 d e \left (c d^4-a e^4\right ) x+5 e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{32 a^2 \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac{c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{8 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac{c e^4 \left (8 a d^3 e^3+x \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac{\sqrt{c} d e^9 \left (5 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt{c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac{3 \sqrt{c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} \left (15 c d^4 e^2-5 a e^6+\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{\sqrt [4]{c} \left (15 c d^4 e^2-5 a e^6+\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6+\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{12 c d^3 e^{11} \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt [4]{c} \left (\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}-5 e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} e^8 \left (11 c d^4 e^2-a e^6-\frac{3 \sqrt{c} d^2 \left (3 c d^4-a e^4\right )}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac{\sqrt [4]{c} \left (\frac{21 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt{a}}-5 e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} \left (c d^4+a e^4\right )^2}-\frac{3 c d^3 e^{11} \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}\\ \end{align*}

Mathematica [A]  time = 1.69329, size = 1115, normalized size = 0.61 \[ \frac{3072 c d^3 \log (d+e x) e^{11}-768 c d^3 \log \left (c x^4+a\right ) e^{11}-\frac{256 \left (c d^4+a e^4\right ) e^{11}}{d+e x}+\frac{8 c \left (c d^4+a e^4\right ) \left (c^2 x \left (7 d^2-12 e x d+15 e^2 x^2\right ) d^8+2 a c e^4 x \left (13 d^2-24 e x d+33 e^2 x^2\right ) d^4+a^2 e^7 \left (64 d^3-45 e x d^2+28 e^2 x^2 d-13 e^3 x^3\right )\right )}{a^2 \left (c x^4+a\right )}+\frac{32 c \left (c d^4+a e^4\right )^2 \left (c x \left (d^2-2 e x d+3 e^2 x^2\right ) d^4+a e^3 \left (4 d^3-3 e x d^2+2 e^2 x^2 d-e^3 x^3\right )\right )}{a \left (c x^4+a\right )^2}-\frac{6 \sqrt [4]{c} \left (7 \sqrt{2} c^{7/2} d^{14}-16 \sqrt [4]{a} c^{13/4} e d^{13}+5 \sqrt{2} \sqrt{a} c^3 e^2 d^{12}+33 \sqrt{2} a c^{5/2} e^4 d^{10}-80 a^{5/4} c^{9/4} e^5 d^9+27 \sqrt{2} a^{3/2} c^2 e^6 d^8+77 \sqrt{2} a^2 c^{3/2} e^8 d^6-240 a^{9/4} c^{5/4} e^9 d^5+135 \sqrt{2} a^{5/2} c e^{10} d^4-77 \sqrt{2} a^3 \sqrt{c} e^{12} d^2+80 a^{13/4} \sqrt [4]{c} e^{13} d-15 \sqrt{2} a^{7/2} e^{14}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac{6 \sqrt [4]{c} \left (7 \sqrt{2} c^{7/2} d^{14}+16 \sqrt [4]{a} c^{13/4} e d^{13}+5 \sqrt{2} \sqrt{a} c^3 e^2 d^{12}+33 \sqrt{2} a c^{5/2} e^4 d^{10}+80 a^{5/4} c^{9/4} e^5 d^9+27 \sqrt{2} a^{3/2} c^2 e^6 d^8+77 \sqrt{2} a^2 c^{3/2} e^8 d^6+240 a^{9/4} c^{5/4} e^9 d^5+135 \sqrt{2} a^{5/2} c e^{10} d^4-77 \sqrt{2} a^3 \sqrt{c} e^{12} d^2-80 a^{13/4} \sqrt [4]{c} e^{13} d-15 \sqrt{2} a^{7/2} e^{14}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{11/4}}-\frac{3 \sqrt{2} \sqrt [4]{c} \left (7 c^{7/2} d^{14}-5 \sqrt{a} c^3 e^2 d^{12}+33 a c^{5/2} e^4 d^{10}-27 a^{3/2} c^2 e^6 d^8+77 a^2 c^{3/2} e^8 d^6-135 a^{5/2} c e^{10} d^4-77 a^3 \sqrt{c} e^{12} d^2+15 a^{7/2} e^{14}\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{11/4}}+\frac{3 \sqrt{2} \sqrt [4]{c} \left (7 c^{7/2} d^{14}-5 \sqrt{a} c^3 e^2 d^{12}+33 a c^{5/2} e^4 d^{10}-27 a^{3/2} c^2 e^6 d^8+77 a^2 c^{3/2} e^8 d^6-135 a^{5/2} c e^{10} d^4-77 a^3 \sqrt{c} e^{12} d^2+15 a^{7/2} e^{14}\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{11/4}}}{256 \left (c d^4+a e^4\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(a + c*x^4)^3),x]

[Out]

((-256*e^11*(c*d^4 + a*e^4))/(d + e*x) + (8*c*(c*d^4 + a*e^4)*(c^2*d^8*x*(7*d^2 - 12*d*e*x + 15*e^2*x^2) + 2*a
*c*d^4*e^4*x*(13*d^2 - 24*d*e*x + 33*e^2*x^2) + a^2*e^7*(64*d^3 - 45*d^2*e*x + 28*d*e^2*x^2 - 13*e^3*x^3)))/(a
^2*(a + c*x^4)) + (32*c*(c*d^4 + a*e^4)^2*(c*d^4*x*(d^2 - 2*d*e*x + 3*e^2*x^2) + a*e^3*(4*d^3 - 3*d^2*e*x + 2*
d*e^2*x^2 - e^3*x^3)))/(a*(a + c*x^4)^2) - (6*c^(1/4)*(7*Sqrt[2]*c^(7/2)*d^14 - 16*a^(1/4)*c^(13/4)*d^13*e + 5
*Sqrt[2]*Sqrt[a]*c^3*d^12*e^2 + 33*Sqrt[2]*a*c^(5/2)*d^10*e^4 - 80*a^(5/4)*c^(9/4)*d^9*e^5 + 27*Sqrt[2]*a^(3/2
)*c^2*d^8*e^6 + 77*Sqrt[2]*a^2*c^(3/2)*d^6*e^8 - 240*a^(9/4)*c^(5/4)*d^5*e^9 + 135*Sqrt[2]*a^(5/2)*c*d^4*e^10
- 77*Sqrt[2]*a^3*Sqrt[c]*d^2*e^12 + 80*a^(13/4)*c^(1/4)*d*e^13 - 15*Sqrt[2]*a^(7/2)*e^14)*ArcTan[1 - (Sqrt[2]*
c^(1/4)*x)/a^(1/4)])/a^(11/4) + (6*c^(1/4)*(7*Sqrt[2]*c^(7/2)*d^14 + 16*a^(1/4)*c^(13/4)*d^13*e + 5*Sqrt[2]*Sq
rt[a]*c^3*d^12*e^2 + 33*Sqrt[2]*a*c^(5/2)*d^10*e^4 + 80*a^(5/4)*c^(9/4)*d^9*e^5 + 27*Sqrt[2]*a^(3/2)*c^2*d^8*e
^6 + 77*Sqrt[2]*a^2*c^(3/2)*d^6*e^8 + 240*a^(9/4)*c^(5/4)*d^5*e^9 + 135*Sqrt[2]*a^(5/2)*c*d^4*e^10 - 77*Sqrt[2
]*a^3*Sqrt[c]*d^2*e^12 - 80*a^(13/4)*c^(1/4)*d*e^13 - 15*Sqrt[2]*a^(7/2)*e^14)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/
a^(1/4)])/a^(11/4) + 3072*c*d^3*e^11*Log[d + e*x] - (3*Sqrt[2]*c^(1/4)*(7*c^(7/2)*d^14 - 5*Sqrt[a]*c^3*d^12*e^
2 + 33*a*c^(5/2)*d^10*e^4 - 27*a^(3/2)*c^2*d^8*e^6 + 77*a^2*c^(3/2)*d^6*e^8 - 135*a^(5/2)*c*d^4*e^10 - 77*a^3*
Sqrt[c]*d^2*e^12 + 15*a^(7/2)*e^14)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(11/4) + (3*Sqrt
[2]*c^(1/4)*(7*c^(7/2)*d^14 - 5*Sqrt[a]*c^3*d^12*e^2 + 33*a*c^(5/2)*d^10*e^4 - 27*a^(3/2)*c^2*d^8*e^6 + 77*a^2
*c^(3/2)*d^6*e^8 - 135*a^(5/2)*c*d^4*e^10 - 77*a^3*Sqrt[c]*d^2*e^12 + 15*a^(7/2)*e^14)*Log[Sqrt[a] + Sqrt[2]*a
^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(11/4) - 768*c*d^3*e^11*Log[a + c*x^4])/(256*(c*d^4 + a*e^4)^4)

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 2769, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*x^4+a)^3,x)

[Out]

1/2*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^11*e^3+5/2*c/(a*e^4+c*d^4)^4/(c*x^4+a)^2*a^2*d^3*e^11-17/32*c/(a*e^4+c*d
^4)^4/(c*x^4+a)^2*e^14*a^2*x^3-45/8*c^2/(a*e^4+c*d^4)^4/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^5*e^9-45/256/(
a*e^4+c*d^4)^4*a/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/
c)^(1/2)))*e^14-45/128/(a*e^4+c*d^4)^4*a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*e^14-45/128/(a*e^
4+c*d^4)^4*a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*e^14+53/32*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)^2*e^
10*x^7*d^4-5/8*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^5*e^9*x^6-19/32*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^6*x^5*e^8+1
01/32*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)^2*e^6*x^3*d^8-17/8*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^9*e^5*x^2+29/32*c^3/(
a*e^4+c*d^4)^4/(c*x^4+a)^2*d^10*x*e^4-13/32*c^2/(a*e^4+c*d^4)^4/(c*x^4+a)^2*e^14*a*x^7+7/32*c^5/(a*e^4+c*d^4)^
4/(c*x^4+a)^2*d^14/a^2*x^5+11/32*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^14/a*x+2*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)^2*x^
4*d^7*e^7+3*c^2/(a*e^4+c*d^4)^4/(c*x^4+a)^2*a*d^7*e^7+12*c*d^3*e^11*ln(e*x+d)/(a*e^4+c*d^4)^4-3*c*d^3*e^11*ln(
c*x^4+a)/(a*e^4+c*d^4)^4+9/8*c/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d*e^13*a^2*x^2-57/32*c/(a*e^4+c*d^4)^4/(c*x^4+a)^2*
d^2*a^2*x*e^12+2*c^2/(a*e^4+c*d^4)^4/(c*x^4+a)^2*x^4*a*d^3*e^11-3/8*c^5/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^13*e/a^2
*x^6+81/32*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)^2*e^6/a*x^7*d^8+15/32*c^5/(a*e^4+c*d^4)^4/(c*x^4+a)^2*e^2/a^2*x^7*d^1
2+7/8*c^2/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d*e^13*a*x^6-15/8*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^9*e^5/a*x^6-45/32*c^
2/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^2*a*x^5*e^12+33/32*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^10/a*x^5*e^4+57/32*c^2/(a
*e^4+c*d^4)^4/(c*x^4+a)^2*e^10*a*x^3*d^4+27/32*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)^2*e^2/a*x^3*d^12-3/8*c^2/(a*e^4+c
*d^4)^4/(c*x^4+a)^2*d^5*e^9*a*x^2-5/8*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^13*e/a*x^2-39/32*c^2/(a*e^4+c*d^4)^4/(
c*x^4+a)^2*d^6*a*x*e^8+405/128*c/(a*e^4+c*d^4)^4/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^4*e^10+
405/256*c/(a*e^4+c*d^4)^4/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^
(1/2)+(a/c)^(1/2)))*d^4*e^10+405/128*c/(a*e^4+c*d^4)^4/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^4
*e^10+15/8*c/(a*e^4+c*d^4)^4*a/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d*e^13-3/8*c^4/(a*e^4+c*d^4)^4/a^2/(a*c)^
(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^13*e-15/8*c^3/(a*e^4+c*d^4)^4/a/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^9*e^
5-231/128*c/(a*e^4+c*d^4)^4*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^2*e^12+21/128*c^4/(a*e^4+c*d
^4)^4/a^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^14-231/128*c/(a*e^4+c*d^4)^4*(a/c)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^2*e^12+21/128*c^4/(a*e^4+c*d^4)^4/a^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a/c)^(1/4)*x-1)*d^14-231/256*c/(a*e^4+c*d^4)^4*(a/c)^(1/4)*2^(1/2)*ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))
/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))*d^2*e^12+21/256*c^4/(a*e^4+c*d^4)^4/a^3*(a/c)^(1/4)*2^(1/2)*ln((x^2+
(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))*d^14-e^11/(a*e^4+c*d^4)^3/(e*x+d)+
15/128*c^3/(a*e^4+c*d^4)^4/a^2/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^12*e^2+81/128*c^2/(a*e^4+
c*d^4)^4/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^8*e^6+231/256*c^2/(a*e^4+c*d^4)^4/a*(a/c)^(1/
4)*2^(1/2)*ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))*d^6*e^8+99/256*
c^3/(a*e^4+c*d^4)^4/a^2*(a/c)^(1/4)*2^(1/2)*ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1
/2)+(a/c)^(1/2)))*d^10*e^4+81/256*c^2/(a*e^4+c*d^4)^4/a/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c
)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))*d^8*e^6+15/256*c^3/(a*e^4+c*d^4)^4/a^2/(a/c)^(1/4)*2^(1/2)*l
n((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))*d^12*e^2+15/128*c^3/(a*e^4+
c*d^4)^4/a^2/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^12*e^2+81/128*c^2/(a*e^4+c*d^4)^4/a/(a/c)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^8*e^6+231/128*c^2/(a*e^4+c*d^4)^4/a*(a/c)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(a/c)^(1/4)*x+1)*d^6*e^8+99/128*c^3/(a*e^4+c*d^4)^4/a^2*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)
*x+1)*d^10*e^4+231/128*c^2/(a*e^4+c*d^4)^4/a*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^6*e^8+99/12
8*c^3/(a*e^4+c*d^4)^4/a^2*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^10*e^4

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^4+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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(1/(e*x+d)^2/(c*x^4+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(1/(e*x+d)**2/(c*x**4+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^4+a)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError