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

Optimal. Leaf size=1384 \[ \text{result too large to display} \]

[Out]

-e^7/(2*(c*d^4 + a*e^4)^2*(d + e*x)^2) - (8*c*d^3*e^7)/((c*d^4 + a*e^4)^3*(d + e*x)) + (c*(2*a*d^2*e^3*(5*c*d^
4 - 3*a*e^4) + x*(d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8) - e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*x + 2*c*
d^3*e^2*(3*c*d^4 - 5*a*e^4)*x^2)))/(4*a*(c*d^4 + a*e^4)^3*(a + c*x^4)) - (Sqrt[c]*e^5*(21*c^2*d^8 - 26*a*c*d^4
*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^4 + a*e^4)^4) - (Sqrt[c]*e*(3*c^2*d^8 - 12*a*c*
d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(3/2)*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*(3*c^2*d^8 - 36*a
*c*d^4*e^4 + 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*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^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 3*(
5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*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^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4 + 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*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^(3/4)*d*e^4*(4*Sqrt[a]*Sqr
t[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*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^2*e^7*(3*c*d^4 - a*e^4)*Log[d + e*x])/(c*d^4 + a*e^4)^
4 - (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sq
rt[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^(3/4)*d*e^4*(4*S
qrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*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^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^
4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*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) - (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4
) - 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*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) - (3*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[a + c*x^4])/(c*d^4 + a*e^4)^4

________________________________________________________________________________________

Rubi [A]  time = 1.96275, antiderivative size = 1384, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 14, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.824, Rules used = {6742, 1854, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 260} \[ \frac{12 c d^2 \left (3 c d^4-a e^4\right ) \log (d+e x) e^7}{\left (c d^4+a e^4\right )^4}-\frac{3 c d^2 \left (3 c d^4-a e^4\right ) \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^4}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}-\frac{e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{\sqrt{c} \left (21 c^2 d^8-26 a c e^4 d^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e^5}{2 \sqrt{a} \left (c d^4+a e^4\right )^4}-\frac{c^{3/4} d \left (4 \sqrt{a} \sqrt{c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac{c^{3/4} d \left (4 \sqrt{a} \sqrt{c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac{c^{3/4} d \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac{c^{3/4} d \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac{\sqrt{c} \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac{c \left (2 a d^2 \left (5 c d^4-3 a e^4\right ) e^3+x \left (2 c e^2 \left (3 c d^4-5 a e^4\right ) x^2 d^3+\left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right ) d-e \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) x\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (c x^4+a\right )}-\frac{c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt{a} \sqrt{c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac{c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt{a} \sqrt{c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^3}-\frac{c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt{a} \sqrt{c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac{c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt{a} \sqrt{c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + c*x^4)^2),x]

[Out]

-e^7/(2*(c*d^4 + a*e^4)^2*(d + e*x)^2) - (8*c*d^3*e^7)/((c*d^4 + a*e^4)^3*(d + e*x)) + (c*(2*a*d^2*e^3*(5*c*d^
4 - 3*a*e^4) + x*(d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8) - e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*x + 2*c*
d^3*e^2*(3*c*d^4 - 5*a*e^4)*x^2)))/(4*a*(c*d^4 + a*e^4)^3*(a + c*x^4)) - (Sqrt[c]*e^5*(21*c^2*d^8 - 26*a*c*d^4
*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^4 + a*e^4)^4) - (Sqrt[c]*e*(3*c^2*d^8 - 12*a*c*
d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(3/2)*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*(3*c^2*d^8 - 36*a
*c*d^4*e^4 + 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*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^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 3*(
5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*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^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4 + 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*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^(3/4)*d*e^4*(4*Sqrt[a]*Sqr
t[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*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^2*e^7*(3*c*d^4 - a*e^4)*Log[d + e*x])/(c*d^4 + a*e^4)^
4 - (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sq
rt[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^(3/4)*d*e^4*(4*S
qrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*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^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^
4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*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) - (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4
) - 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*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) - (3*c*d^2*e^7*(3*c*d^4 - a*e^4)*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 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)^3 \left (a+c x^4\right )^2} \, dx &=\int \left (\frac{e^8}{\left (c d^4+a e^4\right )^2 (d+e x)^3}+\frac{8 c d^3 e^8}{\left (c d^4+a e^4\right )^3 (d+e x)^2}+\frac{12 c d^2 e^8 \left (3 c d^4-a e^4\right )}{\left (c d^4+a e^4\right )^4 (d+e x)}+\frac{c \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )^2}+\frac{c e^4 \left (3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^4 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac{e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^4\right ) \int \frac{3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}+\frac{c \int \frac{d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3}{\left (a+c x^4\right )^2} \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac{e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 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^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^4\right ) \int \left (\frac{3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2}{a+c x^4}+\frac{x \left (-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^4}-\frac{c \int \frac{-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x-2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^3}\\ &=-\frac{e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 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^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^4\right ) \int \frac{3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )+4 c d^3 e^2 \left (7 c d^4-5 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 \frac{x \left (-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}-\frac{c \int \left (\frac{2 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x}{a+c x^4}+\frac{-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{4 a \left (c d^4+a e^4\right )^3}\\ &=-\frac{e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 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^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{\left (c e^4\right ) \operatorname{Subst}\left (\int \frac{-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^4}-\frac{c \int \frac{-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-2 c d^3 e^2 \left (3 c d^4-5 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 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )\right ) \int \frac{x}{a+c x^4} \, dx}{2 a \left (c d^4+a e^4\right )^3}-\frac{\left (\sqrt{c} d e^4 \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 \sqrt{a} \left (c d^4+a e^4\right )^4}+\frac{\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac{3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 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^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}-\frac{\left (6 c^2 d^2 e^7 \left (3 c d^4-a e^4\right )\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 e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^4}-\frac{\left (c e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^4+a e^4\right )^3}+\frac{\left (c^{3/4} d e^4 \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\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}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac{\left (c^{3/4} d e^4 \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\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}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac{\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac{3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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 (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac{3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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 (c d \left (6 c d^6 e^2-10 a d^2 e^6-\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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 (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 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} e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt{c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac{12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{c^{3/4} d e^4 \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac{c^{3/4} d e^4 \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac{3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}+\frac{\left (c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac{3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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 (c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac{3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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 (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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 (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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 (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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 (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 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} e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt{c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac{c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac{3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac{3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{c^{3/4} d e^4 \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac{c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{c^{3/4} d e^4 \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac{c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}+\frac{\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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 (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac{c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 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} e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (c d^4+a e^4\right )^4}-\frac{\sqrt{c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac{c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac{3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac{3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac{c^{3/4} d e^4 \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac{c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{c^{3/4} d e^4 \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac{c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac{3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt{a} \sqrt{c}}\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{3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}\\ \end{align*}

Mathematica [A]  time = 1.41356, size = 996, normalized size = 0.72 \[ \frac{384 c d^2 \left (3 c d^4-a e^4\right ) \log (d+e x) e^7-96 c d^2 \left (3 c d^4-a e^4\right ) \log \left (c x^4+a\right ) e^7-\frac{256 c d^3 \left (c d^4+a e^4\right ) e^7}{d+e x}-\frac{16 \left (c d^4+a e^4\right )^2 e^7}{(d+e x)^2}+\frac{8 c \left (c d^4+a e^4\right ) \left (c^2 x \left (d^2-3 e x d+6 e^2 x^2\right ) d^7+2 a c e^3 \left (5 d^3-6 e x d^2+6 e^2 x^2 d-5 e^3 x^3\right ) d^3-a^2 e^7 \left (6 d^2-3 e x d+e^2 x^2\right )\right )}{a \left (c x^4+a\right )}-\frac{6 \sqrt{c} \left (\sqrt{2} c^{13/4} d^{13}-4 \sqrt [4]{a} c^3 e d^{12}+2 \sqrt{2} \sqrt{a} c^{11/4} e^2 d^{11}+9 \sqrt{2} a c^{9/4} e^4 d^9-44 a^{5/4} c^2 e^5 d^8+36 \sqrt{2} a^{3/2} c^{7/4} e^6 d^7-49 \sqrt{2} a^2 c^{5/4} e^8 d^5+84 a^{9/4} c e^9 d^4-30 \sqrt{2} a^{5/2} c^{3/4} e^{10} d^3+7 \sqrt{2} a^3 \sqrt [4]{c} e^{12} d-4 a^{13/4} e^{13}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{6 \sqrt{c} \left (\sqrt{2} c^{13/4} d^{13}+4 \sqrt [4]{a} c^3 e d^{12}+2 \sqrt{2} \sqrt{a} c^{11/4} e^2 d^{11}+9 \sqrt{2} a c^{9/4} e^4 d^9+44 a^{5/4} c^2 e^5 d^8+36 \sqrt{2} a^{3/2} c^{7/4} e^6 d^7-49 \sqrt{2} a^2 c^{5/4} e^8 d^5-84 a^{9/4} c e^9 d^4-30 \sqrt{2} a^{5/2} c^{3/4} e^{10} d^3+7 \sqrt{2} a^3 \sqrt [4]{c} e^{12} d+4 a^{13/4} e^{13}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac{3 \sqrt{2} c^{3/4} \left (c^3 d^{13}-2 \sqrt{a} c^{5/2} e^2 d^{11}+9 a c^2 e^4 d^9-36 a^{3/2} c^{3/2} e^6 d^7-49 a^2 c e^8 d^5+30 a^{5/2} \sqrt{c} e^{10} d^3+7 a^3 e^{12} d\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{7/4}}+\frac{3 \sqrt{2} c^{3/4} \left (c^3 d^{13}-2 \sqrt{a} c^{5/2} e^2 d^{11}+9 a c^2 e^4 d^9-36 a^{3/2} c^{3/2} e^6 d^7-49 a^2 c e^8 d^5+30 a^{5/2} \sqrt{c} e^{10} d^3+7 a^3 e^{12} d\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{7/4}}}{32 \left (c d^4+a e^4\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-16*e^7*(c*d^4 + a*e^4)^2)/(d + e*x)^2 - (256*c*d^3*e^7*(c*d^4 + a*e^4))/(d + e*x) + (8*c*(c*d^4 + a*e^4)*(-
(a^2*e^7*(6*d^2 - 3*d*e*x + e^2*x^2)) + c^2*d^7*x*(d^2 - 3*d*e*x + 6*e^2*x^2) + 2*a*c*d^3*e^3*(5*d^3 - 6*d^2*e
*x + 6*d*e^2*x^2 - 5*e^3*x^3)))/(a*(a + c*x^4)) - (6*Sqrt[c]*(Sqrt[2]*c^(13/4)*d^13 - 4*a^(1/4)*c^3*d^12*e + 2
*Sqrt[2]*Sqrt[a]*c^(11/4)*d^11*e^2 + 9*Sqrt[2]*a*c^(9/4)*d^9*e^4 - 44*a^(5/4)*c^2*d^8*e^5 + 36*Sqrt[2]*a^(3/2)
*c^(7/4)*d^7*e^6 - 49*Sqrt[2]*a^2*c^(5/4)*d^5*e^8 + 84*a^(9/4)*c*d^4*e^9 - 30*Sqrt[2]*a^(5/2)*c^(3/4)*d^3*e^10
+ 7*Sqrt[2]*a^3*c^(1/4)*d*e^12 - 4*a^(13/4)*e^13)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) + (6*Sqrt[
c]*(Sqrt[2]*c^(13/4)*d^13 + 4*a^(1/4)*c^3*d^12*e + 2*Sqrt[2]*Sqrt[a]*c^(11/4)*d^11*e^2 + 9*Sqrt[2]*a*c^(9/4)*d
^9*e^4 + 44*a^(5/4)*c^2*d^8*e^5 + 36*Sqrt[2]*a^(3/2)*c^(7/4)*d^7*e^6 - 49*Sqrt[2]*a^2*c^(5/4)*d^5*e^8 - 84*a^(
9/4)*c*d^4*e^9 - 30*Sqrt[2]*a^(5/2)*c^(3/4)*d^3*e^10 + 7*Sqrt[2]*a^3*c^(1/4)*d*e^12 + 4*a^(13/4)*e^13)*ArcTan[
1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) + 384*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[d + e*x] - (3*Sqrt[2]*c^(3/4)*
(c^3*d^13 - 2*Sqrt[a]*c^(5/2)*d^11*e^2 + 9*a*c^2*d^9*e^4 - 36*a^(3/2)*c^(3/2)*d^7*e^6 - 49*a^2*c*d^5*e^8 + 30*
a^(5/2)*Sqrt[c]*d^3*e^10 + 7*a^3*d*e^12)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (3*
Sqrt[2]*c^(3/4)*(c^3*d^13 - 2*Sqrt[a]*c^(5/2)*d^11*e^2 + 9*a*c^2*d^9*e^4 - 36*a^(3/2)*c^(3/2)*d^7*e^6 - 49*a^2
*c*d^5*e^8 + 30*a^(5/2)*Sqrt[c]*d^3*e^10 + 7*a^3*d*e^12)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2
])/a^(7/4) - 96*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[a + c*x^4])/(32*(c*d^4 + a*e^4)^4)

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 2121, 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)^3/(c*x^4+a)^2,x)

[Out]

36*e^7*c^2*d^6/(a*e^4+c*d^4)^4*ln(e*x+d)+5/2*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)*d^10*e^3-9*c^2/(a*e^4+c*d^4)^4*ln(c
*x^4+a)*d^6*e^7-33/4*c^3/(a*e^4+c*d^4)^4/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^8*e^5+3*c/(a*e^4+c*d^4)^4*a*l
n(c*x^4+a)*d^2*e^11-3/4*c/(a*e^4+c*d^4)^4*a^2/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*e^13-12*e^11*c*d^2/(a*e^4+
c*d^4)^4*ln(e*x+d)*a-c^3/(a*e^4+c*d^4)^4/(c*x^4+a)*d^7*e^6*x^3+9/4*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)*e^5*x^2*d^8-1
1/4*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)*d^9*x*e^4+1/4*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)*d^13/a*x+c^2/(a*e^4+c*d^4)^4/(c*
x^4+a)*a*d^6*e^7-1/4*c/(a*e^4+c*d^4)^4/(c*x^4+a)*e^13*a^2*x^2-3/2*c/(a*e^4+c*d^4)^4/(c*x^4+a)*a^2*d^2*e^11-8*c
*d^3*e^7/(a*e^4+c*d^4)^3/(e*x+d)+63/4*c^2/(a*e^4+c*d^4)^4*a/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^4*e^9+3/16
*c^4/(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^13+3/16*c^4/(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^13+3/32*c^4/(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^13-3/4*c^4/(a*e^4+c*d^4)^4/
a/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^12*e+27/8*c^2/(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^7*e^6+27/4*c^2/(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^7*e^6+27/4*c^2/(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^7*e^6-147/16*c^2/(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^5*e^8
-147/16*c^2/(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^5*e^8-147/32*c^2/(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
^5*e^8+3/4*c/(a*e^4+c*d^4)^4/(c*x^4+a)*d*a^2*x*e^12+3/2*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)*d^11*e^2/a*x^3+11/4*c^2/
(a*e^4+c*d^4)^4/(c*x^4+a)*e^9*a*x^2*d^4-3/4*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)*e/a*x^2*d^12-9/4*c^2/(a*e^4+c*d^4)^4
/(c*x^4+a)*d^5*a*x*e^8-5/2*c^2/(a*e^4+c*d^4)^4/(c*x^4+a)*d^3*e^10*a*x^3+3/8*c^3/(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^11*e^2+3/8*c^3/(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^11*e^2+27/16*c^3/(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^9*
e^4+27/16*c^3/(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^9*e^4+21/16*c/(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*e^12+21/16*c/(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*e^12+21/32*c/(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*e^12-45/16*c/(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^3*e^10-1/2*e^7/(a*e
^4+c*d^4)^2/(e*x+d)^2-45/8*c/(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^3*e^10-45
/8*c/(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^3*e^10+27/32*c^3/(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^9*
e^4+3/16*c^3/(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^11*e^2

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.70529, size = 1993, normalized size = 1.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(2*sqrt(2)*sqrt(a*c)*c^3*d^4*e + (a*c^3)^(1/4)*c^3*d^5 + 4*sqrt(2)*a*c^3*d^2*e^3 + 2*(a*c^3)^(3/4)*c*d^3*e
^2 - 9*(a*c^3)^(1/4)*a*c^2*d*e^4 + 2*sqrt(2)*sqrt(a*c)*a*c^2*e^5)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4
))/(a/c)^(1/4))/(sqrt(2)*a^2*c^4*d^8 - 8*(a*c^3)^(1/4)*a^2*c^3*d^7*e + 16*sqrt(2)*sqrt(a*c)*a^2*c^3*d^6*e^2 +
34*sqrt(2)*a^3*c^3*d^4*e^4 - 40*(a*c^3)^(3/4)*a^2*c*d^5*e^3 - 40*(a*c^3)^(1/4)*a^3*c^2*d^3*e^5 + 16*sqrt(2)*sq
rt(a*c)*a^3*c^2*d^2*e^6 + sqrt(2)*a^4*c^2*e^8 - 8*(a*c^3)^(3/4)*a^3*d*e^7) + 3/8*(2*sqrt(2)*sqrt(a*c)*c^3*d^4*
e + (a*c^3)^(1/4)*c^3*d^5 - 4*sqrt(2)*a*c^3*d^2*e^3 + 2*(a*c^3)^(3/4)*c*d^3*e^2 - 9*(a*c^3)^(1/4)*a*c^2*d*e^4
+ 2*sqrt(2)*sqrt(a*c)*a*c^2*e^5)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^4*
d^8 + 8*(a*c^3)^(1/4)*a^2*c^3*d^7*e + 16*sqrt(2)*sqrt(a*c)*a^2*c^3*d^6*e^2 + 34*sqrt(2)*a^3*c^3*d^4*e^4 + 40*(
a*c^3)^(3/4)*a^2*c*d^5*e^3 + 40*(a*c^3)^(1/4)*a^3*c^2*d^3*e^5 + 16*sqrt(2)*sqrt(a*c)*a^3*c^2*d^2*e^6 + sqrt(2)
*a^4*c^2*e^8 + 8*(a*c^3)^(3/4)*a^3*d*e^7) + 3/16*((a*c^3)^(1/4)*c^4*d^13 - 2*(a*c^3)^(3/4)*c^2*d^11*e^2 + 9*(a
*c^3)^(1/4)*a*c^3*d^9*e^4 - 36*(a*c^3)^(3/4)*a*c*d^7*e^6 - 49*(a*c^3)^(1/4)*a^2*c^2*d^5*e^8 + 30*(a*c^3)^(3/4)
*a^2*d^3*e^10 + 7*(a*c^3)^(1/4)*a^3*c*d*e^12)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a^2*c^5*d^
16 + 4*sqrt(2)*a^3*c^4*d^12*e^4 + 6*sqrt(2)*a^4*c^3*d^8*e^8 + 4*sqrt(2)*a^5*c^2*d^4*e^12 + sqrt(2)*a^6*c*e^16)
- 3/16*((a*c^3)^(1/4)*c^4*d^13 - 2*(a*c^3)^(3/4)*c^2*d^11*e^2 + 9*(a*c^3)^(1/4)*a*c^3*d^9*e^4 - 36*(a*c^3)^(3
/4)*a*c*d^7*e^6 - 49*(a*c^3)^(1/4)*a^2*c^2*d^5*e^8 + 30*(a*c^3)^(3/4)*a^2*d^3*e^10 + 7*(a*c^3)^(1/4)*a^3*c*d*e
^12)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a^2*c^5*d^16 + 4*sqrt(2)*a^3*c^4*d^12*e^4 + 6*sqrt(
2)*a^4*c^3*d^8*e^8 + 4*sqrt(2)*a^5*c^2*d^4*e^12 + sqrt(2)*a^6*c*e^16) - 3*(3*c^2*d^6*e^7 - a*c*d^2*e^11)*log(a
bs(c*x^4 + a))/(c^4*d^16 + 4*a*c^3*d^12*e^4 + 6*a^2*c^2*d^8*e^8 + 4*a^3*c*d^4*e^12 + a^4*e^16) + 12*(3*c^2*d^6
*e^8 - a*c*d^2*e^12)*log(abs(x*e + d))/(c^4*d^16*e + 4*a*c^3*d^12*e^5 + 6*a^2*c^2*d^8*e^9 + 4*a^3*c*d^4*e^13 +
a^4*e^17) + 1/4*(10*a*c^3*d^12*e^3 - 30*a^2*c^2*d^8*e^7 - 42*a^3*c*d^4*e^11 + 6*(c^4*d^11*e^4 - 6*a*c^3*d^7*e
^8 - 7*a^2*c^2*d^3*e^12)*x^5 + 3*(3*c^4*d^12*e^3 - 11*a*c^3*d^8*e^7 - 15*a^2*c^2*d^4*e^11 - a^3*c*e^15)*x^4 -
2*a^4*e^15 + (c^4*d^13*e^2 + 3*a*c^3*d^9*e^6 + 3*a^2*c^2*d^5*e^10 + a^3*c*d*e^14)*x^3 - (c^4*d^14*e + 3*a*c^3*
d^10*e^5 + 3*a^2*c^2*d^6*e^9 + a^3*c*d^2*e^13)*x^2 + (c^4*d^15 + 9*a*c^3*d^11*e^4 - 33*a^2*c^2*d^7*e^8 - 41*a^
3*c*d^3*e^12)*x)/((c*d^4 + a*e^4)^4*(c*x^4 + a)*(x*e + d)^2*a)