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/(a*e^4+c*d^4)^3/(e*x+d)+1/32*c*x*(7*d^2*(-3*a*e^4+c*d^4)-12*d*e*(-a*e^4+c*d^4)*x+5*e^2*(-a*e^4+3*c*d^4)*
x^2)/a^2/(a*e^4+c*d^4)^2/(c*x^4+a)+1/8*c*(4*a*d^3*e^3+x*(d^2*(-3*a*e^4+c*d^4)-2*d*e*(-a*e^4+c*d^4)*x+e^2*(-a*e
^4+3*c*d^4)*x^2))/a/(a*e^4+c*d^4)^2/(c*x^4+a)^2+1/4*c*e^4*(8*a*d^3*e^3+x*(d^2*(-3*a*e^4+5*c*d^4)-2*d*e*(-a*e^4
+3*c*d^4)*x+e^2*(-a*e^4+7*c*d^4)*x^2))/a/(a*e^4+c*d^4)^3/(c*x^4+a)+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)^4-1/2*d*e^5*(-a*e^4+3*c*d^4)*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/a^(3/2)/(
a*e^4+c*d^4)^3-3/8*d*e*(-a*e^4+c*d^4)*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/a^(5/2)/(a*e^4+c*d^4)^2-d*e^9*(-a*e^
4+5*c*d^4)*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/(a*e^4+c*d^4)^4/a^(1/2)-1/8*c^(1/4)*e^8*ln(-a^(1/4)*c^(1/4)*x*2
^(1/2)+a^(1/2)+x^2*c^(1/2))*(9*c^(3/2)*d^6+a^(3/2)*e^6-11*c*d^4*e^2*a^(1/2)-3*a*d^2*e^4*c^(1/2))/a^(3/4)/(a*e^
4+c*d^4)^4*2^(1/2)+1/8*c^(1/4)*e^8*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(9*c^(3/2)*d^6+a^(3/2)*e^
6-11*c*d^4*e^2*a^(1/2)-3*a*d^2*e^4*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^4*2^(1/2)-1/256*c^(1/4)*ln(-a^(1/4)*c^(1/4)*
x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-5*e^2*(-a*e^4+3*c*d^4)*a^(1/2)+21*d^2*(-3*a*e^4+c*d^4)*c^(1/2))/a^(11/4)/(a*e
^4+c*d^4)^2*2^(1/2)+1/256*c^(1/4)*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-5*e^2*(-a*e^4+3*c*d^4)*a
^(1/2)+21*d^2*(-3*a*e^4+c*d^4)*c^(1/2))/a^(11/4)/(a*e^4+c*d^4)^2*2^(1/2)+1/128*c^(1/4)*arctan(-1+c^(1/4)*x*2^(
1/2)/a^(1/4))*(5*e^2*(-a*e^4+3*c*d^4)*a^(1/2)+21*d^2*(-3*a*e^4+c*d^4)*c^(1/2))/a^(11/4)/(a*e^4+c*d^4)^2*2^(1/2
)+1/128*c^(1/4)*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(5*e^2*(-a*e^4+3*c*d^4)*a^(1/2)+21*d^2*(-3*a*e^4+c*d^4)*c^
(1/2))/a^(11/4)/(a*e^4+c*d^4)^2*2^(1/2)-1/32*c^(1/4)*e^4*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-
e^2*(-a*e^4+7*c*d^4)*a^(1/2)+3*d^2*(-3*a*e^4+5*c*d^4)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/32*c^(1/4)*e^
4*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e^2*(-a*e^4+7*c*d^4)*a^(1/2)+3*d^2*(-3*a*e^4+5*c*d^4)*c^
(1/2))/a^(7/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/16*c^(1/4)*e^4*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*(-a*e^4+7*c*
d^4)*a^(1/2)+3*d^2*(-3*a*e^4+5*c*d^4)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/16*c^(1/4)*e^4*arctan(1+c^(1/
4)*x*2^(1/2)/a^(1/4))*(e^2*(-a*e^4+7*c*d^4)*a^(1/2)+3*d^2*(-3*a*e^4+5*c*d^4)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^3*
2^(1/2)+1/4*c^(1/4)*e^8*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*(-a*e^4+11*c*d^4)*a^(1/2)+3*d^2*(-a*e^4+3*c*
d^4)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^4*2^(1/2)+1/4*c^(1/4)*e^8*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*(-a*e^4
+11*c*d^4)*a^(1/2)+3*d^2*(-a*e^4+3*c*d^4)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^4*2^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.78, antiderivative size = 1830, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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 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 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 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 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 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 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 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 6742

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

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*}

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Mathematica [A]  time = 1.55, 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)

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [A]  time = 0.03, size = 2769, normalized size = 1.51 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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-e^11/(a*e^4+c*d^4)^3/(e*x+d)+
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+231/256*c^2/(a*e^4+c*d^4)^4/a*(a/c)^(1
/4)*2^(1/2)*ln((x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*d^6*e^8+1/2*c^
3/(a*e^4+c*d^4)^4/(c*x^4+a)^2*e^3*d^11+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)*
2^(1/2)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*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+99/128*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+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)*2^(1/2
)*x+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*d^8*e^6+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+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+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+15/256*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)*2^(1/2)*x+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*2^(1/2)*x+(
a/c)^(1/2)))*d^12*e^2+5/2*c/(a*e^4+c*d^4)^4/(c*x^4+a)^2*e^11*d^3*a^2-17/32*c/(a*e^4+c*d^4)^4/(c*x^4+a)^2*e^14*
a^2*x^3+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^7-13/32*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+3*c^2/(a*e^4+c*d^4)^
4/(c*x^4+a)^2*e^7*d^7*a+101/32*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+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^8-45/8*c^2
/(a*e^4+c*d^4)^4/(a*c)^(1/2)*arctan((1/a*c)^(1/2)*x^2)*e^9*d^5-45/256/(a*e^4+c*d^4)^4*a/(a/c)^(1/4)*2^(1/2)*ln
((x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*2^(1/2)*x+(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+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^12-15/8*c^3/(a*e^4+c*d^4)^4/a/(a*c)^(1/2)*arctan((1/a*c)^(1/2)*x^2)*e^5*d^9-3/8*c^4/(a*e^4+c
*d^4)^4/a^2/(a*c)^(1/2)*arctan((1/a*c)^(1/2)*x^2)*e*d^13+405/256*c/(a*e^4+c*d^4)^4/(a/c)^(1/4)*2^(1/2)*ln((x^2
-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*2^(1/2)*x+(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+405/128*c/(a*e^4+c*d^4)^4/(a/c)^(1/4)*2^(1/2)*a
rctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^4*e^10+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+2*c^2/(a*e^4+c*d^4)^4/(c*x^4+a)^2*x^4*a*d^3*e^11+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^12+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^6-3/8*c^5/(a*e^4+c*d^4)^4/(c*x^4+a)^2*d^13*e/a^2*x^6-4
5/32*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+15/8*c/(a*e^4+c*d^4)^4*a/(a*c)^(1/2)*arctan((1/a*c)^(1/2)*x^2)
*e^13*d-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)*2^(1/2)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*2^(1/2)*x+(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)*2^(1/2)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*2^(1/2)
*x+(a/c)^(1/2)))*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+2
1/128*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

________________________________________________________________________________________

maxima [A]  time = 3.10, size = 1564, normalized size = 0.85 \[ \text {result too large to display} \]

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]

12*c*d^3*e^11*log(e*x + d)/(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) - 3
/256*c*(sqrt(2)*(128*sqrt(2)*a^(11/4)*c^(5/4)*d^3*e^11 - 7*c^4*d^14 + 5*sqrt(a)*c^(7/2)*d^12*e^2 - 33*a*c^3*d^
10*e^4 + 27*a^(3/2)*c^(5/2)*d^8*e^6 - 77*a^2*c^2*d^6*e^8 + 135*a^(5/2)*c^(3/2)*d^4*e^10 + 77*a^3*c*d^2*e^12 -
15*a^(7/2)*sqrt(c)*e^14)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)*(1
28*sqrt(2)*a^(11/4)*c^(5/4)*d^3*e^11 + 7*c^4*d^14 - 5*sqrt(a)*c^(7/2)*d^12*e^2 + 33*a*c^3*d^10*e^4 - 27*a^(3/2
)*c^(5/2)*d^8*e^6 + 77*a^2*c^2*d^6*e^8 - 135*a^(5/2)*c^(3/2)*d^4*e^10 - 77*a^3*c*d^2*e^12 + 15*a^(7/2)*sqrt(c)
*e^14)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) - 2*(7*sqrt(2)*a^(1/4)*c^(17/4
)*d^14 + 5*sqrt(2)*a^(3/4)*c^(15/4)*d^12*e^2 + 33*sqrt(2)*a^(5/4)*c^(13/4)*d^10*e^4 + 27*sqrt(2)*a^(7/4)*c^(11
/4)*d^8*e^6 + 77*sqrt(2)*a^(9/4)*c^(9/4)*d^6*e^8 + 135*sqrt(2)*a^(11/4)*c^(7/4)*d^4*e^10 - 77*sqrt(2)*a^(13/4)
*c^(5/4)*d^2*e^12 - 15*sqrt(2)*a^(15/4)*c^(3/4)*e^14 + 16*sqrt(a)*c^4*d^13*e + 80*a^(3/2)*c^3*d^9*e^5 + 240*a^
(5/2)*c^2*d^5*e^9 - 80*a^(7/2)*c*d*e^13)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(
a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(5/4)) - 2*(7*sqrt(2)*a^(1/4)*c^(17/4)*d^14 + 5*sqrt(2)*a^(3/4)*
c^(15/4)*d^12*e^2 + 33*sqrt(2)*a^(5/4)*c^(13/4)*d^10*e^4 + 27*sqrt(2)*a^(7/4)*c^(11/4)*d^8*e^6 + 77*sqrt(2)*a^
(9/4)*c^(9/4)*d^6*e^8 + 135*sqrt(2)*a^(11/4)*c^(7/4)*d^4*e^10 - 77*sqrt(2)*a^(13/4)*c^(5/4)*d^2*e^12 - 15*sqrt
(2)*a^(15/4)*c^(3/4)*e^14 - 16*sqrt(a)*c^4*d^13*e - 80*a^(3/2)*c^3*d^9*e^5 - 240*a^(5/2)*c^2*d^5*e^9 + 80*a^(7
/2)*c*d*e^13)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(
sqrt(a)*sqrt(c))*c^(5/4)))/(a^2*c^4*d^16 + 4*a^3*c^3*d^12*e^4 + 6*a^4*c^2*d^8*e^8 + 4*a^5*c*d^4*e^12 + a^6*e^1
6) + 1/32*(16*a^2*c^2*d^8*e^3 + 80*a^3*c*d^4*e^7 - 32*a^4*e^11 + 3*(5*c^4*d^8*e^3 + 22*a*c^3*d^4*e^7 - 15*a^2*
c^2*e^11)*x^8 + 3*(c^4*d^9*e^2 + 6*a*c^3*d^5*e^6 + 5*a^2*c^2*d*e^10)*x^7 - (5*c^4*d^10*e + 22*a*c^3*d^6*e^5 +
17*a^2*c^2*d^2*e^9)*x^6 + (7*c^4*d^11 + 26*a*c^3*d^7*e^4 + 19*a^2*c^2*d^3*e^8)*x^5 + 3*(9*a*c^3*d^8*e^3 + 46*a
^2*c^2*d^4*e^7 - 27*a^3*c*e^11)*x^4 + (7*a*c^3*d^9*e^2 + 26*a^2*c^2*d^5*e^6 + 19*a^3*c*d*e^10)*x^3 - 3*(3*a*c^
3*d^10*e + 10*a^2*c^2*d^6*e^5 + 7*a^3*c*d^2*e^9)*x^2 + (11*a*c^3*d^11 + 34*a^2*c^2*d^7*e^4 + 23*a^3*c*d^3*e^8)
*x)/(a^4*c^3*d^13 + 3*a^5*c^2*d^9*e^4 + 3*a^6*c*d^5*e^8 + a^7*d*e^12 + (a^2*c^5*d^12*e + 3*a^3*c^4*d^8*e^5 + 3
*a^4*c^3*d^4*e^9 + a^5*c^2*e^13)*x^9 + (a^2*c^5*d^13 + 3*a^3*c^4*d^9*e^4 + 3*a^4*c^3*d^5*e^8 + a^5*c^2*d*e^12)
*x^8 + 2*(a^3*c^4*d^12*e + 3*a^4*c^3*d^8*e^5 + 3*a^5*c^2*d^4*e^9 + a^6*c*e^13)*x^5 + 2*(a^3*c^4*d^13 + 3*a^4*c
^3*d^9*e^4 + 3*a^5*c^2*d^5*e^8 + a^6*c*d*e^12)*x^4 + (a^4*c^3*d^12*e + 3*a^5*c^2*d^8*e^5 + 3*a^6*c*d^4*e^9 + a
^7*e^13)*x)

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mupad [B]  time = 5.75, size = 3572, normalized size = 1.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^4)^3*(d + e*x)^2),x)

[Out]

symsum(log((194481*c^9*d^17*e^6 + 1527012*a*c^8*d^13*e^10 + 4100625*a^4*c^5*d*e^22 + 1926342*a^2*c^7*d^9*e^14
- 3102300*a^3*c^6*d^5*e^18)/(1048576*(a^14*e^24 + a^8*c^6*d^24 + 6*a^13*c*d^4*e^20 + 6*a^9*c^5*d^20*e^4 + 15*a
^10*c^4*d^16*e^8 + 20*a^11*c^3*d^12*e^12 + 15*a^12*c^2*d^8*e^16)) + root(1610612736*a^13*c^2*d^8*e^8*z^4 + 107
3741824*a^12*c^3*d^12*e^4*z^4 + 1073741824*a^14*c*d^4*e^12*z^4 + 268435456*a^11*c^4*d^16*z^4 + 268435456*a^15*
e^16*z^4 + 3221225472*a^11*c*d^3*e^11*z^3 + 239468544*a^7*c^2*d^6*e^6*z^2 + 39518208*a^6*c^3*d^10*e^2*z^2 + 11
53105920*a^8*c*d^2*e^10*z^2 + 32071680*a^4*c^2*d^5*e^5*z + 5419008*a^3*c^3*d^9*e*z + 124416000*a^5*c*d*e^9*z +
1138050*a*c^2*d^4*e^4 + 4100625*a^2*c*e^8 + 194481*c^3*d^8, z, k)*(root(1610612736*a^13*c^2*d^8*e^8*z^4 + 107
3741824*a^12*c^3*d^12*e^4*z^4 + 1073741824*a^14*c*d^4*e^12*z^4 + 268435456*a^11*c^4*d^16*z^4 + 268435456*a^15*
e^16*z^4 + 3221225472*a^11*c*d^3*e^11*z^3 + 239468544*a^7*c^2*d^6*e^6*z^2 + 39518208*a^6*c^3*d^10*e^2*z^2 + 11
53105920*a^8*c*d^2*e^10*z^2 + 32071680*a^4*c^2*d^5*e^5*z + 5419008*a^3*c^3*d^9*e*z + 124416000*a^5*c*d*e^9*z +
1138050*a*c^2*d^4*e^4 + 4100625*a^2*c*e^8 + 194481*c^3*d^8, z, k)*(root(1610612736*a^13*c^2*d^8*e^8*z^4 + 107
3741824*a^12*c^3*d^12*e^4*z^4 + 1073741824*a^14*c*d^4*e^12*z^4 + 268435456*a^11*c^4*d^16*z^4 + 268435456*a^15*
e^16*z^4 + 3221225472*a^11*c*d^3*e^11*z^3 + 239468544*a^7*c^2*d^6*e^6*z^2 + 39518208*a^6*c^3*d^10*e^2*z^2 + 11
53105920*a^8*c*d^2*e^10*z^2 + 32071680*a^4*c^2*d^5*e^5*z + 5419008*a^3*c^3*d^9*e*z + 124416000*a^5*c*d*e^9*z +
1138050*a*c^2*d^4*e^4 + 4100625*a^2*c*e^8 + 194481*c^3*d^8, z, k)*((23592960*a^14*c^4*e^29 - 11010048*a^7*c^1
1*d^28*e + 33030144*a^8*c^10*d^24*e^5 + 504889344*a^9*c^9*d^20*e^9 + 3103260672*a^10*c^8*d^16*e^13 + 679949107
2*a^11*c^7*d^12*e^17 + 6101139456*a^12*c^6*d^8*e^21 + 1967652864*a^13*c^5*d^4*e^25)/(1048576*(a^14*e^24 + a^8*
c^6*d^24 + 6*a^13*c*d^4*e^20 + 6*a^9*c^5*d^20*e^4 + 15*a^10*c^4*d^16*e^8 + 20*a^11*c^3*d^12*e^12 + 15*a^12*c^2
*d^8*e^16)) + root(1610612736*a^13*c^2*d^8*e^8*z^4 + 1073741824*a^12*c^3*d^12*e^4*z^4 + 1073741824*a^14*c*d^4*
e^12*z^4 + 268435456*a^11*c^4*d^16*z^4 + 268435456*a^15*e^16*z^4 + 3221225472*a^11*c*d^3*e^11*z^3 + 239468544*
a^7*c^2*d^6*e^6*z^2 + 39518208*a^6*c^3*d^10*e^2*z^2 + 1153105920*a^8*c*d^2*e^10*z^2 + 32071680*a^4*c^2*d^5*e^5
*z + 5419008*a^3*c^3*d^9*e*z + 124416000*a^5*c*d*e^9*z + 1138050*a*c^2*d^4*e^4 + 4100625*a^2*c*e^8 + 194481*c^
3*d^8, z, k)*((402653184*a^17*c^4*d*e^30 - 134217728*a^10*c^11*d^29*e^2 - 402653184*a^11*c^10*d^25*e^6 + 40265
3184*a^12*c^9*d^21*e^10 + 3355443200*a^13*c^8*d^17*e^14 + 6039797760*a^14*c^7*d^13*e^18 + 5234491392*a^15*c^6*
d^9*e^22 + 2281701376*a^16*c^5*d^5*e^26)/(1048576*(a^14*e^24 + a^8*c^6*d^24 + 6*a^13*c*d^4*e^20 + 6*a^9*c^5*d^
20*e^4 + 15*a^10*c^4*d^16*e^8 + 20*a^11*c^3*d^12*e^12 + 15*a^12*c^2*d^8*e^16)) + (x*(335544320*a^17*c^4*e^31 -
201326592*a^10*c^11*d^28*e^3 - 872415232*a^11*c^10*d^24*e^7 - 1006632960*a^12*c^9*d^20*e^11 + 1006632960*a^13
*c^8*d^16*e^15 + 3690987520*a^14*c^7*d^12*e^19 + 3825205248*a^15*c^6*d^8*e^23 + 1811939328*a^16*c^5*d^4*e^27))
/(1048576*(a^14*e^24 + a^8*c^6*d^24 + 6*a^13*c*d^4*e^20 + 6*a^9*c^5*d^20*e^4 + 15*a^10*c^4*d^16*e^8 + 20*a^11*
c^3*d^12*e^12 + 15*a^12*c^2*d^8*e^16))) + (x*(2554331136*a^10*c^8*d^15*e^14 - 144703488*a^8*c^10*d^23*e^6 - 15
4140672*a^9*c^9*d^19*e^10 - 34603008*a^7*c^11*d^27*e^2 + 7659847680*a^11*c^7*d^11*e^18 + 7556038656*a^12*c^6*d
^7*e^22 + 2494562304*a^13*c^5*d^3*e^26))/(1048576*(a^14*e^24 + a^8*c^6*d^24 + 6*a^13*c*d^4*e^20 + 6*a^9*c^5*d^
20*e^4 + 15*a^10*c^4*d^16*e^8 + 20*a^11*c^3*d^12*e^12 + 15*a^12*c^2*d^8*e^16))) + (12681216*a^5*c^10*d^23*e^4
+ 127107072*a^6*c^9*d^19*e^8 + 674168832*a^7*c^8*d^15*e^12 + 1018626048*a^8*c^7*d^11*e^16 - 446201856*a^9*c^6*
d^7*e^20 + 906854400*a^10*c^5*d^3*e^24)/(1048576*(a^14*e^24 + a^8*c^6*d^24 + 6*a^13*c*d^4*e^20 + 6*a^9*c^5*d^2
0*e^4 + 15*a^10*c^4*d^16*e^8 + 20*a^11*c^3*d^12*e^12 + 15*a^12*c^2*d^8*e^16)) + (x*(516096*a^5*c^10*d^22*e^5 -
1806336*a^4*c^11*d^26*e + 90427392*a^6*c^9*d^18*e^9 + 896090112*a^7*c^8*d^14*e^13 + 1960906752*a^8*c^7*d^10*e
^17 + 1732829184*a^9*c^6*d^6*e^21 + 1183887360*a^10*c^5*d^2*e^25))/(1048576*(a^14*e^24 + a^8*c^6*d^24 + 6*a^13
*c*d^4*e^20 + 6*a^9*c^5*d^20*e^4 + 15*a^10*c^4*d^16*e^8 + 20*a^11*c^3*d^12*e^12 + 15*a^12*c^2*d^8*e^16))) + (3
87072*a^2*c^10*d^22*e^3 + 8004096*a^3*c^9*d^18*e^7 + 49379328*a^4*c^8*d^14*e^11 + 49572864*a^5*c^7*d^10*e^15 -
156930048*a^6*c^6*d^6*e^19 + 125452800*a^7*c^5*d^2*e^23)/(1048576*(a^14*e^24 + a^8*c^6*d^24 + 6*a^13*c*d^4*e^
20 + 6*a^9*c^5*d^20*e^4 + 15*a^10*c^4*d^16*e^8 + 20*a^11*c^3*d^12*e^12 + 15*a^12*c^2*d^8*e^16)) + (x*(12636000
0*a^7*c^5*d*e^24 + 561600*a^2*c^10*d^21*e^4 + 9609408*a^3*c^9*d^17*e^8 + 75731328*a^4*c^8*d^13*e^12 + 11499148
8*a^5*c^7*d^9*e^16 - 80136000*a^6*c^6*d^5*e^20))/(1048576*(a^14*e^24 + a^8*c^6*d^24 + 6*a^13*c*d^4*e^20 + 6*a^
9*c^5*d^20*e^4 + 15*a^10*c^4*d^16*e^8 + 20*a^11*c^3*d^12*e^12 + 15*a^12*c^2*d^8*e^16))) + (x*(4100625*a^4*c^5*
e^23 + 194481*c^9*d^16*e^7 + 1527012*a*c^8*d^12*e^11 - 167994*a^2*c^7*d^8*e^15 - 13988700*a^3*c^6*d^4*e^19))/(
1048576*(a^14*e^24 + a^8*c^6*d^24 + 6*a^13*c*d^4*e^20 + 6*a^9*c^5*d^20*e^4 + 15*a^10*c^4*d^16*e^8 + 20*a^11*c^
3*d^12*e^12 + 15*a^12*c^2*d^8*e^16)))*root(1610612736*a^13*c^2*d^8*e^8*z^4 + 1073741824*a^12*c^3*d^12*e^4*z^4
+ 1073741824*a^14*c*d^4*e^12*z^4 + 268435456*a^11*c^4*d^16*z^4 + 268435456*a^15*e^16*z^4 + 3221225472*a^11*c*d
^3*e^11*z^3 + 239468544*a^7*c^2*d^6*e^6*z^2 + 39518208*a^6*c^3*d^10*e^2*z^2 + 1153105920*a^8*c*d^2*e^10*z^2 +
32071680*a^4*c^2*d^5*e^5*z + 5419008*a^3*c^3*d^9*e*z + 124416000*a^5*c*d*e^9*z + 1138050*a*c^2*d^4*e^4 + 41006
25*a^2*c*e^8 + 194481*c^3*d^8, z, k), k, 1, 4) + ((c^2*d^8*e^3 - 2*a^2*e^11 + 5*a*c*d^4*e^7)/(2*(a*e^4 + c*d^4
)*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) + (3*x^8*(5*c^4*d^8*e^3 - 15*a^2*c^2*e^11 + 22*a*c^3*d^4*e^7))/(32*a^2*
(a^3*e^12 + c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8)) + (x^5*(7*c^3*d^7 + 19*a*c^2*d^3*e^4))/(32*a^2*(a^2
*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) - (3*x^2*(3*c^2*d^6*e + 7*a*c*d^2*e^5))/(32*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*
e^4)) + (x^3*(7*c^2*d^5*e^2 + 19*a*c*d*e^6))/(32*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) + (x*(11*c^2*d^7 + 23*
a*c*d^3*e^4))/(32*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) - (x^6*(5*c^3*d^6*e + 17*a*c^2*d^2*e^5))/(32*a^2*(a^2
*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) + (3*e^2*x^7*(c^3*d^5 + 5*a*c^2*d*e^4))/(32*a^2*(a^2*e^8 + c^2*d^8 + 2*a*c*d^
4*e^4)) + (3*e^2*x^4*(9*c^3*d^8*e - 27*a^2*c*e^9 + 46*a*c^2*d^4*e^5))/(32*a*(a*e^4 + c*d^4)*(a^2*e^8 + c^2*d^8
+ 2*a*c*d^4*e^4)))/(a^2*d + c^2*d*x^8 + c^2*e*x^9 + a^2*e*x + 2*a*c*d*x^4 + 2*a*c*e*x^5) + (12*c*d^3*e^11*log
(d + e*x))/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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