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

Optimal. Leaf size=1352 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.41165, antiderivative size = 1352, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 15, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.882, Rules used = {6742, 1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 260} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(a + c*x^4)^3),x]

[Out]

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

Rule 6742

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

Rule 1854

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

Rule 1855

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

Rule 1876

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

Rule 275

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

Rule 205

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

Rule 1168

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

Rule 1162

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

Rule 617

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1248

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

Rule 635

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

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.762644, size = 835, normalized size = 0.62 \[ \frac{256 \log (d+e x) e^{11}-64 \log \left (c x^4+a\right ) e^{11}+\frac{32 \left (c d^4+a e^4\right )^2 \left (a e^3+c d x \left (d^2-e x d+e^2 x^2\right )\right )}{a \left (c x^4+a\right )^2}+\frac{8 \left (c d^4+a e^4\right ) \left (8 a^2 e^7+a c d x \left (15 d^2-14 e x d+13 e^2 x^2\right ) e^4+c^2 d^5 x \left (7 d^2-6 e x d+5 e^2 x^2\right )\right )}{a^2 \left (c x^4+a\right )}-\frac{2 \sqrt [4]{c} d \left (21 \sqrt{2} c^{5/2} d^{10}-24 \sqrt [4]{a} c^{9/4} e d^9+5 \sqrt{2} \sqrt{a} c^2 e^2 d^8+66 \sqrt{2} a c^{3/2} e^4 d^6-80 a^{5/4} c^{5/4} e^5 d^5+18 \sqrt{2} a^{3/2} c e^6 d^4+77 \sqrt{2} a^2 \sqrt{c} e^8 d^2-120 a^{9/4} \sqrt [4]{c} e^9 d+45 \sqrt{2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac{2 \sqrt [4]{c} d \left (21 \sqrt{2} c^{5/2} d^{10}+24 \sqrt [4]{a} c^{9/4} e d^9+5 \sqrt{2} \sqrt{a} c^2 e^2 d^8+66 \sqrt{2} a c^{3/2} e^4 d^6+80 a^{5/4} c^{5/4} e^5 d^5+18 \sqrt{2} a^{3/2} c e^6 d^4+77 \sqrt{2} a^2 \sqrt{c} e^8 d^2+120 a^{9/4} \sqrt [4]{c} e^9 d+45 \sqrt{2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{11/4}}+\frac{\sqrt{2} \sqrt [4]{c} \left (-21 c^{5/2} d^{11}+5 \sqrt{a} c^2 e^2 d^9-66 a c^{3/2} e^4 d^7+18 a^{3/2} c e^6 d^5-77 a^2 \sqrt{c} e^8 d^3+45 a^{5/2} e^{10} d\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{11/4}}+\frac{\sqrt{2} \sqrt [4]{c} \left (21 c^{5/2} d^{11}-5 \sqrt{a} c^2 e^2 d^9+66 a c^{3/2} e^4 d^7-18 a^{3/2} c e^6 d^5+77 a^2 \sqrt{c} e^8 d^3-45 a^{5/2} e^{10} d\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{11/4}}}{256 \left (c d^4+a e^4\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((32*(c*d^4 + a*e^4)^2*(a*e^3 + c*d*x*(d^2 - d*e*x + e^2*x^2)))/(a*(a + c*x^4)^2) + (8*(c*d^4 + a*e^4)*(8*a^2*
e^7 + c^2*d^5*x*(7*d^2 - 6*d*e*x + 5*e^2*x^2) + a*c*d*e^4*x*(15*d^2 - 14*d*e*x + 13*e^2*x^2)))/(a^2*(a + c*x^4
)) - (2*c^(1/4)*d*(21*Sqrt[2]*c^(5/2)*d^10 - 24*a^(1/4)*c^(9/4)*d^9*e + 5*Sqrt[2]*Sqrt[a]*c^2*d^8*e^2 + 66*Sqr
t[2]*a*c^(3/2)*d^6*e^4 - 80*a^(5/4)*c^(5/4)*d^5*e^5 + 18*Sqrt[2]*a^(3/2)*c*d^4*e^6 + 77*Sqrt[2]*a^2*Sqrt[c]*d^
2*e^8 - 120*a^(9/4)*c^(1/4)*d*e^9 + 45*Sqrt[2]*a^(5/2)*e^10)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4)
+ (2*c^(1/4)*d*(21*Sqrt[2]*c^(5/2)*d^10 + 24*a^(1/4)*c^(9/4)*d^9*e + 5*Sqrt[2]*Sqrt[a]*c^2*d^8*e^2 + 66*Sqrt[
2]*a*c^(3/2)*d^6*e^4 + 80*a^(5/4)*c^(5/4)*d^5*e^5 + 18*Sqrt[2]*a^(3/2)*c*d^4*e^6 + 77*Sqrt[2]*a^2*Sqrt[c]*d^2*
e^8 + 120*a^(9/4)*c^(1/4)*d*e^9 + 45*Sqrt[2]*a^(5/2)*e^10)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4) +
256*e^11*Log[d + e*x] + (Sqrt[2]*c^(1/4)*(-21*c^(5/2)*d^11 + 5*Sqrt[a]*c^2*d^9*e^2 - 66*a*c^(3/2)*d^7*e^4 + 1
8*a^(3/2)*c*d^5*e^6 - 77*a^2*Sqrt[c]*d^3*e^8 + 45*a^(5/2)*d*e^10)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq
rt[c]*x^2])/a^(11/4) + (Sqrt[2]*c^(1/4)*(21*c^(5/2)*d^11 - 5*Sqrt[a]*c^2*d^9*e^2 + 66*a*c^(3/2)*d^7*e^4 - 18*a
^(3/2)*c*d^5*e^6 + 77*a^2*Sqrt[c]*d^3*e^8 - 45*a^(5/2)*d*e^10)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[
c]*x^2])/a^(11/4) - 64*e^11*Log[a + c*x^4])/(256*(c*d^4 + a*e^4)^3)

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Maple [A]  time = 0.023, size = 2098, normalized size = 1.6 \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)/(c*x^4+a)^3,x)

[Out]

1/8*c^2/(a*e^4+c*d^4)^3/(c*x^4+a)^2*e^3*d^8+1/4*c^2/(a*e^4+c*d^4)^3/(c*x^4+a)^2*x^4*d^4*e^7+13/16*c^2/(a*e^4+c
*d^4)^3/(c*x^4+a)^2*d^5*e^6*x^3-7/8*c^2/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^6*e^5*x^2+15/16*c^2/(a*e^4+c*d^4)^3/(c*x
^4+a)^2*d^7*x*e^4+11/32*c^3/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^11/a*x+1/4*c/(a*e^4+c*d^4)^3/(c*x^4+a)^2*x^4*a*e^11+
1/2*c/(a*e^4+c*d^4)^3/(c*x^4+a)^2*a*e^7*d^4-15/16*c/(a*e^4+c*d^4)^3/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^2*
e^9+45/256/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2
^(1/2)+(a/c)^(1/2)))*d*e^10+45/128/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d*e^10+
45/128/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d*e^10+13/32*c^2/(a*e^4+c*d^4)^3/(c
*x^4+a)^2*d*e^10*x^7-7/16*c^2/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^2*e^9*x^6+15/32*c^2/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^
3*x^5*e^8+7/32*c^4/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^11/a^2*x^5+3/8/(a*e^4+c*d^4)^3/(c*x^4+a)^2*a^2*e^11-3/16*c^3/
(a*e^4+c*d^4)^3/a^2/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^10*e+21/128*c^3/(a*e^4+c*d^4)^3/a^3*(a/c)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^11+21/256*c^3/(a*e^4+c*d^4)^3/a^3*(a/c)^(1/4)*2^(1/2)*ln((x^2+(a/c)^(1
/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))*d^11+17/32*c/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d
*e^10*a*x^3-9/16*c/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^2*e^9*a*x^2+19/32*c/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^3*a*x*e^8+9
/16*c^3/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^5*e^6/a*x^7+5/32*c^4/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^9*e^2/a^2*x^7-5/8*c^3
/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^6*e^5/a*x^6-3/16*c^4/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^10*e/a^2*x^6+11/16*c^3/(a*e^
4+c*d^4)^3/(c*x^4+a)^2*d^7/a*x^5*e^4+9/32*c^3/(a*e^4+c*d^4)^3/(c*x^4+a)^2*d^9*e^2/a*x^3-5/16*c^3/(a*e^4+c*d^4)
^3/(c*x^4+a)^2*d^10*e/a*x^2+21/128*c^3/(a*e^4+c*d^4)^3/a^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)
*d^11-5/8*c^2/(a*e^4+c*d^4)^3/a/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^6*e^5+77/128*c/(a*e^4+c*d^4)^3/a*(a/c)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^3*e^8+9/128*c/(a*e^4+c*d^4)^3/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^5*e^6+9/64*c/(a*e^4+c*d^4)^3/a/(a
/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^5*e^6+e^11*ln(e*x+d)/(a*e^4+c*d^4)^3-1/4*e^11*ln(c*x^4+a)/
(a*e^4+c*d^4)^3+33/128*c^2/(a*e^4+c*d^4)^3/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^7*e^4+33/64*c^2/(a*e^4+c*d^4)^3/a^2*(a/c)^(1/4)*2^(1/2)*arctan(2^(1
/2)/(a/c)^(1/4)*x+1)*d^7*e^4+33/64*c^2/(a*e^4+c*d^4)^3/a^2*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)
*d^7*e^4+9/64*c/(a*e^4+c*d^4)^3/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^5*e^6+77/128*c/(a*e^4+
c*d^4)^3/a*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^3*e^8+77/256*c/(a*e^4+c*d^4)^3/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^8+5/256*c^2/
(a*e^4+c*d^4)^3/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^9*e^2+5/128*c^2/(a*e^4+c*d^4)^3/a^2/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^9*e^
2+5/128*c^2/(a*e^4+c*d^4)^3/a^2/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^9*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)/(c*x^4+a)^3,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)/(c*x^4+a)^3,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)/(c*x**4+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.37036, size = 1705, normalized size = 1.26 \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)/(c*x^4+a)^3,x, algorithm="giac")

[Out]

1/64*(51*sqrt(2)*sqrt(a*c)*c^3*d^4*e + 21*(a*c^3)^(1/4)*c^3*d^5 - 75*sqrt(2)*a*c^3*d^2*e^3 + 122*(a*c^3)^(3/4)
*c*d^3*e^2 + 45*(a*c^3)^(1/4)*a*c^2*d*e^4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2
)*a^3*c^4*d^6 - 6*(a*c^3)^(1/4)*a^3*c^3*d^5*e + 9*sqrt(2)*sqrt(a*c)*a^3*c^3*d^4*e^2 + 9*sqrt(2)*a^4*c^3*d^2*e^
4 - 16*(a*c^3)^(3/4)*a^3*c*d^3*e^3 - 6*(a*c^3)^(1/4)*a^4*c^2*d*e^5 + sqrt(2)*sqrt(a*c)*a^4*c^2*e^6) + 1/64*(51
*sqrt(2)*sqrt(a*c)*c^3*d^4*e + 21*(a*c^3)^(1/4)*c^3*d^5 + 75*sqrt(2)*a*c^3*d^2*e^3 + 122*(a*c^3)^(3/4)*c*d^3*e
^2 + 45*(a*c^3)^(1/4)*a*c^2*d*e^4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^3*c^
4*d^6 + 6*(a*c^3)^(1/4)*a^3*c^3*d^5*e + 9*sqrt(2)*sqrt(a*c)*a^3*c^3*d^4*e^2 + 9*sqrt(2)*a^4*c^3*d^2*e^4 + 16*(
a*c^3)^(3/4)*a^3*c*d^3*e^3 + 6*(a*c^3)^(1/4)*a^4*c^2*d*e^5 + sqrt(2)*sqrt(a*c)*a^4*c^2*e^6) + 1/128*(21*(a*c^3
)^(1/4)*c^4*d^11 - 5*(a*c^3)^(3/4)*c^2*d^9*e^2 + 66*(a*c^3)^(1/4)*a*c^3*d^7*e^4 - 18*(a*c^3)^(3/4)*a*c*d^5*e^6
+ 77*(a*c^3)^(1/4)*a^2*c^2*d^3*e^8 - 45*(a*c^3)^(3/4)*a^2*d*e^10)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c)
)/(sqrt(2)*a^3*c^5*d^12 + 3*sqrt(2)*a^4*c^4*d^8*e^4 + 3*sqrt(2)*a^5*c^3*d^4*e^8 + sqrt(2)*a^6*c^2*e^12) - 1/12
8*(21*(a*c^3)^(1/4)*c^4*d^11 - 5*(a*c^3)^(3/4)*c^2*d^9*e^2 + 66*(a*c^3)^(1/4)*a*c^3*d^7*e^4 - 18*(a*c^3)^(3/4)
*a*c*d^5*e^6 + 77*(a*c^3)^(1/4)*a^2*c^2*d^3*e^8 - 45*(a*c^3)^(3/4)*a^2*d*e^10)*log(x^2 - sqrt(2)*x*(a/c)^(1/4)
+ sqrt(a/c))/(sqrt(2)*a^3*c^5*d^12 + 3*sqrt(2)*a^4*c^4*d^8*e^4 + 3*sqrt(2)*a^5*c^3*d^4*e^8 + sqrt(2)*a^6*c^2*
e^12) - 1/4*e^11*log(abs(c*x^4 + a))/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) + e^12*log(abs(
x*e + d))/(c^3*d^12*e + 3*a*c^2*d^8*e^5 + 3*a^2*c*d^4*e^9 + a^3*e^13) + 1/32*(4*a^2*c^2*d^8*e^3 + 16*a^3*c*d^4
*e^7 + (5*c^4*d^9*e^2 + 18*a*c^3*d^5*e^6 + 13*a^2*c^2*d*e^10)*x^7 - 2*(3*c^4*d^10*e + 10*a*c^3*d^6*e^5 + 7*a^2
*c^2*d^2*e^9)*x^6 + (7*c^4*d^11 + 22*a*c^3*d^7*e^4 + 15*a^2*c^2*d^3*e^8)*x^5 + 8*(a^2*c^2*d^4*e^7 + a^3*c*e^11
)*x^4 + 12*a^4*e^11 + (9*a*c^3*d^9*e^2 + 26*a^2*c^2*d^5*e^6 + 17*a^3*c*d*e^10)*x^3 - 2*(5*a*c^3*d^10*e + 14*a^
2*c^2*d^6*e^5 + 9*a^3*c*d^2*e^9)*x^2 + (11*a*c^3*d^11 + 30*a^2*c^2*d^7*e^4 + 19*a^3*c*d^3*e^8)*x)/((c*d^4 + a*
e^4)^3*(c*x^4 + a)^2*a^2)