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

Optimal. Leaf size=1141 \[ \text{result too large to display} \]

[Out]

-(e^7/((c*d^4 + a*e^4)^2*(d + e*x))) + (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)))/(4*a*(c*d^4 + a*e^4)^2*(a + c*x^4)) - (Sqrt[c]*d*e^5*(3*c*d^4 - a*e^4)*ArcTan[(Sq
rt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*(c*d^4 + a*e^4)^3) - (Sqrt[c]*d*e*(c*d^4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])
/(2*a^(3/2)*(c*d^4 + a*e^4)^2) - (c^(1/4)*(3*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) + Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*Ar
cTan[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)*e^4*(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)])/(2*Sqrt[2]*a^(3/4)*(c
*d^4 + a*e^4)^3) + (c^(1/4)*(3*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) + Sqrt[a]*e^2*(3*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)^2) + (c^(1/4)*e^4*(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)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^
3) + (8*c*d^3*e^7*Log[d + e*x])/(c*d^4 + a*e^4)^3 - (c^(1/4)*(3*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) - 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])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^2
) - (c^(1/4)*e^4*(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])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) + (c^(1/4)*(3*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4)
- 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])/(16*Sqrt[2]*a^(7/4)*
(c*d^4 + a*e^4)^2) + (c^(1/4)*e^4*(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])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) - (2*c*d^3*e^7*Log[a + c*x
^4])/(c*d^4 + a*e^4)^3

________________________________________________________________________________________

Rubi [A]  time = 1.65882, antiderivative size = 1141, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 14, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.824, Rules used = {6742, 1854, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 260} \[ \frac{8 c d^3 \log (d+e x) e^7}{\left (c d^4+a e^4\right )^3}-\frac{2 c d^3 \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^3}-\frac{e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac{\sqrt{c} d \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e^5}{\sqrt{a} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} \left (\sqrt{c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt{a} e^2 \left (7 c d^4-a e^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} \left (\sqrt{c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt{a} e^2 \left (7 c d^4-a e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (7 c d^4-a e^4\right )\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac{\sqrt [4]{c} \left (\sqrt{c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (7 c d^4-a e^4\right )\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac{\sqrt{c} d \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e}{2 a^{3/2} \left (c d^4+a e^4\right )^2}+\frac{c \left (4 a d^3 e^3+x \left (\left (c d^4-3 a e^4\right ) d^2-2 e \left (c d^4-a e^4\right ) x d+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (c x^4+a\right )}-\frac{\sqrt [4]{c} \left (3 \sqrt{c} \left (c d^4-3 a e^4\right ) d^2+\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\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} \left (3 \sqrt{c} \left (c d^4-3 a e^4\right ) d^2+\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac{\sqrt [4]{c} \left (3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^2}+\frac{\sqrt [4]{c} \left (3 \sqrt{c} d^2 \left (c d^4-3 a e^4\right )-\sqrt{a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(a + c*x^4)^2),x]

[Out]

-(e^7/((c*d^4 + a*e^4)^2*(d + e*x))) + (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)))/(4*a*(c*d^4 + a*e^4)^2*(a + c*x^4)) - (Sqrt[c]*d*e^5*(3*c*d^4 - a*e^4)*ArcTan[(Sq
rt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*(c*d^4 + a*e^4)^3) - (Sqrt[c]*d*e*(c*d^4 - a*e^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])
/(2*a^(3/2)*(c*d^4 + a*e^4)^2) - (c^(1/4)*(3*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) + Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*Ar
cTan[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)*e^4*(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)])/(2*Sqrt[2]*a^(3/4)*(c
*d^4 + a*e^4)^3) + (c^(1/4)*(3*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) + Sqrt[a]*e^2*(3*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)^2) + (c^(1/4)*e^4*(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)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^
3) + (8*c*d^3*e^7*Log[d + e*x])/(c*d^4 + a*e^4)^3 - (c^(1/4)*(3*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4) - 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])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^2
) - (c^(1/4)*e^4*(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])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) + (c^(1/4)*(3*Sqrt[c]*d^2*(c*d^4 - 3*a*e^4)
- 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])/(16*Sqrt[2]*a^(7/4)*
(c*d^4 + a*e^4)^2) + (c^(1/4)*e^4*(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])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) - (2*c*d^3*e^7*Log[a + c*x
^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 1876

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

Rule 275

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

Rule 205

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

Rule 1168

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

Rule 1162

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

Rule 617

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

Rule 204

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

Rule 1165

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

Rule 628

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

Rule 1248

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

Rule 635

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

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.983025, size = 807, normalized size = 0.71 \[ \frac{256 c d^3 \log (d+e x) e^7-64 c d^3 \log \left (c x^4+a\right ) e^7-\frac{32 \left (c d^4+a e^4\right ) e^7}{d+e x}+\frac{8 c \left (c d^4+a e^4\right ) \left (c x \left (d^2-2 e x d+3 e^2 x^2\right ) d^4+a e^3 \left (4 d^3-3 e x d^2+2 e^2 x^2 d-e^3 x^3\right )\right )}{a \left (c x^4+a\right )}+\frac{2 \sqrt [4]{c} \left (-3 \sqrt{2} c^{5/2} d^{10}+8 \sqrt [4]{a} c^{9/4} e d^9-3 \sqrt{2} \sqrt{a} c^2 e^2 d^8-14 \sqrt{2} a c^{3/2} e^4 d^6+48 a^{5/4} c^{5/4} e^5 d^5-30 \sqrt{2} a^{3/2} c e^6 d^4+21 \sqrt{2} a^2 \sqrt{c} e^8 d^2-24 a^{9/4} \sqrt [4]{c} e^9 d+5 \sqrt{2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{2 \sqrt [4]{c} \left (3 \sqrt{2} c^{5/2} d^{10}+8 \sqrt [4]{a} c^{9/4} e d^9+3 \sqrt{2} \sqrt{a} c^2 e^2 d^8+14 \sqrt{2} a c^{3/2} e^4 d^6+48 a^{5/4} c^{5/4} e^5 d^5+30 \sqrt{2} a^{3/2} c e^6 d^4-21 \sqrt{2} a^2 \sqrt{c} e^8 d^2-24 a^{9/4} \sqrt [4]{c} e^9 d-5 \sqrt{2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac{\sqrt{2} \sqrt [4]{c} \left (3 c^{5/2} d^{10}-3 \sqrt{a} c^2 e^2 d^8+14 a c^{3/2} e^4 d^6-30 a^{3/2} c e^6 d^4-21 a^2 \sqrt{c} e^8 d^2+5 a^{5/2} e^{10}\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{7/4}}+\frac{\sqrt{2} \sqrt [4]{c} \left (3 c^{5/2} d^{10}-3 \sqrt{a} c^2 e^2 d^8+14 a c^{3/2} e^4 d^6-30 a^{3/2} c e^6 d^4-21 a^2 \sqrt{c} e^8 d^2+5 a^{5/2} e^{10}\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{7/4}}}{32 \left (c d^4+a e^4\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-32*e^7*(c*d^4 + a*e^4))/(d + e*x) + (8*c*(c*d^4 + a*e^4)*(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*c^(1/4)*(-3*Sqrt[2]*c^(5/2)*d^10 + 8*a^(1/4)*c^(
9/4)*d^9*e - 3*Sqrt[2]*Sqrt[a]*c^2*d^8*e^2 - 14*Sqrt[2]*a*c^(3/2)*d^6*e^4 + 48*a^(5/4)*c^(5/4)*d^5*e^5 - 30*Sq
rt[2]*a^(3/2)*c*d^4*e^6 + 21*Sqrt[2]*a^2*Sqrt[c]*d^2*e^8 - 24*a^(9/4)*c^(1/4)*d*e^9 + 5*Sqrt[2]*a^(5/2)*e^10)*
ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) + (2*c^(1/4)*(3*Sqrt[2]*c^(5/2)*d^10 + 8*a^(1/4)*c^(9/4)*d^9*
e + 3*Sqrt[2]*Sqrt[a]*c^2*d^8*e^2 + 14*Sqrt[2]*a*c^(3/2)*d^6*e^4 + 48*a^(5/4)*c^(5/4)*d^5*e^5 + 30*Sqrt[2]*a^(
3/2)*c*d^4*e^6 - 21*Sqrt[2]*a^2*Sqrt[c]*d^2*e^8 - 24*a^(9/4)*c^(1/4)*d*e^9 - 5*Sqrt[2]*a^(5/2)*e^10)*ArcTan[1
+ (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) + 256*c*d^3*e^7*Log[d + e*x] - (Sqrt[2]*c^(1/4)*(3*c^(5/2)*d^10 - 3*Sq
rt[a]*c^2*d^8*e^2 + 14*a*c^(3/2)*d^6*e^4 - 30*a^(3/2)*c*d^4*e^6 - 21*a^2*Sqrt[c]*d^2*e^8 + 5*a^(5/2)*e^10)*Log
[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (Sqrt[2]*c^(1/4)*(3*c^(5/2)*d^10 - 3*Sqrt[a]*c^
2*d^8*e^2 + 14*a*c^(3/2)*d^6*e^4 - 30*a^(3/2)*c*d^4*e^6 - 21*a^2*Sqrt[c]*d^2*e^8 + 5*a^(5/2)*e^10)*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) - 64*c*d^3*e^7*Log[a + c*x^4])/(32*(c*d^4 + a*e^4)^3)

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Maple [A]  time = 0.022, size = 1636, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/8*c^2/(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^6*e^4+7/8*c^2/(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^6*e^4+7/16*c^2/(a*e^4+c*d^4)^3/a*(a/c)^(1/4)*2^(1/2)*l
n((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))*d^6*e^4+3/32*c^2/(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^8*e^2+3/16*c^2/(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^8*e^2+3/16*c^2/(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^8*e^2+c^2/(a*e^4+c*d^4)^3/(c*x^4+a)*d^7*e^
3+1/4*c^3/(a*e^4+c*d^4)^3/(c*x^4+a)*d^10/a*x-1/4*c/(a*e^4+c*d^4)^3/(c*x^4+a)*e^10*a*x^3+1/2*c^2/(a*e^4+c*d^4)^
3/(c*x^4+a)*e^6*x^3*d^4-1/2*c^2/(a*e^4+c*d^4)^3/(c*x^4+a)*d^6*x*e^4+c/(a*e^4+c*d^4)^3/(c*x^4+a)*a*d^3*e^7-3*c^
2/(a*e^4+c*d^4)^3/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^5*e^5-5/16/(a*e^4+c*d^4)^3*a/(a/c)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(a/c)^(1/4)*x+1)*e^10-5/16/(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)*e
^10-5/32/(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)))*e^10+8*c*d^3*e^7*ln(e*x+d)/(a*e^4+c*d^4)^3-2*c*d^3*e^7*ln(c*x^4+a)/(a*e^4+c*d^4)^3-1/2*c^
3/(a*e^4+c*d^4)^3/(c*x^4+a)*d^9*e/a*x^2+15/8*c/(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^4*e^6+3/32*c^3/(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^10-1/2*c^3/(a*e^4+c*d^4)^3/a/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*d^9*e
+15/16*c/(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^4*e^6-21/16*c/(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^2*e^8
+3/16*c^3/(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^10-21/16*c/(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^2*e^8+3/16*c^3/(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^10-21/32*c/(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^2*e^8+1/2*c/(a*e^4+c*d^4)^3/(c*x^4+a)*d*e^9*a*x^2-3/4
*c/(a*e^4+c*d^4)^3/(c*x^4+a)*d^2*a*x*e^8+3/4*c^3/(a*e^4+c*d^4)^3/(c*x^4+a)*e^2/a*x^3*d^8+3/2*c/(a*e^4+c*d^4)^3
*a/(a*c)^(1/2)*arctan(x^2*(1/a*c)^(1/2))*e^9*d+15/8*c/(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^4*e^6-e^7/(a*e^4+c*d^4)^2/(e*x+d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out