\(\int \frac {1}{(d+e x)^2 (a+c x^4)^2} \, dx\) [406]

Optimal result

Integrand size = 17, antiderivative size = 1141 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^2} \, dx=-\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 ) \arctan \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 ) \arctan \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 \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 ) \arctan \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} e^4 \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 ) \arctan \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} \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 ) \arctan \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} e^4 \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 ) \arctan \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 {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 \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 {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} e^4 \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 {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} \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 {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} e^4 \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 {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 {2 c d^3 e^7 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^3} \]

[Out]

-e^7/(a*e^4+c*d^4)^2/(e*x+d)+1/4*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)+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*d*e*(-a*e^4+c*d^4)*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/a^(3/2)/(a*e^4+c*d^4)^2-d*e^5*(-a*e^4+3*c*d
^4)*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/(a*e^4+c*d^4)^3/a^(1/2)-1/32*c^(1/4)*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(
1/2)+x^2*c^(1/2))*(-e^2*(-a*e^4+3*c*d^4)*a^(1/2)+3*d^2*(-3*a*e^4+c*d^4)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^2*2^(1/
2)+1/32*c^(1/4)*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e^2*(-a*e^4+3*c*d^4)*a^(1/2)+3*d^2*(-3*a*e
^4+c*d^4)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^2*2^(1/2)+1/16*c^(1/4)*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*(-a*
e^4+3*c*d^4)*a^(1/2)+3*d^2*(-3*a*e^4+c*d^4)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^2*2^(1/2)+1/16*c^(1/4)*arctan(1+c^(
1/4)*x*2^(1/2)/a^(1/4))*(e^2*(-a*e^4+3*c*d^4)*a^(1/2)+3*d^2*(-3*a*e^4+c*d^4)*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)^2*
2^(1/2)-1/8*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)+d^2*
(-3*a*e^4+5*c*d^4)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/8*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)+d^2*(-3*a*e^4+5*c*d^4)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)
+1/4*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)+d^2*(-3*a*e^4+5*c*d^4)*c^(
1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/4*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)+d^2*(-3*a*e^4+5*c*d^4)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 1141, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {6874, 1868, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 266} \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^2} \, dx=\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 ) \arctan \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 ) \arctan \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 ) \arctan \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 ) \arctan \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 ) \arctan \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 ) \arctan \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} \]

[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 210

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

Rule 211

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

Rule 266

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 281

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 631

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 642

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 649

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 1176

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 1179

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 1182

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 1262

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 1868

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 1890

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.49 (sec) , antiderivative size = 807, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^2} \, dx=\frac {-\frac {32 e^7 \left (c d^4+a e^4\right )}{d+e x}+\frac {8 c \left (c d^4+a e^4\right ) \left (c d^4 x \left (d^2-2 d e x+3 e^2 x^2\right )+a e^3 \left (4 d^3-3 d^2 e x+2 d e^2 x^2-e^3 x^3\right )\right )}{a \left (a+c x^4\right )}+\frac {2 \sqrt [4]{c} \left (-3 \sqrt {2} c^{5/2} d^{10}+8 \sqrt [4]{a} c^{9/4} d^9 e-3 \sqrt {2} \sqrt {a} c^2 d^8 e^2-14 \sqrt {2} a c^{3/2} d^6 e^4+48 a^{5/4} c^{5/4} d^5 e^5-30 \sqrt {2} a^{3/2} c d^4 e^6+21 \sqrt {2} a^2 \sqrt {c} d^2 e^8-24 a^{9/4} \sqrt [4]{c} d e^9+5 \sqrt {2} a^{5/2} e^{10}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {2 \sqrt [4]{c} \left (3 \sqrt {2} c^{5/2} d^{10}+8 \sqrt [4]{a} c^{9/4} d^9 e+3 \sqrt {2} \sqrt {a} c^2 d^8 e^2+14 \sqrt {2} a c^{3/2} d^6 e^4+48 a^{5/4} c^{5/4} d^5 e^5+30 \sqrt {2} a^{3/2} c d^4 e^6-21 \sqrt {2} a^2 \sqrt {c} d^2 e^8-24 a^{9/4} \sqrt [4]{c} d e^9-5 \sqrt {2} a^{5/2} e^{10}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+256 c d^3 e^7 \log (d+e x)-\frac {\sqrt {2} \sqrt [4]{c} \left (3 c^{5/2} d^{10}-3 \sqrt {a} c^2 d^8 e^2+14 a c^{3/2} d^6 e^4-30 a^{3/2} c d^4 e^6-21 a^2 \sqrt {c} d^2 e^8+5 a^{5/2} e^{10}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (3 c^{5/2} d^{10}-3 \sqrt {a} c^2 d^8 e^2+14 a c^{3/2} d^6 e^4-30 a^{3/2} c d^4 e^6-21 a^2 \sqrt {c} d^2 e^8+5 a^{5/2} e^{10}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}-64 c d^3 e^7 \log \left (a+c x^4\right )}{32 \left (c d^4+a e^4\right )^3} \]

[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)

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 534, normalized size of antiderivative = 0.47

method	result	size
default	\(-\frac {c \left (\frac {\frac {e^{2} \left (a^{2} e^{8}-2 a c \,d^{4} e^{4}-3 c^{2} d^{8}\right ) x^{3}}{4 a}-\frac {e d \left (a^{2} e^{8}-c^{2} d^{8}\right ) x^{2}}{2 a}+\frac {d^{2} \left (3 a^{2} e^{8}+2 a c \,d^{4} e^{4}-c^{2} d^{8}\right ) x}{4 a}-d^{3} e^{3} \left (e^{4} a +d^{4} c \right )}{c \,x^{4}+a}+\frac {\frac {\left (21 a^{2} d^{2} e^{8}-14 a c \,d^{6} e^{4}-3 c^{2} d^{10}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-12 a^{2} d \,e^{9}+24 a c \,d^{5} e^{5}+4 c^{2} d^{9} e \right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {\left (5 a^{2} e^{10}-30 a c \,d^{4} e^{6}-3 c^{2} d^{8} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}+8 a \,d^{3} e^{7} \ln \left (c \,x^{4}+a \right )}{4 a}\right )}{\left (e^{4} a +d^{4} c \right )^{3}}-\frac {e^{7}}{\left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )}+\frac {8 c \,d^{3} e^{7} \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{3}}\)	\(534\)
risch	\(\frac {-\frac {e^{3} c \left (5 e^{4} a -3 d^{4} c \right ) x^{4}}{4 a \left (e^{4} a +d^{4} c \right )^{2}}+\frac {c d \,e^{2} x^{3}}{4 a \left (e^{4} a +d^{4} c \right )}-\frac {d^{2} c e \,x^{2}}{4 a \left (e^{4} a +d^{4} c \right )}+\frac {d^{3} c x}{4 a \left (e^{4} a +d^{4} c \right )}-\frac {e^{3} \left (e^{4} a -d^{4} c \right )}{\left (e^{4} a +d^{4} c \right )^{2}}}{\left (e x +d \right ) \left (c \,x^{4}+a \right )}+\frac {8 d^{3} e^{7} c \ln \left (e x +d \right )}{a^{3} e^{12}+3 a^{2} c \,d^{4} e^{8}+3 a \,c^{2} d^{8} e^{4}+c^{3} d^{12}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{10} e^{12}+3 a^{9} c \,d^{4} e^{8}+3 a^{8} c^{2} d^{8} e^{4}+a^{7} c^{3} d^{12}\right ) \textit {\_Z}^{4}+128 a^{7} c \,d^{3} e^{7} \textit {\_Z}^{3}+\left (708 a^{5} c \,d^{2} e^{6}+68 a^{4} c^{2} d^{6} e^{2}\right ) \textit {\_Z}^{2}+\left (1200 a^{3} c d \,e^{5}+144 a^{2} c^{2} d^{5} e \right ) \textit {\_Z} +625 a c \,e^{4}+81 c^{2} d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{11} e^{22}+17 a^{10} c \,d^{4} e^{18}+18 a^{9} c^{2} d^{8} e^{14}+2 a^{8} c^{3} d^{12} e^{10}-7 a^{7} c^{4} d^{16} e^{6}-3 a^{6} c^{5} d^{20} e^{2}\right ) \textit {\_R}^{4}+\left (382 a^{8} c \,d^{3} e^{17}+752 a^{7} c^{2} d^{7} e^{13}+348 a^{6} c^{3} d^{11} e^{9}-32 a^{5} c^{4} d^{15} e^{5}-10 a^{4} c^{5} d^{19} e \right ) \textit {\_R}^{3}+\left (2871 a^{6} c \,d^{2} e^{16}+3468 a^{5} c^{2} d^{6} e^{12}+1642 a^{4} c^{3} d^{10} e^{8}+12 a^{3} c^{4} d^{14} e^{4}-9 a^{2} c^{5} d^{18}\right ) \textit {\_R}^{2}+\left (4850 a^{4} c d \,e^{15}+534 a^{3} c^{2} d^{5} e^{11}+1878 a^{2} c^{3} d^{9} e^{7}+50 a \,c^{4} d^{13} e^{3}\right ) \textit {\_R} +2500 a^{2} c \,e^{14}-3576 a \,c^{2} d^{4} e^{10}+324 c^{3} d^{8} e^{6}\right ) x +\left (6 a^{11} d \,e^{21}+22 a^{10} c \,d^{5} e^{17}+28 a^{9} c^{2} d^{9} e^{13}+12 a^{8} c^{3} d^{13} e^{9}-2 a^{7} c^{4} d^{17} e^{5}-2 a^{6} c^{5} d^{21} e \right ) \textit {\_R}^{4}+\left (5 a^{9} e^{20}+297 a^{8} c \,d^{4} e^{16}+602 a^{7} c^{2} d^{8} e^{12}+330 a^{6} c^{3} d^{12} e^{8}+17 a^{5} c^{4} d^{16} e^{4}-3 a^{4} c^{5} d^{20}\right ) \textit {\_R}^{3}+\left (2256 a^{6} c \,d^{3} e^{15}-32 a^{5} c^{2} d^{7} e^{11}+848 a^{4} c^{3} d^{11} e^{7}+64 a^{3} c^{4} d^{15} e^{3}\right ) \textit {\_R}^{2}+\left (4830 a^{4} c \,d^{2} e^{14}-2726 a^{3} c^{2} d^{6} e^{10}+666 a^{2} c^{3} d^{10} e^{6}+30 a \,c^{4} d^{14} e^{2}\right ) \textit {\_R} +2500 a^{2} c d \,e^{13}-1016 a \,c^{2} d^{5} e^{9}+324 c^{3} d^{9} e^{5}\right )\right )}{16}\)	\(979\)

[In]

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

[Out]

-c/(a*e^4+c*d^4)^3*((1/4*e^2*(a^2*e^8-2*a*c*d^4*e^4-3*c^2*d^8)/a*x^3-1/2*e*d*(a^2*e^8-c^2*d^8)/a*x^2+1/4*d^2*(
3*a^2*e^8+2*a*c*d^4*e^4-c^2*d^8)/a*x-d^3*e^3*(a*e^4+c*d^4))/(c*x^4+a)+1/4/a*(1/8*(21*a^2*d^2*e^8-14*a*c*d^6*e^
4-3*c^2*d^10)*(a/c)^(1/4)/a*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)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/2*(-12*a^2*d*e^9+24*a*c*d^5*
e^5+4*c^2*d^9*e)/(a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))+1/8*(5*a^2*e^10-30*a*c*d^4*e^6-3*c^2*d^8*e^2)/c/(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)))+2*arctan(2^(1/
2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+8*a*d^3*e^7*ln(c*x^4+a)))-e^7/(a*e^4+c*d^4)^2/(e*x+d)+8
*c*d^3*e^7*ln(e*x+d)/(a*e^4+c*d^4)^3

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 961, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

8*c*d^3*e^7*log(e*x + d)/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) - 1/32*c*(sqrt(2)*(32*sqrt(
2)*a^(7/4)*c^(5/4)*d^3*e^7 - 3*c^3*d^10 + 3*sqrt(a)*c^(5/2)*d^8*e^2 - 14*a*c^2*d^6*e^4 + 30*a^(3/2)*c^(3/2)*d^
4*e^6 + 21*a^2*c*d^2*e^8 - 5*a^(5/2)*sqrt(c)*e^10)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(
3/4)*c^(5/4)) + sqrt(2)*(32*sqrt(2)*a^(7/4)*c^(5/4)*d^3*e^7 + 3*c^3*d^10 - 3*sqrt(a)*c^(5/2)*d^8*e^2 + 14*a*c^
2*d^6*e^4 - 30*a^(3/2)*c^(3/2)*d^4*e^6 - 21*a^2*c*d^2*e^8 + 5*a^(5/2)*sqrt(c)*e^10)*log(sqrt(c)*x^2 - sqrt(2)*
a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) - 2*(3*sqrt(2)*a^(1/4)*c^(13/4)*d^10 + 3*sqrt(2)*a^(3/4)*c^(11/
4)*d^8*e^2 + 14*sqrt(2)*a^(5/4)*c^(9/4)*d^6*e^4 + 30*sqrt(2)*a^(7/4)*c^(7/4)*d^4*e^6 - 21*sqrt(2)*a^(9/4)*c^(5
/4)*d^2*e^8 - 5*sqrt(2)*a^(11/4)*c^(3/4)*e^10 + 8*sqrt(a)*c^3*d^9*e + 48*a^(3/2)*c^2*d^5*e^5 - 24*a^(5/2)*c*d*
e^9)*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)*s
qrt(c))*c^(5/4)) - 2*(3*sqrt(2)*a^(1/4)*c^(13/4)*d^10 + 3*sqrt(2)*a^(3/4)*c^(11/4)*d^8*e^2 + 14*sqrt(2)*a^(5/4
)*c^(9/4)*d^6*e^4 + 30*sqrt(2)*a^(7/4)*c^(7/4)*d^4*e^6 - 21*sqrt(2)*a^(9/4)*c^(5/4)*d^2*e^8 - 5*sqrt(2)*a^(11/
4)*c^(3/4)*e^10 - 8*sqrt(a)*c^3*d^9*e - 48*a^(3/2)*c^2*d^5*e^5 + 24*a^(5/2)*c*d*e^9)*arctan(1/2*sqrt(2)*(2*sqr
t(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*c^3*d^12
+ 3*a^2*c^2*d^8*e^4 + 3*a^3*c*d^4*e^8 + a^4*e^12) + 1/4*(4*a*c*d^4*e^3 - 4*a^2*e^7 + (3*c^2*d^4*e^3 - 5*a*c*e^
7)*x^4 + (c^2*d^5*e^2 + a*c*d*e^6)*x^3 - (c^2*d^6*e + a*c*d^2*e^5)*x^2 + (c^2*d^7 + a*c*d^3*e^4)*x)/(a^2*c^2*d
^9 + 2*a^3*c*d^5*e^4 + a^4*d*e^8 + (a*c^3*d^8*e + 2*a^2*c^2*d^4*e^5 + a^3*c*e^9)*x^5 + (a*c^3*d^9 + 2*a^2*c^2*
d^5*e^4 + a^3*c*d*e^8)*x^4 + (a^2*c^2*d^8*e + 2*a^3*c*d^4*e^5 + a^4*e^9)*x)

Giac [A] (verification not implemented)

none

Time = 6.61 (sec) , antiderivative size = 1145, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

8*c*d^3*e^8*log(abs(e*x + 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) - 2*c*d^3*e^7*log(ab
s(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) + 1/8*(3*sqrt(2)*a*c^2*d*e^3 + 5*sqrt(
2)*sqrt(a*c)*c^2*d^3*e + 3*(a*c^3)^(1/4)*c^2*d^4 - 5*(a*c^3)^(1/4)*a*c*e^4 + 6*(a*c^3)^(3/4)*d^2*e^2)*arctan(1
/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^3*d^6 + 9*sqrt(2)*a^3*c^2*d^2*e^4 + 9*sqrt(
2)*sqrt(a*c)*a^2*c^2*d^4*e^2 + sqrt(2)*sqrt(a*c)*a^3*c*e^6 - 6*(a*c^3)^(1/4)*a^2*c^2*d^5*e - 6*(a*c^3)^(1/4)*a
^3*c*d*e^5 - 16*(a*c^3)^(3/4)*a^2*d^3*e^3) - 1/8*(3*sqrt(2)*a*c^2*d*e^3 - 5*sqrt(2)*sqrt(a*c)*c^2*d^3*e - 3*(a
*c^3)^(1/4)*c^2*d^4 + 5*(a*c^3)^(1/4)*a*c*e^4 - 6*(a*c^3)^(3/4)*d^2*e^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/
c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^3*d^6 + 9*sqrt(2)*a^3*c^2*d^2*e^4 + 9*sqrt(2)*sqrt(a*c)*a^2*c^2*d^4*e^2
+ sqrt(2)*sqrt(a*c)*a^3*c*e^6 + 6*(a*c^3)^(1/4)*a^2*c^2*d^5*e + 6*(a*c^3)^(1/4)*a^3*c*d*e^5 + 16*(a*c^3)^(3/4)
*a^2*d^3*e^3) + 1/32*(3*sqrt(2)*(a*c^3)^(1/4)*c^4*d^10 + 14*sqrt(2)*(a*c^3)^(1/4)*a*c^3*d^6*e^4 - 21*sqrt(2)*(
a*c^3)^(1/4)*a^2*c^2*d^2*e^8 - 3*sqrt(2)*(a*c^3)^(3/4)*c^2*d^8*e^2 - 30*sqrt(2)*(a*c^3)^(3/4)*a*c*d^4*e^6 + 5*
sqrt(2)*(a*c^3)^(3/4)*a^2*e^10)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^5*d^12 + 3*a^3*c^4*d^8*e^4
+ 3*a^4*c^3*d^4*e^8 + a^5*c^2*e^12) - 1/32*(3*sqrt(2)*(a*c^3)^(1/4)*c^4*d^10 + 14*sqrt(2)*(a*c^3)^(1/4)*a*c^3
*d^6*e^4 - 21*sqrt(2)*(a*c^3)^(1/4)*a^2*c^2*d^2*e^8 - 3*sqrt(2)*(a*c^3)^(3/4)*c^2*d^8*e^2 - 30*sqrt(2)*(a*c^3)
^(3/4)*a*c*d^4*e^6 + 5*sqrt(2)*(a*c^3)^(3/4)*a^2*e^10)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^5*d
^12 + 3*a^3*c^4*d^8*e^4 + 3*a^4*c^3*d^4*e^8 + a^5*c^2*e^12) + 1/4*(3*c^2*d^4*e^3*x^4 - 5*a*c*e^7*x^4 + c^2*d^5
*e^2*x^3 + a*c*d*e^6*x^3 - c^2*d^6*e*x^2 - a*c*d^2*e^5*x^2 + c^2*d^7*x + a*c*d^3*e^4*x + 4*a*c*d^4*e^3 - 4*a^2
*e^7)/((a*c^2*d^8 + 2*a^2*c*d^4*e^4 + a^3*e^8)*(c*e*x^5 + c*d*x^4 + a*e*x + a*d))

Mupad [B] (verification not implemented)

Time = 10.59 (sec) , antiderivative size = 2246, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/((a + c*x^4)^2*(d + e*x)^2),x)

[Out]

symsum(log(root(196608*a^9*c*d^4*e^8*z^4 + 196608*a^8*c^2*d^8*e^4*z^4 + 65536*a^7*c^3*d^12*z^4 + 65536*a^10*e^
12*z^4 + 524288*a^7*c*d^3*e^7*z^3 + 181248*a^5*c*d^2*e^6*z^2 + 17408*a^4*c^2*d^6*e^2*z^2 + 2304*a^2*c^2*d^5*e*
z + 19200*a^3*c*d*e^5*z + 625*a*c*e^4 + 81*c^2*d^4, z, k)*((120*a*c^8*d^14*e^3 + 2664*a^2*c^7*d^10*e^7 - 10904
*a^3*c^6*d^6*e^11 + 19320*a^4*c^5*d^2*e^15)/(256*(a^8*e^16 + a^4*c^4*d^16 + 4*a^7*c*d^4*e^12 + 4*a^5*c^3*d^12*
e^4 + 6*a^6*c^2*d^8*e^8)) + root(196608*a^9*c*d^4*e^8*z^4 + 196608*a^8*c^2*d^8*e^4*z^4 + 65536*a^7*c^3*d^12*z^
4 + 65536*a^10*e^12*z^4 + 524288*a^7*c*d^3*e^7*z^3 + 181248*a^5*c*d^2*e^6*z^2 + 17408*a^4*c^2*d^6*e^2*z^2 + 23
04*a^2*c^2*d^5*e*z + 19200*a^3*c*d*e^5*z + 625*a*c*e^4 + 81*c^2*d^4, z, k)*((4096*a^3*c^8*d^15*e^4 + 54272*a^4
*c^7*d^11*e^8 - 2048*a^5*c^6*d^7*e^12 + 144384*a^6*c^5*d^3*e^16)/(256*(a^8*e^16 + a^4*c^4*d^16 + 4*a^7*c*d^4*e
^12 + 4*a^5*c^3*d^12*e^4 + 6*a^6*c^2*d^8*e^8)) + root(196608*a^9*c*d^4*e^8*z^4 + 196608*a^8*c^2*d^8*e^4*z^4 +
65536*a^7*c^3*d^12*z^4 + 65536*a^10*e^12*z^4 + 524288*a^7*c*d^3*e^7*z^3 + 181248*a^5*c*d^2*e^6*z^2 + 17408*a^4
*c^2*d^6*e^2*z^2 + 2304*a^2*c^2*d^5*e*z + 19200*a^3*c*d*e^5*z + 625*a*c*e^4 + 81*c^2*d^4, z, k)*(root(196608*a
^9*c*d^4*e^8*z^4 + 196608*a^8*c^2*d^8*e^4*z^4 + 65536*a^7*c^3*d^12*z^4 + 65536*a^10*e^12*z^4 + 524288*a^7*c*d^
3*e^7*z^3 + 181248*a^5*c*d^2*e^6*z^2 + 17408*a^4*c^2*d^6*e^2*z^2 + 2304*a^2*c^2*d^5*e*z + 19200*a^3*c*d*e^5*z
+ 625*a*c*e^4 + 81*c^2*d^4, z, k)*((98304*a^11*c^4*d*e^22 - 32768*a^6*c^9*d^21*e^2 - 32768*a^7*c^8*d^17*e^6 +
196608*a^8*c^7*d^13*e^10 + 458752*a^9*c^6*d^9*e^14 + 360448*a^10*c^5*d^5*e^18)/(256*(a^8*e^16 + a^4*c^4*d^16 +
4*a^7*c*d^4*e^12 + 4*a^5*c^3*d^12*e^4 + 6*a^6*c^2*d^8*e^8)) + (x*(81920*a^11*c^4*e^23 - 49152*a^6*c^9*d^20*e^
3 - 114688*a^7*c^8*d^16*e^7 + 32768*a^8*c^7*d^12*e^11 + 294912*a^9*c^6*d^8*e^15 + 278528*a^10*c^5*d^4*e^19))/(
256*(a^8*e^16 + a^4*c^4*d^16 + 4*a^7*c*d^4*e^12 + 4*a^5*c^3*d^12*e^4 + 6*a^6*c^2*d^8*e^8))) + (5120*a^9*c^4*e^
21 - 3072*a^4*c^9*d^20*e + 17408*a^5*c^8*d^16*e^5 + 337920*a^6*c^7*d^12*e^9 + 616448*a^7*c^6*d^8*e^13 + 304128
*a^8*c^5*d^4*e^17)/(256*(a^8*e^16 + a^4*c^4*d^16 + 4*a^7*c*d^4*e^12 + 4*a^5*c^3*d^12*e^4 + 6*a^6*c^2*d^8*e^8))
+ (x*(356352*a^6*c^7*d^11*e^10 - 32768*a^5*c^8*d^15*e^6 - 10240*a^4*c^9*d^19*e^2 + 770048*a^7*c^6*d^7*e^14 +
391168*a^8*c^5*d^3*e^18))/(256*(a^8*e^16 + a^4*c^4*d^16 + 4*a^7*c*d^4*e^12 + 4*a^5*c^3*d^12*e^4 + 6*a^6*c^2*d^
8*e^8))) + (x*(768*a^3*c^8*d^14*e^5 - 576*a^2*c^9*d^18*e + 105088*a^4*c^7*d^10*e^9 + 221952*a^5*c^6*d^6*e^13 +
183744*a^6*c^5*d^2*e^17))/(256*(a^8*e^16 + a^4*c^4*d^16 + 4*a^7*c*d^4*e^12 + 4*a^5*c^3*d^12*e^4 + 6*a^6*c^2*d
^8*e^8))) + (x*(200*a*c^8*d^13*e^4 + 19400*a^4*c^5*d*e^16 + 7512*a^2*c^7*d^9*e^8 + 2136*a^3*c^6*d^5*e^12))/(25
6*(a^8*e^16 + a^4*c^4*d^16 + 4*a^7*c*d^4*e^12 + 4*a^5*c^3*d^12*e^4 + 6*a^6*c^2*d^8*e^8))) + (81*c^7*d^9*e^6 -
254*a*c^6*d^5*e^10 + 625*a^2*c^5*d*e^14)/(256*(a^8*e^16 + a^4*c^4*d^16 + 4*a^7*c*d^4*e^12 + 4*a^5*c^3*d^12*e^4
+ 6*a^6*c^2*d^8*e^8)) + (x*(625*a^2*c^5*e^15 + 81*c^7*d^8*e^7 - 894*a*c^6*d^4*e^11))/(256*(a^8*e^16 + a^4*c^4
*d^16 + 4*a^7*c*d^4*e^12 + 4*a^5*c^3*d^12*e^4 + 6*a^6*c^2*d^8*e^8)))*root(196608*a^9*c*d^4*e^8*z^4 + 196608*a^
8*c^2*d^8*e^4*z^4 + 65536*a^7*c^3*d^12*z^4 + 65536*a^10*e^12*z^4 + 524288*a^7*c*d^3*e^7*z^3 + 181248*a^5*c*d^2
*e^6*z^2 + 17408*a^4*c^2*d^6*e^2*z^2 + 2304*a^2*c^2*d^5*e*z + 19200*a^3*c*d*e^5*z + 625*a*c*e^4 + 81*c^2*d^4,
z, k), k, 1, 4) + ((x^4*(3*c^2*d^4*e^3 - 5*a*c*e^7))/(4*a*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) - (a*e^7 - c*d^
4*e^3)/(a*e^4 + c*d^4)^2 + (c*d^3*x)/(4*a*(a*e^4 + c*d^4)) - (c*d^2*e*x^2)/(4*a*(a*e^4 + c*d^4)) + (c*d*e^2*x^
3)/(4*a*(a*e^4 + c*d^4)))/(a*d + a*e*x + c*d*x^4 + c*e*x^5) + (8*c*d^3*e^7*log(d + e*x))/(a^3*e^12 + c^3*d^12
+ 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8)