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

Optimal. Leaf size=864 \[ \frac{x e^4}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac{4 c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{\left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c x \left (c d^2-2 c e x^2 d-a e^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (c x^4+a\right )} \]

[Out]

(e^4*x)/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) + (c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^2))/(4*a*(c*d^2 + a*e^2)^2*(a +
c*x^4)) + (4*c*Sqrt[d]*e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c*d^2 + a*e^2)^3 + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sq
rt[d]])/(2*d^(3/2)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 - 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 - (Sqr
t[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e -
3*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d^
2 - 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)
^3) + (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]
*a^(7/4)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 + 4*Sqrt[a]*Sqrt[c]*d*e - 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^2 + a*e^2)^3) - (c^(3/4)*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d
*e - 3*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^2 + a*e^2)^2) +
(c^(3/4)*e^2*(3*c*d^2 + 4*Sqrt[a]*Sqrt[c]*d*e - 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^2 + a*e^2)^3) + (c^(3/4)*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2)

________________________________________________________________________________________

Rubi [A]  time = 0.906108, antiderivative size = 864, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1239, 199, 205, 1179, 1168, 1162, 617, 204, 1165, 628} \[ \frac{x e^4}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac{4 c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{\left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c x \left (c d^2-2 c e x^2 d-a e^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (c x^4+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*x)/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) + (c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^2))/(4*a*(c*d^2 + a*e^2)^2*(a +
c*x^4)) + (4*c*Sqrt[d]*e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c*d^2 + a*e^2)^3 + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sq
rt[d]])/(2*d^(3/2)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 - 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 - (Sqr
t[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e -
3*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d^
2 - 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)
^3) + (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]
*a^(7/4)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 + 4*Sqrt[a]*Sqrt[c]*d*e - 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^2 + a*e^2)^3) - (c^(3/4)*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d
*e - 3*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^2 + a*e^2)^2) +
(c^(3/4)*e^2*(3*c*d^2 + 4*Sqrt[a]*Sqrt[c]*d*e - 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^2 + a*e^2)^3) + (c^(3/4)*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2)

Rule 1239

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a +
c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, p, q}, x] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 1179

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)*(a + c*x^4)^(p + 1))/(
4*a*(p + 1)), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x
] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

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]

Rubi steps

\begin{align*} \int \frac{1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^2} \, dx &=\int \left (\frac{e^4}{\left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^2}+\frac{4 c d e^4}{\left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}+\frac{c \left (c d^2-a e^2-2 c d e x^2\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^4\right )^2}-\frac{c e^2 \left (-3 c d^2+a e^2+4 c d e x^2\right )}{\left (c d^2+a e^2\right )^3 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac{\left (c e^2\right ) \int \frac{-3 c d^2+a e^2+4 c d e x^2}{a+c x^4} \, dx}{\left (c d^2+a e^2\right )^3}+\frac{\left (4 c d e^4\right ) \int \frac{1}{d+e x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac{c \int \frac{c d^2-a e^2-2 c d e x^2}{\left (a+c x^4\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac{e^4 \int \frac{1}{\left (d+e x^2\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{\left (\sqrt{c} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{2 \sqrt{a} \left (c d^2+a e^2\right )^3}+\frac{\left (\sqrt{c} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 \sqrt{a} \left (c d^2+a e^2\right )^3}-\frac{c \int \frac{-3 \left (c d^2-a e^2\right )+2 c d e x^2}{a+c x^4} \, dx}{4 a \left (c d^2+a e^2\right )^2}+\frac{e^4 \int \frac{1}{d+e x^2} \, dx}{2 d \left (c d^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac{\left (\sqrt{c} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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 \sqrt{a} \left (c d^2+a e^2\right )^3}+\frac{\left (\sqrt{c} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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 \sqrt{a} \left (c d^2+a e^2\right )^3}-\frac{\left (c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{\left (c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{\left (\sqrt{c} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac{\left (\sqrt{c} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} \left (c d^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{\left (c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{\left (c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{\left (\sqrt{c} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^{3/2} \left (c d^2+a e^2\right )^2}+\frac{\left (\sqrt{c} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^{3/2} \left (c d^2+a e^2\right )^2}-\frac{\left (c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}-\frac{\left (c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}+\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}+\frac{\left (c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}-\frac{\left (c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac{c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac{4 c \sqrt{d} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}+\frac{c^{3/4} e^2 \left (3 c d^2-4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}-\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}+\frac{c^{3/4} e^2 \left (3 c d^2+4 \sqrt{a} \sqrt{c} d e-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^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} d e-3 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^2+a e^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.623756, size = 540, normalized size = 0.62 \[ \frac{-\frac{\sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3-7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}+\frac{\sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3-7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}+\frac{2 \sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3+7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac{2 \sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3+7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac{16 e^4 x \left (a e^2+c d^2\right )}{d \left (d+e x^2\right )}+\frac{8 c x \left (a e^2+c d^2\right ) \left (c d \left (d-2 e x^2\right )-a e^2\right )}{a \left (a+c x^4\right )}+\frac{16 e^{7/2} \left (a e^2+9 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2}}}{32 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((16*e^4*(c*d^2 + a*e^2)*x)/(d*(d + e*x^2)) + (8*c*(c*d^2 + a*e^2)*x*(-(a*e^2) + c*d*(d - 2*e*x^2)))/(a*(a + c
*x^4)) + (16*e^(7/2)*(9*c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 +
2*Sqrt[a]*c^(3/2)*d^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*
x)/a^(1/4)])/a^(7/4) - (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*
Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (Sqrt[2]*c^(3/4)*(3*c^2*d^4 + 2*
Sqrt[a]*c^(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2*e^4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c
^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (Sqrt[2]*c^(3/4)*(3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18
*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2*e^4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4))/(32*(c*d^
2 + a*e^2)^3)

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Maple [A]  time = 0.066, size = 1169, 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^2+d)^2/(c*x^4+a)^2,x)

[Out]

-1/2*c^2/(a*e^2+c*d^2)^3/(c*x^4+a)*d*e^3*x^3-1/2*c^3/(a*e^2+c*d^2)^3/(c*x^4+a)*d^3*e/a*x^3-1/4*c/(a*e^2+c*d^2)
^3/(c*x^4+a)*x*a*e^4+1/4*c^3/(a*e^2+c*d^2)^3/(c*x^4+a)*x/a*d^4-7/16*c/(a*e^2+c*d^2)^3*(a/c)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(a/c)^(1/4)*x+1)*e^4+3/4*c^2/(a*e^2+c*d^2)^3/a*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*
d^2*e^2+3/16*c^3/(a*e^2+c*d^2)^3/a^2*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^4-7/16*c/(a*e^2+c*d
^2)^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*e^4+3/4*c^2/(a*e^2+c*d^2)^3/a*(a/c)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^2*e^2+3/16*c^3/(a*e^2+c*d^2)^3/a^2*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1
/4)*x-1)*d^4-7/32*c/(a*e^2+c*d^2)^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)))*e^4+3/8*c^2/(a*e^2+c*d^2)^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^2*e^2+3/32*c^3/(a*e^2+c*d^2)^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^4-9/16*c/(a*e^2+c*d^2)^
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^
3-1/16*c^2/(a*e^2+c*d^2)^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-9/8*c/(a*e^2+c*d^2)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d*e^3-1
/8*c^2/(a*e^2+c*d^2)^3/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^3*e-9/8*c/(a*e^2+c*d^2)^3/(a/c)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d*e^3-1/8*c^2/(a*e^2+c*d^2)^3/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1
/2)/(a/c)^(1/4)*x-1)*d^3*e+1/2*e^6/(a*e^2+c*d^2)^3/d*x/(e*x^2+d)*a+1/2*e^4/(a*e^2+c*d^2)^3*d*x/(e*x^2+d)*c+1/2
*e^6/(a*e^2+c*d^2)^3/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*a+9/2*e^4/(a*e^2+c*d^2)^3*d/(d*e)^(1/2)*arctan(e*x/
(d*e)^(1/2))*c

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.15424, size = 1146, normalized size = 1.33 \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^2+d)^2/(c*x^4+a)^2,x, algorithm="giac")

[Out]

1/2*(9*c*d^2*e^4 + a*e^6)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/((c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a
^3*d*e^6)*sqrt(d)) + 1/8*(3*(a*c^3)^(1/4)*c^3*d^4 + 12*(a*c^3)^(1/4)*a*c^2*d^2*e^2 - 2*(a*c^3)^(3/4)*c*d^3*e -
7*(a*c^3)^(1/4)*a^2*c*e^4 - 18*(a*c^3)^(3/4)*a*d*e^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1
/4))/(sqrt(2)*a^2*c^4*d^6 + 3*sqrt(2)*a^3*c^3*d^4*e^2 + 3*sqrt(2)*a^4*c^2*d^2*e^4 + sqrt(2)*a^5*c*e^6) + 1/8*(
3*(a*c^3)^(1/4)*c^3*d^4 + 12*(a*c^3)^(1/4)*a*c^2*d^2*e^2 - 2*(a*c^3)^(3/4)*c*d^3*e - 7*(a*c^3)^(1/4)*a^2*c*e^4
- 18*(a*c^3)^(3/4)*a*d*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^4*d^6
+ 3*sqrt(2)*a^3*c^3*d^4*e^2 + 3*sqrt(2)*a^4*c^2*d^2*e^4 + sqrt(2)*a^5*c*e^6) + 1/16*(3*(a*c^3)^(1/4)*c^3*d^4 +
12*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + 2*(a*c^3)^(3/4)*c*d^3*e - 7*(a*c^3)^(1/4)*a^2*c*e^4 + 18*(a*c^3)^(3/4)*a*d*e
^3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a^2*c^4*d^6 + 3*sqrt(2)*a^3*c^3*d^4*e^2 + 3*sqrt(2)*
a^4*c^2*d^2*e^4 + sqrt(2)*a^5*c*e^6) - 1/16*(3*(a*c^3)^(1/4)*c^3*d^4 + 12*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + 2*(a*c
^3)^(3/4)*c*d^3*e - 7*(a*c^3)^(1/4)*a^2*c*e^4 + 18*(a*c^3)^(3/4)*a*d*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sq
rt(a/c))/(sqrt(2)*a^2*c^4*d^6 + 3*sqrt(2)*a^3*c^3*d^4*e^2 + 3*sqrt(2)*a^4*c^2*d^2*e^4 + sqrt(2)*a^5*c*e^6) - 1
/4*(2*c^2*d^2*x^5*e^2 + c^2*d^3*x^3*e - 2*a*c*x^5*e^4 - c^2*d^4*x + a*c*d*x^3*e^3 + a*c*d^2*x*e^2 - 2*a^2*x*e^
4)/((a*c^2*d^5 + 2*a^2*c*d^3*e^2 + a^3*d*e^4)*(c*x^6*e + c*d*x^4 + a*x^2*e + a*d))