3.40 \(\int \frac{d+\frac{e}{x^4}}{c+\frac{a}{x^8}} \, dx\)

Optimal. Leaf size=753 \[ -\frac{\left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \log \left (-\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \log \left (\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (-\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2+\sqrt{2}} \left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}+\frac{\sqrt{2+\sqrt{2}} \left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}+\frac{d x}{c} \]

[Out]

(d*x)/c + (Sqrt[2 - Sqrt[2]]*((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) - 2*c^(1/
8)*x)/(Sqrt[2 + Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) - (Sqrt[2 + Sqrt[2]]*(Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c
]*e)*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8) - 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) - (Sqr
t[2 - Sqrt[2]]*((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2
+ Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) + (Sqrt[2 + Sqrt[2]]*(Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c]*e)*ArcTan[(S
qrt[2 + Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) - ((Sqrt[a]*(d - Sqr
t[2]*d) + Sqrt[c]*e)*Log[a^(1/4) - Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 - Sqrt[2])
]*a^(3/8)*c^(9/8)) + ((Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c]*e)*Log[a^(1/4) + Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x
+ c^(1/4)*x^2])/(8*Sqrt[2*(2 - Sqrt[2])]*a^(3/8)*c^(9/8)) + (((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*Log[a^(1/4)
- Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 + Sqrt[2])]*a^(3/8)*c^(9/8)) - (((1 + Sqrt
[2])*Sqrt[a]*d + Sqrt[c]*e)*Log[a^(1/4) + Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 + S
qrt[2])]*a^(3/8)*c^(9/8))

________________________________________________________________________________________

Rubi [A]  time = 1.43632, antiderivative size = 753, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {1394, 1503, 1415, 1169, 634, 618, 204, 628} \[ -\frac{\left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \log \left (-\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \log \left (\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (-\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2+\sqrt{2}} \left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}+\frac{\sqrt{2+\sqrt{2}} \left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}+\frac{d x}{c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e/x^4)/(c + a/x^8),x]

[Out]

(d*x)/c + (Sqrt[2 - Sqrt[2]]*((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) - 2*c^(1/
8)*x)/(Sqrt[2 + Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) - (Sqrt[2 + Sqrt[2]]*(Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c
]*e)*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8) - 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) - (Sqr
t[2 - Sqrt[2]]*((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2
+ Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) + (Sqrt[2 + Sqrt[2]]*(Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c]*e)*ArcTan[(S
qrt[2 + Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) - ((Sqrt[a]*(d - Sqr
t[2]*d) + Sqrt[c]*e)*Log[a^(1/4) - Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 - Sqrt[2])
]*a^(3/8)*c^(9/8)) + ((Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c]*e)*Log[a^(1/4) + Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x
+ c^(1/4)*x^2])/(8*Sqrt[2*(2 - Sqrt[2])]*a^(3/8)*c^(9/8)) + (((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*Log[a^(1/4)
- Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 + Sqrt[2])]*a^(3/8)*c^(9/8)) - (((1 + Sqrt
[2])*Sqrt[a]*d + Sqrt[c]*e)*Log[a^(1/4) + Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 + S
qrt[2])]*a^(3/8)*c^(9/8))

Rule 1394

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[x^(n*(2*p + q))*(e + d/x
^n)^q*(c + a/x^(2*n))^p, x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]

Rule 1503

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(e*f^(n - 1)
*(f*x)^(m - n + 1)*(a + c*x^(2*n))^(p + 1))/(c*(m + n*(2*p + 1) + 1)), x] - Dist[f^n/(c*(m + n*(2*p + 1) + 1))
, Int[(f*x)^(m - n)*(a + c*x^(2*n))^p*(a*e*(m - n + 1) - c*d*(m + n*(2*p + 1) + 1)*x^n), x], x] /; FreeQ[{a, c
, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]

Rule 1415

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[a/c, 4]}, Dist[1/(2*Sqrt[2]*
c*q^3), Int[(Sqrt[2]*d*q - (d - e*q^2)*x^(n/2))/(q^2 - Sqrt[2]*q*x^(n/2) + x^n), x], x] + Dist[1/(2*Sqrt[2]*c*
q^3), Int[(Sqrt[2]*d*q + (d - e*q^2)*x^(n/2))/(q^2 + Sqrt[2]*q*x^(n/2) + x^n), x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && PosQ[a*c]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 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{d+\frac{e}{x^4}}{c+\frac{a}{x^8}} \, dx &=\int \frac{x^4 \left (e+d x^4\right )}{a+c x^8} \, dx\\ &=\frac{d x}{c}-\frac{\int \frac{a d-c e x^4}{a+c x^8} \, dx}{c}\\ &=\frac{d x}{c}-\frac{\int \frac{\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}+\left (-a d-\sqrt{a} \sqrt{c} e\right ) x^2}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+x^4} \, dx}{2 \sqrt{2} a^{3/4} c^{5/4}}-\frac{\int \frac{\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}+\left (a d+\sqrt{a} \sqrt{c} e\right ) x^2}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+x^4} \, dx}{2 \sqrt{2} a^{3/4} c^{5/4}}\\ &=\frac{d x}{c}-\frac{\int \frac{\frac{\sqrt{2 \left (2-\sqrt{2}\right )} a^{11/8} d}{c^{3/8}}-\left (\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}-\frac{\sqrt [4]{a} \left (a d+\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right ) x}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{4 \sqrt{2 \left (2-\sqrt{2}\right )} a^{9/8} c^{7/8}}-\frac{\int \frac{\frac{\sqrt{2 \left (2-\sqrt{2}\right )} a^{11/8} d}{c^{3/8}}+\left (\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}-\frac{\sqrt [4]{a} \left (a d+\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right ) x}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{4 \sqrt{2 \left (2-\sqrt{2}\right )} a^{9/8} c^{7/8}}-\frac{\int \frac{\frac{\sqrt{2 \left (2+\sqrt{2}\right )} a^{11/8} d}{c^{3/8}}-\left (\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}-\frac{\sqrt [4]{a} \left (-a d-\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right ) x}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{4 \sqrt{2 \left (2+\sqrt{2}\right )} a^{9/8} c^{7/8}}-\frac{\int \frac{\frac{\sqrt{2 \left (2+\sqrt{2}\right )} a^{11/8} d}{c^{3/8}}+\left (\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}-\frac{\sqrt [4]{a} \left (-a d-\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right ) x}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{4 \sqrt{2 \left (2+\sqrt{2}\right )} a^{9/8} c^{7/8}}\\ &=\frac{d x}{c}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \int \frac{1}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \int \frac{1}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \int \frac{-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \int \frac{\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}-\frac{\left (\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}-\frac{\sqrt [4]{a} \left (a d+\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right ) \int \frac{\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{9/8} c^{7/8}}-\frac{\left (-\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}+\frac{\sqrt [4]{a} \left (a d+\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right ) \int \frac{-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{9/8} c^{7/8}}-\frac{\left (\frac{2 \sqrt{2 \left (2+\sqrt{2}\right )} a^{11/8} d}{c^{3/8}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} \left (\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}-\frac{\sqrt [4]{a} \left (-a d-\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right )}{\sqrt [8]{c}}\right ) \int \frac{1}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{9/8} c^{7/8}}-\frac{\left (\frac{2 \sqrt{2 \left (2+\sqrt{2}\right )} a^{11/8} d}{c^{3/8}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} \left (-\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}+\frac{\sqrt [4]{a} \left (-a d-\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right )}{\sqrt [8]{c}}\right ) \int \frac{1}{\frac{\sqrt [4]{a}}{\sqrt [4]{c}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{9/8} c^{7/8}}\\ &=\frac{d x}{c}-\frac{\left (\left (1-\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt [4]{a}-\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\left (1-\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt [4]{a}+\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt [4]{a}-\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt [4]{a}+\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{\left (2+\sqrt{2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}-x^2} \, dx,x,-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{\left (2+\sqrt{2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}-x^2} \, dx,x,\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\frac{2 \sqrt{2 \left (2+\sqrt{2}\right )} a^{11/8} d}{c^{3/8}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} \left (\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}-\frac{\sqrt [4]{a} \left (-a d-\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right )}{\sqrt [8]{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{\left (2-\sqrt{2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}-x^2} \, dx,x,\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )} a^{9/8} c^{7/8}}+\frac{\left (\frac{2 \sqrt{2 \left (2+\sqrt{2}\right )} a^{11/8} d}{c^{3/8}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a} \left (-\frac{\sqrt{2} a^{5/4} d}{\sqrt [4]{c}}+\frac{\sqrt [4]{a} \left (-a d-\sqrt{a} \sqrt{c} e\right )}{\sqrt [4]{c}}\right )}{\sqrt [8]{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{\left (2-\sqrt{2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}-x^2} \, dx,x,-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )} a^{9/8} c^{7/8}}\\ &=\frac{d x}{c}+\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}-\frac{\sqrt{2+\sqrt{2}} \left (\left (1-\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\sqrt{2+\sqrt{2}} \left (\left (1-\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac{\left (\left (1-\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt [4]{a}-\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\left (1-\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt [4]{a}+\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt [4]{a}-\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt [4]{a}+\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}\\ \end{align*}

Mathematica [A]  time = 0.917837, size = 551, normalized size = 0.73 \[ \frac{\log \left (2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (a^{5/8} c e \cos \left (\frac{\pi }{8}\right )-a^{9/8} \sqrt{c} d \sin \left (\frac{\pi }{8}\right )\right )+\log \left (-2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (a^{9/8} \sqrt{c} d \sin \left (\frac{\pi }{8}\right )-a^{5/8} c e \cos \left (\frac{\pi }{8}\right )\right )+\log \left (-2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (a^{9/8} \sqrt{c} d \cos \left (\frac{\pi }{8}\right )+a^{5/8} c e \sin \left (\frac{\pi }{8}\right )\right )-\log \left (2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (a^{9/8} \sqrt{c} d \cos \left (\frac{\pi }{8}\right )+a^{5/8} c e \sin \left (\frac{\pi }{8}\right )\right )-2 \tan ^{-1}\left (\frac{\sqrt [8]{c} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right ) \left (a^{9/8} \sqrt{c} d \cos \left (\frac{\pi }{8}\right )+a^{5/8} c e \sin \left (\frac{\pi }{8}\right )\right )-2 \tan ^{-1}\left (\frac{\sqrt [8]{c} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right ) \left (a^{9/8} \sqrt{c} d \cos \left (\frac{\pi }{8}\right )+a^{5/8} c e \sin \left (\frac{\pi }{8}\right )\right )+2 \left (a^{5/8} c e \cos \left (\frac{\pi }{8}\right )-a^{9/8} \sqrt{c} d \sin \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+2 \left (a^{9/8} \sqrt{c} d \sin \left (\frac{\pi }{8}\right )-a^{5/8} c e \cos \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+8 a c^{5/8} d x}{8 a c^{13/8}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e/x^4)/(c + a/x^8),x]

[Out]

(8*a*c^(5/8)*d*x + 2*ArcTan[Cot[Pi/8] + (c^(1/8)*x*Csc[Pi/8])/a^(1/8)]*(a^(5/8)*c*e*Cos[Pi/8] - a^(9/8)*Sqrt[c
]*d*Sin[Pi/8]) + Log[a^(1/4) + c^(1/4)*x^2 + 2*a^(1/8)*c^(1/8)*x*Sin[Pi/8]]*(a^(5/8)*c*e*Cos[Pi/8] - a^(9/8)*S
qrt[c]*d*Sin[Pi/8]) + 2*ArcTan[Cot[Pi/8] - (c^(1/8)*x*Csc[Pi/8])/a^(1/8)]*(-(a^(5/8)*c*e*Cos[Pi/8]) + a^(9/8)*
Sqrt[c]*d*Sin[Pi/8]) + Log[a^(1/4) + c^(1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Sin[Pi/8]]*(-(a^(5/8)*c*e*Cos[Pi/8]) +
a^(9/8)*Sqrt[c]*d*Sin[Pi/8]) - 2*ArcTan[(c^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8
] + a^(5/8)*c*e*Sin[Pi/8]) - 2*ArcTan[(c^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8]
+ a^(5/8)*c*e*Sin[Pi/8]) + Log[a^(1/4) + c^(1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Cos[Pi/8]]*(a^(9/8)*Sqrt[c]*d*Cos[P
i/8] + a^(5/8)*c*e*Sin[Pi/8]) - Log[a^(1/4) + c^(1/4)*x^2 + 2*a^(1/8)*c^(1/8)*x*Cos[Pi/8]]*(a^(9/8)*Sqrt[c]*d*
Cos[Pi/8] + a^(5/8)*c*e*Sin[Pi/8]))/(8*a*c^(13/8))

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Maple [C]  time = 0.003, size = 45, normalized size = 0.1 \begin{align*}{\frac{dx}{c}}+{\frac{1}{8\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{ \left ({{\it \_R}}^{4}ce-ad \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e/x^4)/(c+a/x^8),x)

[Out]

d*x/c+1/8/c^2*sum((_R^4*c*e-a*d)/_R^7*ln(x-_R),_R=RootOf(_Z^8*c+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e/x^4)/(c+a/x^8),x, algorithm="maxima")

[Out]

d*x/c + integrate((c*e*x^4 - a*d)/(c*x^8 + a), x)/c

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Fricas [B]  time = 5.35588, size = 6888, normalized size = 9.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e/x^4)/(c+a/x^8),x, algorithm="fricas")

[Out]

-1/8*(4*c*(-(a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c
^9)) - 4*a*d^3*e + 4*c*d*e^3)/(a*c^4))^(1/4)*arctan(-((3*a^4*c^4*d^6*e - 19*a^3*c^5*d^4*e^3 + 9*a^2*c^6*d^2*e^
5 - a*c^7*e^7 + (a^4*c^8*d^3 - 3*a^3*c^9*d*e^2)*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*
c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)))*sqrt(((a^4*d^8 - 4*a^3*c*d^6*e^2 - 10*a^2*c^2*d^4*e^4 - 4*a*c^3*d^2*e^6 + c
^4*e^8)*x^2 - (2*a^3*c^7*d*e*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e
^8)/(a^3*c^9)) - a^4*c^2*d^6 + 7*a^3*c^3*d^4*e^2 - 7*a^2*c^4*d^2*e^4 + a*c^5*e^6)*sqrt(-(a*c^4*sqrt(-(a^4*d^8
- 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - 4*a*d^3*e + 4*c*d*e^3)/(a*c
^4)))/(a^4*d^8 - 4*a^3*c*d^6*e^2 - 10*a^2*c^2*d^4*e^4 - 4*a*c^3*d^2*e^6 + c^4*e^8))*sqrt(-(a*c^4*sqrt(-(a^4*d^
8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - 4*a*d^3*e + 4*c*d*e^3)/(a
*c^4)) - ((a^4*c^8*d^3 - 3*a^3*c^9*d*e^2)*x*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*
d^2*e^6 + c^4*e^8)/(a^3*c^9)) + (3*a^4*c^4*d^6*e - 19*a^3*c^5*d^4*e^3 + 9*a^2*c^6*d^2*e^5 - a*c^7*e^7)*x)*sqrt
(-(a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - 4*a
*d^3*e + 4*c*d*e^3)/(a*c^4)))*(-(a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e
^6 + c^4*e^8)/(a^3*c^9)) - 4*a*d^3*e + 4*c*d*e^3)/(a*c^4))^(1/4)/(a^5*d^10 - 3*a^4*c*d^8*e^2 - 14*a^3*c^2*d^6*
e^4 - 14*a^2*c^3*d^4*e^6 - 3*a*c^4*d^2*e^8 + c^5*e^10)) - 4*c*((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a
^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/(a*c^4))^(1/4)*arctan(((3*a^4
*c^4*d^6*e - 19*a^3*c^5*d^4*e^3 + 9*a^2*c^6*d^2*e^5 - a*c^7*e^7 - (a^4*c^8*d^3 - 3*a^3*c^9*d*e^2)*sqrt(-(a^4*d
^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)))*sqrt(((a^4*d^8 - 4*a^3*c*
d^6*e^2 - 10*a^2*c^2*d^4*e^4 - 4*a*c^3*d^2*e^6 + c^4*e^8)*x^2 + (2*a^3*c^7*d*e*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e
^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + a^4*c^2*d^6 - 7*a^3*c^3*d^4*e^2 + 7*a^2*c^4
*d^2*e^4 - a*c^5*e^6)*sqrt((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 +
c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/(a*c^4)))/(a^4*d^8 - 4*a^3*c*d^6*e^2 - 10*a^2*c^2*d^4*e^4 - 4*a*c
^3*d^2*e^6 + c^4*e^8))*((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4
*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/(a*c^4))^(3/4) + ((a^4*c^8*d^3 - 3*a^3*c^9*d*e^2)*x*sqrt(-(a^4*d^8 -
12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - (3*a^4*c^4*d^6*e - 19*a^3*c^
5*d^4*e^3 + 9*a^2*c^6*d^2*e^5 - a*c^7*e^7)*x)*((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 -
12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/(a*c^4))^(3/4))/(a^5*d^10 - 3*a^4*c*d^8*e^2 -
14*a^3*c^2*d^6*e^4 - 14*a^2*c^3*d^4*e^6 - 3*a*c^4*d^2*e^8 + c^5*e^10)) + c*((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*
d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/(a*c^4))^(1/4)*
log((a^3*d^6 - 5*a^2*c*d^4*e^2 - 5*a*c^2*d^2*e^4 + c^3*e^6)*x + (a^2*c^6*e*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 +
38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + a^3*c*d^5 - 6*a^2*c^2*d^3*e^2 + a*c^3*d*e^4)*((
a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^
3*e - 4*c*d*e^3)/(a*c^4))^(1/4)) - c*((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3
*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/(a*c^4))^(1/4)*log((a^3*d^6 - 5*a^2*c*d^4*e^2 - 5*a*c^
2*d^2*e^4 + c^3*e^6)*x - (a^2*c^6*e*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6
+ c^4*e^8)/(a^3*c^9)) + a^3*c*d^5 - 6*a^2*c^2*d^3*e^2 + a*c^3*d*e^4)*((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2
+ 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/(a*c^4))^(1/4)) - c*(-
(a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - 4*a*d
^3*e + 4*c*d*e^3)/(a*c^4))^(1/4)*log((a^3*d^6 - 5*a^2*c*d^4*e^2 - 5*a*c^2*d^2*e^4 + c^3*e^6)*x + (a^2*c^6*e*sq
rt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - a^3*c*d^5 + 6*
a^2*c^2*d^3*e^2 - a*c^3*d*e^4)*(-(a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*
e^6 + c^4*e^8)/(a^3*c^9)) - 4*a*d^3*e + 4*c*d*e^3)/(a*c^4))^(1/4)) + c*(-(a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*
e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - 4*a*d^3*e + 4*c*d*e^3)/(a*c^4))^(1/4)*log(
(a^3*d^6 - 5*a^2*c*d^4*e^2 - 5*a*c^2*d^2*e^4 + c^3*e^6)*x - (a^2*c^6*e*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*
a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - a^3*c*d^5 + 6*a^2*c^2*d^3*e^2 - a*c^3*d*e^4)*(-(a*c
^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - 4*a*d^3*e
+ 4*c*d*e^3)/(a*c^4))^(1/4)) - 8*d*x)/c

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Sympy [A]  time = 18.9418, size = 204, normalized size = 0.27 \begin{align*} \operatorname{RootSum}{\left (16777216 t^{8} a^{3} c^{9} + t^{4} \left (- 32768 a^{3} c^{5} d^{3} e + 32768 a^{2} c^{6} d e^{3}\right ) + a^{4} d^{8} + 4 a^{3} c d^{6} e^{2} + 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{2} e^{6} + c^{4} e^{8}, \left ( t \mapsto t \log{\left (x + \frac{- 32768 t^{5} a^{2} c^{6} e - 8 t a^{3} c d^{5} + 80 t a^{2} c^{2} d^{3} e^{2} - 40 t a c^{3} d e^{4}}{a^{3} d^{6} - 5 a^{2} c d^{4} e^{2} - 5 a c^{2} d^{2} e^{4} + c^{3} e^{6}} \right )} \right )\right )} + \frac{d x}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e/x**4)/(c+a/x**8),x)

[Out]

RootSum(16777216*_t**8*a**3*c**9 + _t**4*(-32768*a**3*c**5*d**3*e + 32768*a**2*c**6*d*e**3) + a**4*d**8 + 4*a*
*3*c*d**6*e**2 + 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**2*e**6 + c**4*e**8, Lambda(_t, _t*log(x + (-32768*_t**5*a
**2*c**6*e - 8*_t*a**3*c*d**5 + 80*_t*a**2*c**2*d**3*e**2 - 40*_t*a*c**3*d*e**4)/(a**3*d**6 - 5*a**2*c*d**4*e*
*2 - 5*a*c**2*d**2*e**4 + c**3*e**6)))) + d*x/c

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Giac [A]  time = 1.22948, size = 873, normalized size = 1.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e/x^4)/(c+a/x^8),x, algorithm="giac")

[Out]

d*x/c - 1/8*(c*sqrt(-sqrt(2) + 2)*(a/c)^(5/8)*e + a*d*sqrt(sqrt(2) + 2)*(a/c)^(1/8))*arctan((2*x + sqrt(-sqrt(
2) + 2)*(a/c)^(1/8))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a*c) - 1/8*(c*sqrt(-sqrt(2) + 2)*(a/c)^(5/8)*e + a*d*sq
rt(sqrt(2) + 2)*(a/c)^(1/8))*arctan((2*x - sqrt(-sqrt(2) + 2)*(a/c)^(1/8))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a
*c) + 1/8*(c*sqrt(sqrt(2) + 2)*(a/c)^(5/8)*e - a*d*sqrt(-sqrt(2) + 2)*(a/c)^(1/8))*arctan((2*x + sqrt(sqrt(2)
+ 2)*(a/c)^(1/8))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*c) + 1/8*(c*sqrt(sqrt(2) + 2)*(a/c)^(5/8)*e - a*d*sqrt(
-sqrt(2) + 2)*(a/c)^(1/8))*arctan((2*x - sqrt(sqrt(2) + 2)*(a/c)^(1/8))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*c
) - 1/16*(c*sqrt(-sqrt(2) + 2)*(a/c)^(5/8)*e + a*d*sqrt(sqrt(2) + 2)*(a/c)^(1/8))*log(x^2 + x*sqrt(sqrt(2) + 2
)*(a/c)^(1/8) + (a/c)^(1/4))/(a*c) + 1/16*(c*sqrt(-sqrt(2) + 2)*(a/c)^(5/8)*e + a*d*sqrt(sqrt(2) + 2)*(a/c)^(1
/8))*log(x^2 - x*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + (a/c)^(1/4))/(a*c) + 1/16*(c*sqrt(sqrt(2) + 2)*(a/c)^(5/8)*e
- a*d*sqrt(-sqrt(2) + 2)*(a/c)^(1/8))*log(x^2 + x*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + (a/c)^(1/4))/(a*c) - 1/16*(
c*sqrt(sqrt(2) + 2)*(a/c)^(5/8)*e - a*d*sqrt(-sqrt(2) + 2)*(a/c)^(1/8))*log(x^2 - x*sqrt(-sqrt(2) + 2)*(a/c)^(
1/8) + (a/c)^(1/4))/(a*c)