\(\int \frac {d+e x^4}{a+c x^8} \, dx\) [3]

Optimal result

Integrand size = 17, antiderivative size = 754 \[ \int \frac {d+e x^4}{a+c x^8} \, dx=-\frac {\sqrt {2-\sqrt {2}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac {\sqrt {2+\sqrt {2}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [4]{a}-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [4]{a}+\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [4]{a}-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} c^{5/8}}+\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [4]{a}+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} \sqrt [8]{c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1429, 1183, 648, 632, 210, 642} \[ \int \frac {d+e x^4}{a+c x^8} \, dx=-\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right ) \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right )}{8 a^{7/8} c^{5/8}}+\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right ) \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right )}{8 a^{7/8} c^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right ) \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right )}{8 a^{7/8} c^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right ) \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right )}{8 a^{7/8} c^{5/8}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} c^{5/8}}+\frac {\left (-\frac {\sqrt {a} e}{\sqrt {c}}+\sqrt {2} d+d\right ) \log \left (\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} \sqrt [8]{c}} \]

[In]

Int[(d + e*x^4)/(a + c*x^8),x]

[Out]

-1/8*(Sqrt[2 - Sqrt[2]]*((1 + Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) - 2*c^(1/8)*x)
/(Sqrt[2 + Sqrt[2]]*a^(1/8))])/(a^(7/8)*c^(5/8)) + (Sqrt[2 + Sqrt[2]]*((1 - Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*Ar
cTan[(Sqrt[2 + Sqrt[2]]*a^(1/8) - 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(7/8)*c^(5/8)) + (Sqrt[2 - S
qrt[2]]*((1 + Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2 + Sqrt[
2]]*a^(1/8))])/(8*a^(7/8)*c^(5/8)) - (Sqrt[2 + Sqrt[2]]*((1 - Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[2 +
Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(7/8)*c^(5/8)) + (((1 - Sqrt[2])*Sqrt[c]*d
- Sqrt[a]*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^(7/
8)*c^(5/8)) - (((1 - Sqrt[2])*Sqrt[c]*d - Sqrt[a]*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^(7/8)*c^(5/8)) - (((1 + Sqrt[2])*Sqrt[c]*d - Sqrt[a]*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^(7/8)*c^(5/8)) + ((d + Sqrt[2]*d -
(Sqrt[a]*e)/Sqrt[c])*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^(7/8)*c^(1/8))

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 632

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 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 648

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 1183

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 1429

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}+\left (-d+\frac {\sqrt {a} e}{\sqrt {c}}\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} \sqrt [4]{c}}+\frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}+\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\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} \sqrt [4]{c}} \\ & = \frac {\sqrt [8]{c} \int \frac {\frac {\sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} d}{c^{3/8}}-\left (\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\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}}+\frac {\sqrt [8]{c} \int \frac {\frac {\sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} d}{c^{3/8}}+\left (\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\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}}+\frac {\sqrt [8]{c} \int \frac {\frac {\sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} d}{c^{3/8}}-\left (\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (-d+\frac {\sqrt {a} e}{\sqrt {c}}\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}}+\frac {\sqrt [8]{c} \int \frac {\frac {\sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} d}{c^{3/8}}+\left (\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (-d+\frac {\sqrt {a} e}{\sqrt {c}}\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}} \\ & = -\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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} a^{3/4} c^{3/4}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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} a^{3/4} c^{3/4}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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} a^{3/4} c^{3/4}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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} a^{3/4} c^{3/4}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {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^{7/8} \sqrt [8]{c}} \\ & = \frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {c}}\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^{7/8} \sqrt [8]{c}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \text {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} a^{3/4} c^{3/4}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \text {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} a^{3/4} c^{3/4}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \text {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} a^{3/4} c^{3/4}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \text {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} a^{3/4} c^{3/4}} \\ & = -\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {c}}\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^{7/8} \sqrt [8]{c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 534, normalized size of antiderivative = 0.71 \[ \int \frac {d+e x^4}{a+c x^8} \, dx=\frac {-2 \sqrt [8]{a} \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{c} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \left (\sqrt {a} e \cos \left (\frac {\pi }{8}\right )+\sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )+2 \sqrt [8]{a} \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{c} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \left (\sqrt {a} e \cos \left (\frac {\pi }{8}\right )+\sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{c} x^2-2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac {\pi }{8}\right )\right ) \left (\sqrt {a} e \cos \left (\frac {\pi }{8}\right )+\sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )+\sqrt [8]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{c} x^2+2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac {\pi }{8}\right )\right ) \left (\sqrt {a} e \cos \left (\frac {\pi }{8}\right )+\sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )+\sqrt [8]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{c} x^2-2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac {\pi }{8}\right )\right ) \left (-\sqrt {c} d \cos \left (\frac {\pi }{8}\right )+\sqrt {a} e \sin \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{c} x^2+2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac {\pi }{8}\right )\right ) \left (-\sqrt {c} d \cos \left (\frac {\pi }{8}\right )+\sqrt {a} e \sin \left (\frac {\pi }{8}\right )\right )+2 \arctan \left (\frac {\sqrt [8]{c} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \left (\sqrt [8]{a} \sqrt {c} d \cos \left (\frac {\pi }{8}\right )-a^{5/8} e \sin \left (\frac {\pi }{8}\right )\right )+2 \arctan \left (\frac {\sqrt [8]{c} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \left (\sqrt [8]{a} \sqrt {c} d \cos \left (\frac {\pi }{8}\right )-a^{5/8} e \sin \left (\frac {\pi }{8}\right )\right )}{8 a c^{5/8}} \]

[In]

Integrate[(d + e*x^4)/(a + c*x^8),x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.05

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c}\)	\(34\)
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c}\)	\(34\)

[In]

int((e*x^4+d)/(c*x^8+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2749 vs. \(2 (514) = 1028\).

Time = 0.56 (sec) , antiderivative size = 2749, normalized size of antiderivative = 3.65 \[ \int \frac {d+e x^4}{a+c x^8} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x^4}{a+c x^8} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {d+e x^4}{a+c x^8} \, dx=\int { \frac {e x^{4} + d}{c x^{8} + a} \,d x } \]

[In]

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

[Out]

integrate((e*x^4 + d)/(c*x^8 + a), x)

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 593, normalized size of antiderivative = 0.79 \[ \int \frac {d+e x^4}{a+c x^8} \, dx=-\frac {{\left (e \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} - d \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{8 \, a} - \frac {{\left (e \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} - d \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{8 \, a} + \frac {{\left (e \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} + d \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{8 \, a} + \frac {{\left (e \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} + d \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{8 \, a} - \frac {{\left (e \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} - d \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{16 \, a} + \frac {{\left (e \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} - d \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{16 \, a} + \frac {{\left (e \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} + d \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{16 \, a} - \frac {{\left (e \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} + d \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{16 \, a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.09 (sec) , antiderivative size = 2510, normalized size of antiderivative = 3.33 \[ \int \frac {d+e x^4}{a+c x^8} \, dx=\text {Too large to display} \]

[In]

int((d + e*x^4)/(a + c*x^8),x)

[Out]

(atan((c^3*d^6*x - a^3*e^6*x + a*c^2*d^4*e^2*x - a^2*c*d^2*e^4*x + (2*d*e*x*(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^
4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2)))/(a^3*c^2))/(a*c^3*d^
5*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-
a^7*c^5)^(1/2))/(a^7*c^5))^(1/4) + a^5*c^3*e*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4
*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(5/4) - 2*a^2*c^2*d^3*e^2*((a^2*e^4*(-a^
7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/
(a^7*c^5))^(1/4) - 3*a^3*c*d*e^4*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a
^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(1/4)))*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c
^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(1/4))/4 - (atan((a
^3*e^6*x - c^3*d^6*x - a*c^2*d^4*e^2*x + a^2*c*d^2*e^4*x + (2*d*e*x*(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*
c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2)))/(a^3*c^2))/(a*c^3*d^5*(-(a^2
*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5
)^(1/2))/(a^7*c^5))^(1/4) + a^5*c^3*e*(-(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e
- 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(5/4) - 2*a^2*c^2*d^3*e^2*(-(a^2*e^4*(-a^7*c^5
)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*
c^5))^(1/4) - 3*a^3*c*d*e^4*(-(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c
^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(1/4)))*(-(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)
^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(1/4))/4 - atan((c^3*d
^6*x*1i - a^3*e^6*x*1i + a*c^2*d^4*e^2*x*1i - a^2*c*d^2*e^4*x*1i + (d*e*x*(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*
(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))*2i)/(a^3*c^2))/(a*c^3*d
^5*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(
-a^7*c^5)^(1/2))/(a^7*c^5))^(1/4) + a^5*c^3*e*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^
4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(5/4) - 2*a^2*c^2*d^3*e^2*((a^2*e^4*(-a
^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))
/(a^7*c^5))^(1/4) - 3*a^3*c*d*e^4*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*
a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(1/4)))*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*
c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(4096*a^7*c^5))^(1/4)*2i + at
an((a^3*e^6*x*1i - c^3*d^6*x*1i - a*c^2*d^4*e^2*x*1i + a^2*c*d^2*e^4*x*1i + (d*e*x*(a^2*e^4*(-a^7*c^5)^(1/2) +
c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))*2i)/(a^3*c^2))
/(a*c^3*d^5*(-(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c
*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(1/4) + a^5*c^3*e*(-(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2)
+ 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(5/4) - 2*a^2*c^2*d^3*e^2*(-
(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7
*c^5)^(1/2))/(a^7*c^5))^(1/4) - 3*a^3*c*d*e^4*(-(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c
^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(1/4)))*(-(a^2*e^4*(-a^7*c^5)^(1/2) +
c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(4096*a^7*c^5))
^(1/4)*2i