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

Optimal. Leaf size=754 \[ \frac{\left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{7/8} c^{5/8}}-\frac{\left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{7/8} c^{5/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{7/8} c^{5/8}}-\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2+\sqrt{2}} \left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac{\left (-\frac{\sqrt{a} e}{\sqrt{c}}+\sqrt{2} d+d\right ) \log \left (\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{7/8} \sqrt [8]{c}} \]

[Out]

-(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)/(Sq
rt[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)*ArcT
an[(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 - Sqr
t[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 + S
qrt[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 - (S
qrt[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))

________________________________________________________________________________________

Rubi [A]  time = 1.24683, antiderivative size = 754, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {1415, 1169, 634, 618, 204, 628} \[ \frac{\left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{7/8} c^{5/8}}-\frac{\left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{7/8} c^{5/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{7/8} c^{5/8}}-\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2+\sqrt{2}} \left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac{\left (-\frac{\sqrt{a} e}{\sqrt{c}}+\sqrt{2} d+d\right ) \log \left (\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{7/8} \sqrt [8]{c}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^4)/(a + c*x^8),x]

[Out]

-(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)/(Sq
rt[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)*ArcT
an[(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 - Sqr
t[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 + S
qrt[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 - (S
qrt[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 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+e x^4}{a+c x^8} \, dx &=\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 ) \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} a^{3/4} c^{3/4}}+\frac{\left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} 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} a^{3/4} c^{3/4}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} 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} a^{3/4} c^{3/4}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} 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} 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]  time = 0.623322, size = 534, normalized size = 0.71 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt [8]{c} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right ) \left (\sqrt [8]{a} \sqrt{c} d \cos \left (\frac{\pi }{8}\right )-a^{5/8} e \sin \left (\frac{\pi }{8}\right )\right )+2 \tan ^{-1}\left (\frac{\sqrt [8]{c} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right ) \left (\sqrt [8]{a} \sqrt{c} d \cos \left (\frac{\pi }{8}\right )-a^{5/8} e \sin \left (\frac{\pi }{8}\right )\right )-\sqrt [8]{a} \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (\sqrt{a} e \cos \left (\frac{\pi }{8}\right )+\sqrt{c} d \sin \left (\frac{\pi }{8}\right )\right )+\sqrt [8]{a} \log \left (2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (\sqrt{a} e \cos \left (\frac{\pi }{8}\right )+\sqrt{c} d \sin \left (\frac{\pi }{8}\right )\right )+\sqrt [8]{a} \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (\sqrt{a} e \sin \left (\frac{\pi }{8}\right )-\sqrt{c} d \cos \left (\frac{\pi }{8}\right )\right )-\sqrt [8]{a} \log \left (2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (\sqrt{a} e \sin \left (\frac{\pi }{8}\right )-\sqrt{c} d \cos \left (\frac{\pi }{8}\right )\right )-2 \sqrt [8]{a} \left (\sqrt{a} e \cos \left (\frac{\pi }{8}\right )+\sqrt{c} d \sin \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sqrt [8]{a} \left (\sqrt{a} e \cos \left (\frac{\pi }{8}\right )+\sqrt{c} d \sin \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{8 a c^{5/8}} \]

Antiderivative was successfully verified.

[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))

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Maple [C]  time = 0.023, size = 34, normalized size = 0.1 \begin{align*}{\frac{1}{8\,c}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+a \right ) }{\frac{ \left ({{\it \_R}}^{4}e+d \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/c*sum((_R^4*e+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*} \int \frac{e x^{4} + d}{c x^{8} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x^4 + d)/(c*x^8 + a), x)

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Fricas [B]  time = 3.36605, size = 6978, normalized size = 9.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 12.6218, size = 199, normalized size = 0.26 \begin{align*} \operatorname{RootSum}{\left (16777216 t^{8} a^{7} c^{5} + t^{4} \left (- 32768 a^{5} c^{3} d e^{3} + 32768 a^{4} c^{4} d^{3} e\right ) + a^{4} e^{8} + 4 a^{3} c d^{2} e^{6} + 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{6} e^{2} + c^{4} d^{8}, \left ( t \mapsto t \log{\left (x + \frac{- 32768 t^{5} a^{5} c^{3} e + 40 t a^{3} c d e^{4} - 80 t a^{2} c^{2} d^{3} e^{2} + 8 t a c^{3} d^{5}}{a^{3} e^{6} - 5 a^{2} c d^{2} e^{4} - 5 a c^{2} d^{4} e^{2} + c^{3} d^{6}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(16777216*_t**8*a**7*c**5 + _t**4*(-32768*a**5*c**3*d*e**3 + 32768*a**4*c**4*d**3*e) + a**4*e**8 + 4*a*
*3*c*d**2*e**6 + 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 + c**4*d**8, Lambda(_t, _t*log(x + (-32768*_t**5*a
**5*c**3*e + 40*_t*a**3*c*d*e**4 - 80*_t*a**2*c**2*d**3*e**2 + 8*_t*a*c**3*d**5)/(a**3*e**6 - 5*a**2*c*d**2*e*
*4 - 5*a*c**2*d**4*e**2 + c**3*d**6))))

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Giac [A]  time = 1.1892, size = 811, normalized size = 1.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(sqrt(-sqrt(2) + 2)*(a/c)^(5/8)*e - 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*(sqrt(-sqrt(2) + 2)*(a/c)^(5/8)*e - 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*(sqrt(sqrt(2)
+ 2)*(a/c)^(5/8)*e + 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*(sqrt(sqrt(2) + 2)*(a/c)^(5/8)*e + 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*(sqrt(-sqrt(2) + 2)*(a/c)^(5/8
)*e - 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*(sqrt
(-sqrt(2) + 2)*(a/c)^(5/8)*e - 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*(sqrt(sqrt(2) + 2)*(a/c)^(5/8)*e + 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*(sqrt(sqrt(2) + 2)*(a/c)^(5/8)*e + 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