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)) - (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 + Sq
rt[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 - 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)) + ((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 + Sqr
t[c]*e)*Log[a^(1/4) + Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqr
t[2*(2 + Sqrt[2])]*a^(3/8)*c^(9/8))

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Rubi [A]  time = 3.00955, 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 \[ -\frac{\left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \log \left (-\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \log \left (\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (-\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{3/8} c^{9/8}}+\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2+\sqrt{2}} \left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{a} d+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}+\frac{\sqrt{2+\sqrt{2}} \left (\sqrt{a} \left (d-\sqrt{2} d\right )+\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}+\frac{d x}{c} \]

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)) - (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 + Sq
rt[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 - 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)) + ((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 + Sqr
t[c]*e)*Log[a^(1/4) + Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqr
t[2*(2 + Sqrt[2])]*a^(3/8)*c^(9/8))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d+e/x**4)/(c+a/x**8),x)

[Out]

Timed out

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Mathematica [A]  time = 2.9331, 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)*Sqrt[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[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[Pi/8] + a^(5/8)*c*e*Sin[Pi/8]))/(8*a*c^(13/8))

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Maple [C]  time = 0.007, size = 45, normalized size = 0.1 \[{\frac{dx}{c}}+{\frac{1}{8\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+a \right ) }{\frac{ \left ({{\it \_R}}^{4}ce-ad \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d+e/x^4)/(c+a/x^8),x)

[Out]

1/c*d*x+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. \[ \frac{d x}{c} + \frac{\int \frac{c e x^{4} - a d}{c x^{8} + a}\,{d x}}{c} \]

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 [A]  time = 0.515055, size = 3742, normalized size = 4.97 \[ \text{result too large to display} \]

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(
(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)/((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^3*d^6 - 5*a^2*c*d^4*e^2 - 5*a*c^2*d^2*
e^4 + c^3*e^6)*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)))) - 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((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
)/((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^3*d^6 - 5*a^2*
c*d^4*e^2 - 5*a*c^2*d^2*e^4 + c^3*e^6)*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*s
qrt(-(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 - 1
0*a^2*c^2*d^4*e^4 - 4*a*c^3*d^2*e^6 + c^4*e^8)))) - c*((a*c^4*sqrt(-(a^4*d^8 - 1
2*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*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*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)) - 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 = 47.5118, size = 204, normalized size = 0.27 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{3} c^{9} + t^{4} \left (- 32768 a^{3} c^{5} d^{3} e + 32768 a^{2} c^{6} d e^{3}\right ) + a^{4} d^{8} + 4 a^{3} c d^{6} e^{2} + 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{2} e^{6} + c^{4} e^{8}, \left ( t \mapsto t \log{\left (x + \frac{- 32768 t^{5} a^{2} c^{6} e - 8 t a^{3} c d^{5} + 80 t a^{2} c^{2} d^{3} e^{2} - 40 t a c^{3} d e^{4}}{a^{3} d^{6} - 5 a^{2} c d^{4} e^{2} - 5 a c^{2} d^{2} e^{4} + c^{3} e^{6}} \right )} \right )\right )} + \frac{d x}{c} \]

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/XCAS [A]  time = 0.311153, size = 873, normalized size = 1.16 \[ \text{result too large to display} \]

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*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) + 1/8*(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))*ln(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))*ln(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))*ln(x^2 + x*sqr
t(-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))*ln(x^2 - x*sqrt(-sqrt(2) + 2)*(
a/c)^(1/8) + (a/c)^(1/4))/(a*c)