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)/(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)) + (Sq
rt[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)) - (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])*Sq
rt[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 - S
qrt[a]*e)*Log[a^(1/4) - Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*S
qrt[2*(2 + Sqrt[2])]*a^(7/8)*c^(5/8)) + ((d + Sqrt[2]*d - (Sqrt[a]*e)/Sqrt[c])*L
og[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))

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Rubi [A]  time = 2.79363, 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 \[ \frac{\left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{7/8} c^{5/8}}-\frac{\left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt{2-\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )} a^{7/8} c^{5/8}}-\frac{\left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{7/8} c^{5/8}}-\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2+\sqrt{2}} \left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \left (\left (1+\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \left (\left (1-\sqrt{2}\right ) \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac{\left (-\frac{\sqrt{a} e}{\sqrt{c}}+\sqrt{2} d+d\right ) \log \left (\sqrt{2+\sqrt{2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )} a^{7/8} \sqrt [8]{c}} \]

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)/(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)) + (Sq
rt[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)) - (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])*Sq
rt[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 - S
qrt[a]*e)*Log[a^(1/4) - Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*S
qrt[2*(2 + Sqrt[2])]*a^(7/8)*c^(5/8)) + ((d + Sqrt[2]*d - (Sqrt[a]*e)/Sqrt[c])*L
og[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))

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

[Out]

Timed out

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Mathematica [A]  time = 1.59334, 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*Sin[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[Pi/8]) + 2*ArcTan[(c^(1/8)*x*Se
c[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 = 0.027, size = 34, normalized size = 0.1 \[{\frac{1}{8\,c}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+a \right ) }{\frac{ \left ({{\it \_R}}^{4}e+d \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \]

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

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 [A]  time = 0.485574, size = 3762, normalized size = 4.99 \[ \text{result too large to display} \]

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
((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)/((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 + (c^3*d^6 - 5*a*c^2*d^4*e^2 - 5*a^2*
c*d^2*e^4 + a^3*e^6)*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)))) + 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((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)/((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 +
(c^3*d^6 - 5*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^4 + a^3*e^6)*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)))) + 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*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))

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Sympy [A]  time = 37.0573, size = 199, normalized size = 0.26 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{7} c^{5} + t^{4} \left (- 32768 a^{5} c^{3} d e^{3} + 32768 a^{4} c^{4} d^{3} e\right ) + a^{4} e^{8} + 4 a^{3} c d^{2} e^{6} + 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{6} e^{2} + c^{4} d^{8}, \left ( t \mapsto t \log{\left (x + \frac{- 32768 t^{5} a^{5} c^{3} e + 40 t a^{3} c d e^{4} - 80 t a^{2} c^{2} d^{3} e^{2} + 8 t a c^{3} d^{5}}{a^{3} e^{6} - 5 a^{2} c d^{2} e^{4} - 5 a c^{2} d^{4} e^{2} + c^{3} d^{6}} \right )} \right )\right )} \]

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/XCAS [A]  time = 0.31174, size = 811, normalized size = 1.08 \[ \text{result too large to display} \]

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