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

Optimal. Leaf size=329 \[ \frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}} \]

[Out]

((Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(c^(1/8)*x)/a^(1/8)])/(4*a^(7/8)*c^(5/8)) - ((d
- (Sqrt[a]*e)/Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/8)*x)/a^(1/8)])/(4*Sqrt[2]*a^(7/
8)*c^(1/8)) + ((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/8)*x)/a^(1/8)]
)/(4*Sqrt[2]*a^(7/8)*c^(1/8)) + ((Sqrt[c]*d + Sqrt[a]*e)*ArcTanh[(c^(1/8)*x)/a^(
1/8)])/(4*a^(7/8)*c^(5/8)) - ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) - Sqrt[2]*a^
(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a^(7/8)*c^(1/8)) + ((d - (Sqrt[a]*e)/
Sqrt[c])*Log[a^(1/4) + Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a^(7
/8)*c^(1/8))

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Rubi [A]  time = 0.456416, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556 \[ \frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{c}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^4)/(a - c*x^8),x]

[Out]

((Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(c^(1/8)*x)/a^(1/8)])/(4*a^(7/8)*c^(5/8)) - ((d
- (Sqrt[a]*e)/Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/8)*x)/a^(1/8)])/(4*Sqrt[2]*a^(7/
8)*c^(1/8)) + ((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/8)*x)/a^(1/8)]
)/(4*Sqrt[2]*a^(7/8)*c^(1/8)) + ((Sqrt[c]*d + Sqrt[a]*e)*ArcTanh[(c^(1/8)*x)/a^(
1/8)])/(4*a^(7/8)*c^(5/8)) - ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) - Sqrt[2]*a^
(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a^(7/8)*c^(1/8)) + ((d - (Sqrt[a]*e)/
Sqrt[c])*Log[a^(1/4) + Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a^(7
/8)*c^(1/8))

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Rubi in Sympy [A]  time = 77.9307, size = 304, normalized size = 0.92 \[ \frac{\sqrt{2} \left (\sqrt{a} e - \sqrt{c} d\right ) \log{\left (- \sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x + \sqrt [4]{a} + \sqrt [4]{c} x^{2} \right )}}{16 a^{\frac{7}{8}} c^{\frac{5}{8}}} - \frac{\sqrt{2} \left (\sqrt{a} e - \sqrt{c} d\right ) \log{\left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x + \sqrt [4]{a} + \sqrt [4]{c} x^{2} \right )}}{16 a^{\frac{7}{8}} c^{\frac{5}{8}}} + \frac{\sqrt{2} \left (\sqrt{a} e - \sqrt{c} d\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}} \right )}}{8 a^{\frac{7}{8}} c^{\frac{5}{8}}} - \frac{\sqrt{2} \left (\sqrt{a} e - \sqrt{c} d\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [8]{c} x}{\sqrt [8]{a}} \right )}}{8 a^{\frac{7}{8}} c^{\frac{5}{8}}} + \frac{\left (\sqrt{a} e + \sqrt{c} d\right ) \operatorname{atan}{\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}} \right )}}{4 a^{\frac{7}{8}} c^{\frac{5}{8}}} + \frac{\left (\sqrt{a} e + \sqrt{c} d\right ) \operatorname{atanh}{\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}} \right )}}{4 a^{\frac{7}{8}} c^{\frac{5}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*(sqrt(a)*e - sqrt(c)*d)*log(-sqrt(2)*a**(1/8)*c**(1/8)*x + a**(1/4) + c*
*(1/4)*x**2)/(16*a**(7/8)*c**(5/8)) - sqrt(2)*(sqrt(a)*e - sqrt(c)*d)*log(sqrt(2
)*a**(1/8)*c**(1/8)*x + a**(1/4) + c**(1/4)*x**2)/(16*a**(7/8)*c**(5/8)) + sqrt(
2)*(sqrt(a)*e - sqrt(c)*d)*atan(1 - sqrt(2)*c**(1/8)*x/a**(1/8))/(8*a**(7/8)*c**
(5/8)) - sqrt(2)*(sqrt(a)*e - sqrt(c)*d)*atan(1 + sqrt(2)*c**(1/8)*x/a**(1/8))/(
8*a**(7/8)*c**(5/8)) + (sqrt(a)*e + sqrt(c)*d)*atan(c**(1/8)*x/a**(1/8))/(4*a**(
7/8)*c**(5/8)) + (sqrt(a)*e + sqrt(c)*d)*atanh(c**(1/8)*x/a**(1/8))/(4*a**(7/8)*
c**(5/8))

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Mathematica [A]  time = 0.233701, size = 425, normalized size = 1.29 \[ \frac{\left (a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a c^{5/8}}-\frac{\left (a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt{2} a c^{5/8}}-\frac{\left (a^{5/8} e+\sqrt [8]{a} \sqrt{c} d\right ) \log \left (\sqrt [8]{a}-\sqrt [8]{c} x\right )}{8 a c^{5/8}}-\frac{\left (-a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \log \left (\sqrt [8]{a}+\sqrt [8]{c} x\right )}{8 a c^{5/8}}+\frac{\left (a^{5/8} e+\sqrt [8]{a} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a c^{5/8}}-\frac{\left (a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{2 \sqrt [8]{c} x-\sqrt{2} \sqrt [8]{a}}{\sqrt{2} \sqrt [8]{a}}\right )}{4 \sqrt{2} a c^{5/8}}-\frac{\left (a^{5/8} e-\sqrt [8]{a} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt{2} \sqrt [8]{a}}\right )}{4 \sqrt{2} a c^{5/8}} \]

Antiderivative was successfully verified.

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

[Out]

((a^(1/8)*Sqrt[c]*d + a^(5/8)*e)*ArcTan[(c^(1/8)*x)/a^(1/8)])/(4*a*c^(5/8)) - ((
-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*ArcTan[(-(Sqrt[2]*a^(1/8)) + 2*c^(1/8)*x)/(Sqr
t[2]*a^(1/8))])/(4*Sqrt[2]*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*ArcT
an[(Sqrt[2]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2]*a^(1/8))])/(4*Sqrt[2]*a*c^(5/8)) - (
(a^(1/8)*Sqrt[c]*d + a^(5/8)*e)*Log[a^(1/8) - c^(1/8)*x])/(8*a*c^(5/8)) - ((-(a^
(1/8)*Sqrt[c]*d) - a^(5/8)*e)*Log[a^(1/8) + c^(1/8)*x])/(8*a*c^(5/8)) + ((-(a^(1
/8)*Sqrt[c]*d) + a^(5/8)*e)*Log[a^(1/4) - Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^
2])/(8*Sqrt[2]*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*Log[a^(1/4) + Sq
rt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a*c^(5/8))

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Maple [C]  time = 0.02, size = 39, 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 = 0.492967, size = 3752, normalized size = 11.4 \[ \text{result too large to display} \]

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*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)) - 1/8*(-(a^3*c^2*sqr
t((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 = 36.8716, size = 202, normalized size = 0.61 \[ - \operatorname{RootSum}{\left (16777216 t^{8} a^{7} c^{5} + t^{4} \left (- 32768 a^{5} c^{3} d e^{3} - 32768 a^{4} c^{4} d^{3} e\right ) - a^{4} e^{8} + 4 a^{3} c d^{2} e^{6} - 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{6} e^{2} - c^{4} d^{8}, \left ( t \mapsto t \log{\left (x + \frac{- 32768 t^{5} a^{5} c^{3} e + 40 t a^{3} c d e^{4} + 80 t a^{2} c^{2} d^{3} e^{2} + 8 t a c^{3} d^{5}}{a^{3} e^{6} + 5 a^{2} c d^{2} e^{4} - 5 a c^{2} d^{4} e^{2} - c^{3} d^{6}} \right )} \right )\right )} \]

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 = 0.310602, size = 855, normalized size = 2.6 \[ \text{result too large to display} \]

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))*arct
an((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))*arct
an((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))*arct
an((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))*arct
an((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))*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(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