3.1466 \(\int \frac{1}{x^6 \left (a+b x^8\right )} \, dx\)

Optimal. Leaf size=277 \[ -\frac{b^{5/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{13/8}}+\frac{b^{5/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{13/8}}+\frac{b^{5/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}+\frac{b^{5/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{13/8}}-\frac{b^{5/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{13/8}}-\frac{b^{5/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac{1}{5 a x^5} \]

[Out]

-1/(5*a*x^5) + (b^(5/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(13/8)) + (b^(5/
8)*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(13/8)) - (b^(5/8
)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(13/8)) - (b^(5/8)
*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(13/8)) - (b^(5/8)*Log[(-a)^(1/4) - Sq
rt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(13/8)) + (b^(5/8)*Lo
g[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(13/
8))

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Rubi [A]  time = 0.542072, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.846 \[ -\frac{b^{5/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{13/8}}+\frac{b^{5/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{13/8}}+\frac{b^{5/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}+\frac{b^{5/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{13/8}}-\frac{b^{5/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{13/8}}-\frac{b^{5/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac{1}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^8)),x]

[Out]

-1/(5*a*x^5) + (b^(5/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(13/8)) + (b^(5/
8)*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(13/8)) - (b^(5/8
)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(13/8)) - (b^(5/8)
*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(13/8)) - (b^(5/8)*Log[(-a)^(1/4) - Sq
rt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(13/8)) + (b^(5/8)*Lo
g[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(13/
8))

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Rubi in Sympy [A]  time = 117.637, size = 255, normalized size = 0.92 \[ - \frac{\sqrt{2} b^{\frac{5}{8}} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{13}{8}}} + \frac{\sqrt{2} b^{\frac{5}{8}} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{13}{8}}} + \frac{b^{\frac{5}{8}} \operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{13}{8}}} - \frac{\sqrt{2} b^{\frac{5}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} - 1 \right )}}{8 \left (- a\right )^{\frac{13}{8}}} - \frac{\sqrt{2} b^{\frac{5}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} + 1 \right )}}{8 \left (- a\right )^{\frac{13}{8}}} - \frac{b^{\frac{5}{8}} \operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{13}{8}}} - \frac{1}{5 a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**8+a),x)

[Out]

-sqrt(2)*b**(5/8)*log(-sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b**(1/4)*x**2 + (-a)**(1
/4))/(16*(-a)**(13/8)) + sqrt(2)*b**(5/8)*log(sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b
**(1/4)*x**2 + (-a)**(1/4))/(16*(-a)**(13/8)) + b**(5/8)*atan(b**(1/8)*x/(-a)**(
1/8))/(4*(-a)**(13/8)) - sqrt(2)*b**(5/8)*atan(sqrt(2)*b**(1/8)*x/(-a)**(1/8) -
1)/(8*(-a)**(13/8)) - sqrt(2)*b**(5/8)*atan(sqrt(2)*b**(1/8)*x/(-a)**(1/8) + 1)/
(8*(-a)**(13/8)) - b**(5/8)*atanh(b**(1/8)*x/(-a)**(1/8))/(4*(-a)**(13/8)) - 1/(
5*a*x**5)

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Mathematica [A]  time = 0.344398, size = 395, normalized size = 1.43 \[ \frac{-8 a^{5/8}+10 b^{5/8} x^5 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+10 b^{5/8} x^5 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+10 b^{5/8} x^5 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-10 b^{5/8} x^5 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )-5 b^{5/8} x^5 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+5 b^{5/8} x^5 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+5 b^{5/8} x^5 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-5 b^{5/8} x^5 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{40 a^{13/8} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^8)),x]

[Out]

(-8*a^(5/8) + 10*b^(5/8)*x^5*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*C
os[Pi/8] - 10*b^(5/8)*x^5*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Cos[
Pi/8] - 5*b^(5/8)*x^5*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*
Sin[Pi/8]] + 5*b^(5/8)*x^5*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/
8)*x*Sin[Pi/8]] + 10*b^(5/8)*x^5*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8
]]*Sin[Pi/8] + 10*b^(5/8)*x^5*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*
Sin[Pi/8] + 5*b^(5/8)*x^5*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi
/8]]*Sin[Pi/8] - 5*b^(5/8)*x^5*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*C
os[Pi/8]]*Sin[Pi/8])/(40*a^(13/8)*x^5)

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Maple [C]  time = 0.008, size = 36, normalized size = 0.1 \[ -{\frac{1}{8\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}}}}-{\frac{1}{5\,a{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^8+a),x)

[Out]

-1/8/a*sum(1/_R^5*ln(x-_R),_R=RootOf(_Z^8*b+a))-1/5/a/x^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{b \int \frac{x^{2}}{b x^{8} + a}\,{d x}}{a} - \frac{1}{5 \, a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^6),x, algorithm="maxima")

[Out]

-b*integrate(x^2/(b*x^8 + a), x)/a - 1/5/(a*x^5)

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Fricas [A]  time = 0.238346, size = 703, normalized size = 2.54 \[ \frac{\sqrt{2}{\left (20 \, \sqrt{2} a x^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}}}{b^{2} x + b^{2} \sqrt{\frac{a^{10} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{4}} + b^{4} x^{2}}{b^{4}}}}\right ) + 5 \, \sqrt{2} a x^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{1}{8}} \log \left (a^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}} + b^{2} x\right ) - 5 \, \sqrt{2} a x^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{1}{8}} \log \left (-a^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}} + b^{2} x\right ) - 20 \, a x^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}}}{a^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}} + \sqrt{2} b^{2} x + \sqrt{2} b^{2} \sqrt{\frac{a^{10} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{4}} + \sqrt{2} a^{5} b^{2} x \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}} + b^{4} x^{2}}{b^{4}}}}\right ) - 20 \, a x^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}}}{a^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}} - \sqrt{2} b^{2} x - \sqrt{2} b^{2} \sqrt{\frac{a^{10} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{4}} - \sqrt{2} a^{5} b^{2} x \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}} + b^{4} x^{2}}{b^{4}}}}\right ) - 5 \, a x^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{1}{8}} \log \left (a^{10} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{4}} + \sqrt{2} a^{5} b^{2} x \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}} + b^{4} x^{2}\right ) + 5 \, a x^{5} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{1}{8}} \log \left (a^{10} \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{4}} - \sqrt{2} a^{5} b^{2} x \left (-\frac{b^{5}}{a^{13}}\right )^{\frac{3}{8}} + b^{4} x^{2}\right ) - 8 \, \sqrt{2}\right )}}{80 \, a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^6),x, algorithm="fricas")

[Out]

1/80*sqrt(2)*(20*sqrt(2)*a*x^5*(-b^5/a^13)^(1/8)*arctan(a^5*(-b^5/a^13)^(3/8)/(b
^2*x + b^2*sqrt((a^10*(-b^5/a^13)^(3/4) + b^4*x^2)/b^4))) + 5*sqrt(2)*a*x^5*(-b^
5/a^13)^(1/8)*log(a^5*(-b^5/a^13)^(3/8) + b^2*x) - 5*sqrt(2)*a*x^5*(-b^5/a^13)^(
1/8)*log(-a^5*(-b^5/a^13)^(3/8) + b^2*x) - 20*a*x^5*(-b^5/a^13)^(1/8)*arctan(a^5
*(-b^5/a^13)^(3/8)/(a^5*(-b^5/a^13)^(3/8) + sqrt(2)*b^2*x + sqrt(2)*b^2*sqrt((a^
10*(-b^5/a^13)^(3/4) + sqrt(2)*a^5*b^2*x*(-b^5/a^13)^(3/8) + b^4*x^2)/b^4))) - 2
0*a*x^5*(-b^5/a^13)^(1/8)*arctan(-a^5*(-b^5/a^13)^(3/8)/(a^5*(-b^5/a^13)^(3/8) -
sqrt(2)*b^2*x - sqrt(2)*b^2*sqrt((a^10*(-b^5/a^13)^(3/4) - sqrt(2)*a^5*b^2*x*(-
b^5/a^13)^(3/8) + b^4*x^2)/b^4))) - 5*a*x^5*(-b^5/a^13)^(1/8)*log(a^10*(-b^5/a^1
3)^(3/4) + sqrt(2)*a^5*b^2*x*(-b^5/a^13)^(3/8) + b^4*x^2) + 5*a*x^5*(-b^5/a^13)^
(1/8)*log(a^10*(-b^5/a^13)^(3/4) - sqrt(2)*a^5*b^2*x*(-b^5/a^13)^(3/8) + b^4*x^2
) - 8*sqrt(2))/(a*x^5)

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Sympy [A]  time = 5.4171, size = 36, normalized size = 0.13 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{13} + b^{5}, \left ( t \mapsto t \log{\left (\frac{512 t^{3} a^{5}}{b^{2}} + x \right )} \right )\right )} - \frac{1}{5 a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**8+a),x)

[Out]

RootSum(16777216*_t**8*a**13 + b**5, Lambda(_t, _t*log(512*_t**3*a**5/b**2 + x))
) - 1/(5*a*x**5)

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GIAC/XCAS [A]  time = 0.254901, size = 601, normalized size = 2.17 \[ \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{3}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{3}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{3}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{3}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{3}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{3}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} + \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{3}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{3}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{1}{5 \, a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^6),x, algorithm="giac")

[Out]

1/8*b*sqrt(-sqrt(2) + 2)*(a/b)^(3/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(a/b)^(1/8
))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)))/a^2 + 1/8*b*sqrt(-sqrt(2) + 2)*(a/b)^(3/8)*a
rctan((2*x - sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)))/a^
2 - 1/8*b*sqrt(sqrt(2) + 2)*(a/b)^(3/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(a/b)^(1
/8))/(sqrt(-sqrt(2) + 2)*(a/b)^(1/8)))/a^2 - 1/8*b*sqrt(sqrt(2) + 2)*(a/b)^(3/8)
*arctan((2*x - sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(-sqrt(2) + 2)*(a/b)^(1/8)))/
a^2 - 1/16*b*sqrt(-sqrt(2) + 2)*(a/b)^(3/8)*ln(x^2 + x*sqrt(sqrt(2) + 2)*(a/b)^(
1/8) + (a/b)^(1/4))/a^2 + 1/16*b*sqrt(-sqrt(2) + 2)*(a/b)^(3/8)*ln(x^2 - x*sqrt(
sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/a^2 + 1/16*b*sqrt(sqrt(2) + 2)*(a/b)^(3/
8)*ln(x^2 + x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/a^2 - 1/16*b*sqrt(sq
rt(2) + 2)*(a/b)^(3/8)*ln(x^2 - x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/
a^2 - 1/5/(a*x^5)