3.758 \(\int \frac{x^{5/2}}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=329 \[ \frac{9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac{9 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{17/8} c^{7/8}}-\frac{9 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{17/8} c^{7/8}}-\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{17/8} c^{7/8}}+\frac{9 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{17/8} c^{7/8}}+\frac{9 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac{9 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\frac{x^{7/2}}{8 a \left (a+c x^4\right )^2} \]

[Out]

x^(7/2)/(8*a*(a + c*x^4)^2) + (9*x^(7/2))/(64*a^2*(a + c*x^4)) - (9*ArcTan[1 - (
Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(17/8)*c^(7/8)) + (9*Arc
Tan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(17/8)*c^(7/8))
+ (9*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(17/8)*c^(7/8)) - (9*ArcTa
nh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(17/8)*c^(7/8)) + (9*Log[(-a)^(1/4)
- Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(17/8)*c^(7
/8)) - (9*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512
*Sqrt[2]*(-a)^(17/8)*c^(7/8))

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Rubi [A]  time = 0.62463, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8 \[ \frac{9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac{9 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{17/8} c^{7/8}}-\frac{9 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{17/8} c^{7/8}}-\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{17/8} c^{7/8}}+\frac{9 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{17/8} c^{7/8}}+\frac{9 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac{9 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\frac{x^{7/2}}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a + c*x^4)^3,x]

[Out]

x^(7/2)/(8*a*(a + c*x^4)^2) + (9*x^(7/2))/(64*a^2*(a + c*x^4)) - (9*ArcTan[1 - (
Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(17/8)*c^(7/8)) + (9*Arc
Tan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(17/8)*c^(7/8))
+ (9*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(17/8)*c^(7/8)) - (9*ArcTa
nh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(17/8)*c^(7/8)) + (9*Log[(-a)^(1/4)
- Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(17/8)*c^(7
/8)) - (9*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512
*Sqrt[2]*(-a)^(17/8)*c^(7/8))

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Rubi in Sympy [A]  time = 128.337, size = 309, normalized size = 0.94 \[ \frac{9 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{7}{8}} \left (- a\right )^{\frac{17}{8}}} - \frac{9 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{7}{8}} \left (- a\right )^{\frac{17}{8}}} + \frac{9 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{7}{8}} \left (- a\right )^{\frac{17}{8}}} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{512 c^{\frac{7}{8}} \left (- a\right )^{\frac{17}{8}}} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{512 c^{\frac{7}{8}} \left (- a\right )^{\frac{17}{8}}} - \frac{9 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{7}{8}} \left (- a\right )^{\frac{17}{8}}} + \frac{x^{\frac{7}{2}}}{8 a \left (a + c x^{4}\right )^{2}} + \frac{9 x^{\frac{7}{2}}}{64 a^{2} \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(c*x**4+a)**3,x)

[Out]

9*sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/
(1024*c**(7/8)*(-a)**(17/8)) - 9*sqrt(2)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8
) + c**(1/4)*x + (-a)**(1/4))/(1024*c**(7/8)*(-a)**(17/8)) + 9*atan(c**(1/8)*sqr
t(x)/(-a)**(1/8))/(256*c**(7/8)*(-a)**(17/8)) + 9*sqrt(2)*atan(sqrt(2)*c**(1/8)*
sqrt(x)/(-a)**(1/8) - 1)/(512*c**(7/8)*(-a)**(17/8)) + 9*sqrt(2)*atan(sqrt(2)*c*
*(1/8)*sqrt(x)/(-a)**(1/8) + 1)/(512*c**(7/8)*(-a)**(17/8)) - 9*atanh(c**(1/8)*s
qrt(x)/(-a)**(1/8))/(256*c**(7/8)*(-a)**(17/8)) + x**(7/2)/(8*a*(a + c*x**4)**2)
+ 9*x**(7/2)/(64*a**2*(a + c*x**4))

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Mathematica [A]  time = 0.984868, size = 427, normalized size = 1.3 \[ \frac{\frac{64 a^{9/8} x^{7/2}}{\left (a+c x^4\right )^2}+\frac{9 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{7/8}}-\frac{9 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{7/8}}+\frac{9 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{7/8}}-\frac{9 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{7/8}}+\frac{18 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{c^{7/8}}+\frac{18 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{c^{7/8}}-\frac{18 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{c^{7/8}}+\frac{18 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{c^{7/8}}+\frac{72 \sqrt [8]{a} x^{7/2}}{a+c x^4}}{512 a^{17/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(a + c*x^4)^3,x]

[Out]

((64*a^(9/8)*x^(7/2))/(a + c*x^4)^2 + (72*a^(1/8)*x^(7/2))/(a + c*x^4) + (18*Arc
Tan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8])/c^(7/8) + (18*Ar
cTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8])/c^(7/8) + (9*Co
s[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/c^(7/8)
- (9*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/c
^(7/8) - (18*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8])/
c^(7/8) + (18*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8])
/c^(7/8) + (9*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin
[Pi/8])/c^(7/8) - (9*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/
8]]*Sin[Pi/8])/c^(7/8))/(512*a^(17/8))

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Maple [C]  time = 0.026, size = 62, normalized size = 0.2 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ({\frac{17\,{x}^{7/2}}{128\,a}}+{\frac{9\,c{x}^{15/2}}{128\,{a}^{2}}} \right ) }+{\frac{9}{512\,{a}^{2}c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{\it \_R}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(c*x^4+a)^3,x)

[Out]

2*(17/128*x^(7/2)/a+9/128/a^2*c*x^(15/2))/(c*x^4+a)^2+9/512/a^2/c*sum(1/_R*ln(x^
(1/2)-_R),_R=RootOf(_Z^8*c+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{9 \, c x^{\frac{15}{2}} + 17 \, a x^{\frac{7}{2}}}{64 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + 9 \, \int \frac{x^{\frac{5}{2}}}{128 \,{\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

1/64*(9*c*x^(15/2) + 17*a*x^(7/2))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4) + 9*integra
te(1/128*x^(5/2)/(a^2*c*x^4 + a^3), x)

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Fricas [A]  time = 0.263203, size = 919, normalized size = 2.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

1/1024*sqrt(2)*(36*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^17*c^7))^(1/
8)*arctan(a^15*c^6*(-1/(a^17*c^7))^(7/8)/(sqrt(-a^13*c^5*(-1/(a^17*c^7))^(3/4) +
x) + sqrt(x))) + 9*sqrt(2)*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^17*c^7))^(1
/8)*log(a^15*c^6*(-1/(a^17*c^7))^(7/8) + sqrt(x)) - 9*sqrt(2)*(a^2*c^2*x^8 + 2*a
^3*c*x^4 + a^4)*(-1/(a^17*c^7))^(1/8)*log(-a^15*c^6*(-1/(a^17*c^7))^(7/8) + sqrt
(x)) + 36*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^17*c^7))^(1/8)*arctan(a^15*c^
6*(-1/(a^17*c^7))^(7/8)/(a^15*c^6*(-1/(a^17*c^7))^(7/8) + sqrt(2)*sqrt(x) + sqrt
(2*sqrt(2)*a^15*c^6*sqrt(x)*(-1/(a^17*c^7))^(7/8) - 2*a^13*c^5*(-1/(a^17*c^7))^(
3/4) + 2*x))) + 36*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^17*c^7))^(1/8)*arcta
n(-a^15*c^6*(-1/(a^17*c^7))^(7/8)/(a^15*c^6*(-1/(a^17*c^7))^(7/8) - sqrt(2)*sqrt
(x) - sqrt(-2*sqrt(2)*a^15*c^6*sqrt(x)*(-1/(a^17*c^7))^(7/8) - 2*a^13*c^5*(-1/(a
^17*c^7))^(3/4) + 2*x))) + 9*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^17*c^7))^(
1/8)*log(2*sqrt(2)*a^15*c^6*sqrt(x)*(-1/(a^17*c^7))^(7/8) - 2*a^13*c^5*(-1/(a^17
*c^7))^(3/4) + 2*x) - 9*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^17*c^7))^(1/8)*
log(-2*sqrt(2)*a^15*c^6*sqrt(x)*(-1/(a^17*c^7))^(7/8) - 2*a^13*c^5*(-1/(a^17*c^7
))^(3/4) + 2*x) + 8*sqrt(2)*(9*c*x^7 + 17*a*x^3)*sqrt(x))/(a^2*c^2*x^8 + 2*a^3*c
*x^4 + a^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(c*x**4+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.355832, size = 626, normalized size = 1.9 \[ \frac{9 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{512 \, a^{3}} + \frac{9 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{512 \, a^{3}} + \frac{9 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{512 \, a^{3}} + \frac{9 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{512 \, a^{3}} - \frac{9 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{1024 \, a^{3}} + \frac{9 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{1024 \, a^{3}} - \frac{9 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{1024 \, a^{3}} + \frac{9 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{1024 \, a^{3}} + \frac{9 \, c x^{\frac{15}{2}} + 17 \, a x^{\frac{7}{2}}}{64 \,{\left (c x^{4} + a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

9/512*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*s
qrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/a^3 + 9/512*sqrt(sqrt(2) + 2)*(a/c)^(7/
8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)
^(1/8)))/a^3 + 9/512*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*arctan((sqrt(sqrt(2) + 2)*(a
/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^3 + 9/512*sqrt(-sqrt(
2) + 2)*(a/c)^(7/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-s
qrt(2) + 2)*(a/c)^(1/8)))/a^3 - 9/1024*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*ln(sqrt(x)*
sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 + 9/1024*sqrt(sqrt(2) + 2)*
(a/c)^(7/8)*ln(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 - 9
/1024*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*ln(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) +
x + (a/c)^(1/4))/a^3 + 9/1024*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*ln(-sqrt(x)*sqrt(-
sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 + 1/64*(9*c*x^(15/2) + 17*a*x^(7
/2))/((c*x^4 + a)^2*a^2)