3.752 \(\int \frac{1}{x^{3/2} \left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=320 \[ -\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}-\frac{9 \sqrt [8]{c} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}} \]

[Out]

-9/(4*a^2*Sqrt[x]) + 1/(4*a*Sqrt[x]*(a + c*x^4)) + (9*c^(1/8)*ArcTan[1 - (Sqrt[2
]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*Sqrt[2]*(-a)^(17/8)) - (9*c^(1/8)*ArcTan[1 +
(Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*Sqrt[2]*(-a)^(17/8)) - (9*c^(1/8)*Ar
cTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-a)^(17/8)) + (9*c^(1/8)*ArcTanh[(c^(1/
8)*Sqrt[x])/(-a)^(1/8)])/(16*(-a)^(17/8)) - (9*c^(1/8)*Log[(-a)^(1/4) - Sqrt[2]*
(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(17/8)) + (9*c^(1/8)*L
og[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a
)^(17/8))

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Rubi [A]  time = 0.614727, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.867 \[ -\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}-\frac{9 \sqrt [8]{c} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(a + c*x^4)^2),x]

[Out]

-9/(4*a^2*Sqrt[x]) + 1/(4*a*Sqrt[x]*(a + c*x^4)) + (9*c^(1/8)*ArcTan[1 - (Sqrt[2
]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*Sqrt[2]*(-a)^(17/8)) - (9*c^(1/8)*ArcTan[1 +
(Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*Sqrt[2]*(-a)^(17/8)) - (9*c^(1/8)*Ar
cTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(16*(-a)^(17/8)) + (9*c^(1/8)*ArcTanh[(c^(1/
8)*Sqrt[x])/(-a)^(1/8)])/(16*(-a)^(17/8)) - (9*c^(1/8)*Log[(-a)^(1/4) - Sqrt[2]*
(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a)^(17/8)) + (9*c^(1/8)*L
og[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(32*Sqrt[2]*(-a
)^(17/8))

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Rubi in Sympy [A]  time = 127.681, size = 303, normalized size = 0.95 \[ - \frac{9 \sqrt{2} \sqrt [8]{c} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 \left (- a\right )^{\frac{17}{8}}} + \frac{9 \sqrt{2} \sqrt [8]{c} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{64 \left (- a\right )^{\frac{17}{8}}} - \frac{9 \sqrt [8]{c} \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 \left (- a\right )^{\frac{17}{8}}} - \frac{9 \sqrt{2} \sqrt [8]{c} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{32 \left (- a\right )^{\frac{17}{8}}} - \frac{9 \sqrt{2} \sqrt [8]{c} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{32 \left (- a\right )^{\frac{17}{8}}} + \frac{9 \sqrt [8]{c} \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{16 \left (- a\right )^{\frac{17}{8}}} + \frac{1}{4 a \sqrt{x} \left (a + c x^{4}\right )} - \frac{9}{4 a^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(3/2)/(c*x**4+a)**2,x)

[Out]

-9*sqrt(2)*c**(1/8)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a
)**(1/4))/(64*(-a)**(17/8)) + 9*sqrt(2)*c**(1/8)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-
a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/(64*(-a)**(17/8)) - 9*c**(1/8)*atan(c**(1/
8)*sqrt(x)/(-a)**(1/8))/(16*(-a)**(17/8)) - 9*sqrt(2)*c**(1/8)*atan(sqrt(2)*c**(
1/8)*sqrt(x)/(-a)**(1/8) - 1)/(32*(-a)**(17/8)) - 9*sqrt(2)*c**(1/8)*atan(sqrt(2
)*c**(1/8)*sqrt(x)/(-a)**(1/8) + 1)/(32*(-a)**(17/8)) + 9*c**(1/8)*atanh(c**(1/8
)*sqrt(x)/(-a)**(1/8))/(16*(-a)**(17/8)) + 1/(4*a*sqrt(x)*(a + c*x**4)) - 9/(4*a
**2*sqrt(x))

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Mathematica [A]  time = 1.3212, size = 419, normalized size = 1.31 \[ \frac{-\frac{8 \sqrt [8]{a} c x^{7/2}}{a+c x^4}-9 \sqrt [8]{c} \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+9 \sqrt [8]{c} \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-9 \sqrt [8]{c} \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+9 \sqrt [8]{c} \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-18 \sqrt [8]{c} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )-18 \sqrt [8]{c} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+18 \sqrt [8]{c} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-18 \sqrt [8]{c} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )-\frac{64 \sqrt [8]{a}}{\sqrt{x}}}{32 a^{17/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(3/2)*(a + c*x^4)^2),x]

[Out]

((-64*a^(1/8))/Sqrt[x] - (8*a^(1/8)*c*x^(7/2))/(a + c*x^4) - 18*c^(1/8)*ArcTan[(
c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8] - 18*c^(1/8)*ArcTan[(c
^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8] - 9*c^(1/8)*Cos[Pi/8]*L
og[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]] + 9*c^(1/8)*Cos[Pi
/8]*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]] + 18*c^(1/8)*
ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] - 18*c^(1/8)*A
rcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] - 9*c^(1/8)*Log
[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8] + 9*c^(1/8
)*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*Sin[Pi/8])/(32*
a^(17/8))

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Maple [C]  time = 0.023, size = 56, normalized size = 0.2 \[ -2\,{\frac{1}{{a}^{2}\sqrt{x}}}-{\frac{c}{4\,{a}^{2} \left ( c{x}^{4}+a \right ) }{x}^{{\frac{7}{2}}}}-{\frac{9}{32\,{a}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{\it \_R}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(c*x^4+a)^2,x)

[Out]

-2/a^2/x^(1/2)-1/4/a^2*c*x^(7/2)/(c*x^4+a)-9/32/a^2*sum(1/_R*ln(x^(1/2)-_R),_R=R
ootOf(_Z^8*c+a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*x^(3/2)),x, algorithm="maxima")

[Out]

-9*c*integrate(1/8*x^(5/2)/(a^2*c*x^4 + a^3), x) - 1/4*(9*c*x^(7/2) + 8*a/sqrt(x
))/(a^2*c*x^4 + a^3)

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Fricas [A]  time = 0.279851, size = 755, normalized size = 2.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*x^(3/2)),x, algorithm="fricas")

[Out]

-1/64*sqrt(2)*(36*sqrt(2)*(a^2*c*x^4 + a^3)*sqrt(x)*(-c/a^17)^(1/8)*arctan(47829
69*a^15*(-c/a^17)^(7/8)/(4782969*c*sqrt(x) + sqrt(-22876792454961*a^13*c*(-c/a^1
7)^(3/4) + 22876792454961*c^2*x))) + 9*sqrt(2)*(a^2*c*x^4 + a^3)*sqrt(x)*(-c/a^1
7)^(1/8)*log(4782969*a^15*(-c/a^17)^(7/8) + 4782969*c*sqrt(x)) - 9*sqrt(2)*(a^2*
c*x^4 + a^3)*sqrt(x)*(-c/a^17)^(1/8)*log(-4782969*a^15*(-c/a^17)^(7/8) + 4782969
*c*sqrt(x)) + 36*(a^2*c*x^4 + a^3)*sqrt(x)*(-c/a^17)^(1/8)*arctan(-4782969*a^15*
(-c/a^17)^(7/8)/(4782969*a^15*(-c/a^17)^(7/8) - 4782969*sqrt(2)*c*sqrt(x) - sqrt
(-45753584909922*sqrt(2)*a^15*c*sqrt(x)*(-c/a^17)^(7/8) - 45753584909922*a^13*c*
(-c/a^17)^(3/4) + 45753584909922*c^2*x))) + 36*(a^2*c*x^4 + a^3)*sqrt(x)*(-c/a^1
7)^(1/8)*arctan(a^15*(-c/a^17)^(7/8)/(a^15*(-c/a^17)^(7/8) + sqrt(2)*c*sqrt(x) +
sqrt(2)*sqrt(sqrt(2)*a^15*c*sqrt(x)*(-c/a^17)^(7/8) - a^13*c*(-c/a^17)^(3/4) +
c^2*x))) + 9*(a^2*c*x^4 + a^3)*sqrt(x)*(-c/a^17)^(1/8)*log(45753584909922*sqrt(2
)*a^15*c*sqrt(x)*(-c/a^17)^(7/8) - 45753584909922*a^13*c*(-c/a^17)^(3/4) + 45753
584909922*c^2*x) - 9*(a^2*c*x^4 + a^3)*sqrt(x)*(-c/a^17)^(1/8)*log(-457535849099
22*sqrt(2)*a^15*c*sqrt(x)*(-c/a^17)^(7/8) - 45753584909922*a^13*c*(-c/a^17)^(3/4
) + 45753584909922*c^2*x) + 8*sqrt(2)*(9*c*x^4 + 8*a))/((a^2*c*x^4 + a^3)*sqrt(x
))

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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(1/x**(3/2)/(c*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.323047, size = 639, normalized size = 2. \[ -\frac{9 \, c \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 )}{32 \, a^{3}} - \frac{9 \, c \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 )}{32 \, a^{3}} - \frac{9 \, c \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 )}{32 \, a^{3}} - \frac{9 \, c \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 )}{32 \, a^{3}} + \frac{9 \, c \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 )}{64 \, a^{3}} - \frac{9 \, c \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 )}{64 \, a^{3}} + \frac{9 \, c \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 )}{64 \, a^{3}} - \frac{9 \, c \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 )}{64 \, a^{3}} - \frac{9 \, c x^{4} + 8 \, a}{4 \,{\left (c x^{\frac{9}{2}} + a \sqrt{x}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*x^(3/2)),x, algorithm="giac")

[Out]

-9/32*c*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 - 9/32*c*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 - 9/32*c*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 - 9/32*c*sqrt(-
sqrt(2) + 2)*(a/c)^(7/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sq
rt(-sqrt(2) + 2)*(a/c)^(1/8)))/a^3 + 9/64*c*sqrt(sqrt(2) + 2)*(a/c)^(7/8)*ln(sqr
t(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 - 9/64*c*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 + 9/64*c*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 - 9/64*c*sqrt(-sqrt(2) + 2)*(a/c)^(7/8)*ln(-sqrt(x)*s
qrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/a^3 - 1/4*(9*c*x^4 + 8*a)/((c*x
^(9/2) + a*sqrt(x))*a^2)