3.736 \(\int \frac{x^{9/2}}{a+c x^4} \, dx\)

Optimal. Leaf size=299 \[ -\frac{(-a)^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{11/8}}-\frac{(-a)^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac{(-a)^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}+\frac{2 x^{3/2}}{3 c} \]

[Out]

(2*x^(3/2))/(3*c) + ((-a)^(3/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]
)/(2*Sqrt[2]*c^(11/8)) - ((-a)^(3/8)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(
1/8)])/(2*Sqrt[2]*c^(11/8)) + ((-a)^(3/8)*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/
(2*c^(11/8)) - ((-a)^(3/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*c^(11/8)) -
((-a)^(3/8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(
4*Sqrt[2]*c^(11/8)) + ((-a)^(3/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sq
rt[x] + c^(1/4)*x])/(4*Sqrt[2]*c^(11/8))

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Rubi [A]  time = 0.714572, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8 \[ -\frac{(-a)^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{11/8}}-\frac{(-a)^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} c^{11/8}}+\frac{(-a)^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac{(-a)^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}+\frac{2 x^{3/2}}{3 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(9/2)/(a + c*x^4),x]

[Out]

(2*x^(3/2))/(3*c) + ((-a)^(3/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]
)/(2*Sqrt[2]*c^(11/8)) - ((-a)^(3/8)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(
1/8)])/(2*Sqrt[2]*c^(11/8)) + ((-a)^(3/8)*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/
(2*c^(11/8)) - ((-a)^(3/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*c^(11/8)) -
((-a)^(3/8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(
4*Sqrt[2]*c^(11/8)) + ((-a)^(3/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sq
rt[x] + c^(1/4)*x])/(4*Sqrt[2]*c^(11/8))

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Rubi in Sympy [A]  time = 120.22, size = 274, normalized size = 0.92 \[ \frac{2 x^{\frac{3}{2}}}{3 c} - \frac{\sqrt{2} \left (- a\right )^{\frac{3}{8}} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 c^{\frac{11}{8}}} + \frac{\sqrt{2} \left (- a\right )^{\frac{3}{8}} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 c^{\frac{11}{8}}} + \frac{\left (- a\right )^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 c^{\frac{11}{8}}} - \frac{\sqrt{2} \left (- a\right )^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{4 c^{\frac{11}{8}}} - \frac{\sqrt{2} \left (- a\right )^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{4 c^{\frac{11}{8}}} - \frac{\left (- a\right )^{\frac{3}{8}} \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 c^{\frac{11}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)/(c*x**4+a),x)

[Out]

2*x**(3/2)/(3*c) - sqrt(2)*(-a)**(3/8)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8)
+ c**(1/4)*x + (-a)**(1/4))/(8*c**(11/8)) + sqrt(2)*(-a)**(3/8)*log(sqrt(2)*c**
(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/(8*c**(11/8)) + (-a)**(3/8
)*atan(c**(1/8)*sqrt(x)/(-a)**(1/8))/(2*c**(11/8)) - sqrt(2)*(-a)**(3/8)*atan(sq
rt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) - 1)/(4*c**(11/8)) - sqrt(2)*(-a)**(3/8)*atan
(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) + 1)/(4*c**(11/8)) - (-a)**(3/8)*atanh(c**
(1/8)*sqrt(x)/(-a)**(1/8))/(2*c**(11/8))

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Mathematica [A]  time = 0.501914, size = 397, normalized size = 1.33 \[ \frac{-3 a^{3/8} \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+3 a^{3/8} \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+3 a^{3/8} \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-3 a^{3/8} \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+6 a^{3/8} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+6 a^{3/8} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+6 a^{3/8} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-6 a^{3/8} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+8 c^{3/8} x^{3/2}}{12 c^{11/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(9/2)/(a + c*x^4),x]

[Out]

(8*c^(3/8)*x^(3/2) + 6*a^(3/8)*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^
(1/8)]*Cos[Pi/8] - 6*a^(3/8)*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1
/8)]*Cos[Pi/8] - 3*a^(3/8)*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)
*Sqrt[x]*Sin[Pi/8]] + 3*a^(3/8)*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^
(1/8)*Sqrt[x]*Sin[Pi/8]] + 6*a^(3/8)*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8)
- Tan[Pi/8]]*Sin[Pi/8] + 6*a^(3/8)*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) +
Tan[Pi/8]]*Sin[Pi/8] + 3*a^(3/8)*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqr
t[x]*Cos[Pi/8]]*Sin[Pi/8] - 3*a^(3/8)*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8
)*Sqrt[x]*Cos[Pi/8]]*Sin[Pi/8])/(12*c^(11/8))

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Maple [C]  time = 0.107, size = 39, normalized size = 0.1 \[{\frac{2}{3\,c}{x}^{{\frac{3}{2}}}}-{\frac{a}{4\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+a \right ) }{\frac{1}{{{\it \_R}}^{5}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)/(c*x^4+a),x)

[Out]

2/3*x^(3/2)/c-1/4*a/c^2*sum(1/_R^5*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -a \int \frac{\sqrt{x}}{c^{2} x^{4} + a c}\,{d x} + \frac{2 \, x^{\frac{3}{2}}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + a),x, algorithm="maxima")

[Out]

-a*integrate(sqrt(x)/(c^2*x^4 + a*c), x) + 2/3*x^(3/2)/c

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Fricas [A]  time = 0.259194, size = 636, normalized size = 2.13 \[ \frac{\sqrt{2}{\left (12 \, \sqrt{2} c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \arctan \left (\frac{c^{4} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}}}{a \sqrt{x} + \sqrt{c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} + a^{2} x}}\right ) + 3 \, \sqrt{2} c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \log \left (c^{4} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + a \sqrt{x}\right ) - 3 \, \sqrt{2} c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \log \left (-c^{4} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + a \sqrt{x}\right ) - 12 \, c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \arctan \left (\frac{c^{4} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}}}{c^{4} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + \sqrt{2} a \sqrt{x} + \sqrt{2 \, c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} + 2 \, \sqrt{2} a c^{4} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + 2 \, a^{2} x}}\right ) - 12 \, c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{c^{4} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}}}{c^{4} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} - \sqrt{2} a \sqrt{x} - \sqrt{2 \, c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} - 2 \, \sqrt{2} a c^{4} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + 2 \, a^{2} x}}\right ) - 3 \, c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \log \left (2 \, c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} + 2 \, \sqrt{2} a c^{4} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + 2 \, a^{2} x\right ) + 3 \, c \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{1}{8}} \log \left (2 \, c^{8} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{4}} - 2 \, \sqrt{2} a c^{4} \sqrt{x} \left (-\frac{a^{3}}{c^{11}}\right )^{\frac{3}{8}} + 2 \, a^{2} x\right ) + 8 \, \sqrt{2} x^{\frac{3}{2}}\right )}}{24 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + a),x, algorithm="fricas")

[Out]

1/24*sqrt(2)*(12*sqrt(2)*c*(-a^3/c^11)^(1/8)*arctan(c^4*(-a^3/c^11)^(3/8)/(a*sqr
t(x) + sqrt(c^8*(-a^3/c^11)^(3/4) + a^2*x))) + 3*sqrt(2)*c*(-a^3/c^11)^(1/8)*log
(c^4*(-a^3/c^11)^(3/8) + a*sqrt(x)) - 3*sqrt(2)*c*(-a^3/c^11)^(1/8)*log(-c^4*(-a
^3/c^11)^(3/8) + a*sqrt(x)) - 12*c*(-a^3/c^11)^(1/8)*arctan(c^4*(-a^3/c^11)^(3/8
)/(c^4*(-a^3/c^11)^(3/8) + sqrt(2)*a*sqrt(x) + sqrt(2*c^8*(-a^3/c^11)^(3/4) + 2*
sqrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + 2*a^2*x))) - 12*c*(-a^3/c^11)^(1/8)*ar
ctan(-c^4*(-a^3/c^11)^(3/8)/(c^4*(-a^3/c^11)^(3/8) - sqrt(2)*a*sqrt(x) - sqrt(2*
c^8*(-a^3/c^11)^(3/4) - 2*sqrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + 2*a^2*x))) -
3*c*(-a^3/c^11)^(1/8)*log(2*c^8*(-a^3/c^11)^(3/4) + 2*sqrt(2)*a*c^4*sqrt(x)*(-a
^3/c^11)^(3/8) + 2*a^2*x) + 3*c*(-a^3/c^11)^(1/8)*log(2*c^8*(-a^3/c^11)^(3/4) -
2*sqrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + 2*a^2*x) + 8*sqrt(2)*x^(3/2))/c

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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**(9/2)/(c*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.305306, size = 601, normalized size = 2.01 \[ \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, c} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, c} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, c} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, c} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, c} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, c} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, c} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{3}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, c} + \frac{2 \, x^{\frac{3}{2}}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + a),x, algorithm="giac")

[Out]

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