3.757 \(\int \frac{x^{7/2}}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=332 \[ -\frac{7 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{15/8} c^{9/8}}+\frac{7 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{15/8} c^{9/8}}-\frac{7 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{15/8} c^{9/8}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{15/8} c^{9/8}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac{\sqrt{x}}{64 a c \left (a+c x^4\right )}-\frac{\sqrt{x}}{8 c \left (a+c x^4\right )^2} \]

[Out]

-Sqrt[x]/(8*c*(a + c*x^4)^2) + Sqrt[x]/(64*a*c*(a + c*x^4)) - (7*ArcTan[1 - (Sqr
t[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(15/8)*c^(9/8)) + (7*ArcTan
[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(15/8)*c^(9/8)) +
(7*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(15/8)*c^(9/8)) + (7*ArcTanh[
(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(15/8)*c^(9/8)) - (7*Log[(-a)^(1/4) - S
qrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(15/8)*c^(9/8)
) + (7*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sq
rt[2]*(-a)^(15/8)*c^(9/8))

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Rubi [A]  time = 0.612799, antiderivative size = 332, 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 \[ -\frac{7 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{15/8} c^{9/8}}+\frac{7 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{15/8} c^{9/8}}-\frac{7 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{15/8} c^{9/8}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{15/8} c^{9/8}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac{\sqrt{x}}{64 a c \left (a+c x^4\right )}-\frac{\sqrt{x}}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/(a + c*x^4)^3,x]

[Out]

-Sqrt[x]/(8*c*(a + c*x^4)^2) + Sqrt[x]/(64*a*c*(a + c*x^4)) - (7*ArcTan[1 - (Sqr
t[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(15/8)*c^(9/8)) + (7*ArcTan
[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(15/8)*c^(9/8)) +
(7*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(15/8)*c^(9/8)) + (7*ArcTanh[
(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(15/8)*c^(9/8)) - (7*Log[(-a)^(1/4) - S
qrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(15/8)*c^(9/8)
) + (7*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sq
rt[2]*(-a)^(15/8)*c^(9/8))

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Rubi in Sympy [A]  time = 130.399, size = 308, normalized size = 0.93 \[ - \frac{\sqrt{x}}{8 c \left (a + c x^{4}\right )^{2}} - \frac{7 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{9}{8}} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{9}{8}} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{9}{8}} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{512 c^{\frac{9}{8}} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{512 c^{\frac{9}{8}} \left (- a\right )^{\frac{15}{8}}} + \frac{7 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{9}{8}} \left (- a\right )^{\frac{15}{8}}} + \frac{\sqrt{x}}{64 a c \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)/(c*x**4+a)**3,x)

[Out]

-sqrt(x)/(8*c*(a + c*x**4)**2) - 7*sqrt(2)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(
1/8) + c**(1/4)*x + (-a)**(1/4))/(1024*c**(9/8)*(-a)**(15/8)) + 7*sqrt(2)*log(sq
rt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/(1024*c**(9/8)*(-
a)**(15/8)) + 7*atan(c**(1/8)*sqrt(x)/(-a)**(1/8))/(256*c**(9/8)*(-a)**(15/8)) +
7*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) - 1)/(512*c**(9/8)*(-a)**(1
5/8)) + 7*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a)**(1/8) + 1)/(512*c**(9/8)*(
-a)**(15/8)) + 7*atanh(c**(1/8)*sqrt(x)/(-a)**(1/8))/(256*c**(9/8)*(-a)**(15/8))
+ sqrt(x)/(64*a*c*(a + c*x**4))

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Mathematica [A]  time = 0.916491, size = 430, normalized size = 1.3 \[ \frac{-\frac{7 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{15/8}}+\frac{7 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{15/8}}-\frac{7 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{15/8}}+\frac{7 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{15/8}}+\frac{14 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{a^{15/8}}+\frac{14 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{a^{15/8}}-\frac{14 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{a^{15/8}}+\frac{14 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{a^{15/8}}+\frac{8 \sqrt [8]{c} \sqrt{x}}{a^2+a c x^4}-\frac{64 \sqrt [8]{c} \sqrt{x}}{\left (a+c x^4\right )^2}}{512 c^{9/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/(a + c*x^4)^3,x]

[Out]

((-64*c^(1/8)*Sqrt[x])/(a + c*x^4)^2 + (8*c^(1/8)*Sqrt[x])/(a^2 + a*c*x^4) + (14
*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8])/a^(15/8) + (
14*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8])/a^(15/8) -
(7*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/a^
(15/8) + (7*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi
/8]])/a^(15/8) - (14*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin
[Pi/8])/a^(15/8) + (14*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*S
in[Pi/8])/a^(15/8) - (7*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[
Pi/8]]*Sin[Pi/8])/a^(15/8) + (7*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt
[x]*Sin[Pi/8]]*Sin[Pi/8])/a^(15/8))/(512*c^(9/8))

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Maple [C]  time = 0.03, size = 61, normalized size = 0.2 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ( -{\frac{7\,\sqrt{x}}{128\,c}}+{\frac{{x}^{9/2}}{128\,a}} \right ) }+{\frac{7}{512\,{c}^{2}a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(c*x^4+a)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.274784, size = 905, normalized size = 2.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

-1/1024*sqrt(2)*(28*sqrt(2)*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^
(1/8)*arctan(a^2*c*(-1/(a^15*c^9))^(1/8)/(sqrt(a^4*c^2*(-1/(a^15*c^9))^(1/4) + x
) + sqrt(x))) - 7*sqrt(2)*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1
/8)*log(a^2*c*(-1/(a^15*c^9))^(1/8) + sqrt(x)) + 7*sqrt(2)*(a*c^3*x^8 + 2*a^2*c^
2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*log(-a^2*c*(-1/(a^15*c^9))^(1/8) + sqrt(x))
+ 28*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*arctan(a^2*c*(-1
/(a^15*c^9))^(1/8)/(a^2*c*(-1/(a^15*c^9))^(1/8) + sqrt(2)*sqrt(x) + sqrt(2*a^4*c
^2*(-1/(a^15*c^9))^(1/4) + 2*sqrt(2)*a^2*c*sqrt(x)*(-1/(a^15*c^9))^(1/8) + 2*x))
) + 28*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*arctan(-a^2*c*(
-1/(a^15*c^9))^(1/8)/(a^2*c*(-1/(a^15*c^9))^(1/8) - sqrt(2)*sqrt(x) - sqrt(2*a^4
*c^2*(-1/(a^15*c^9))^(1/4) - 2*sqrt(2)*a^2*c*sqrt(x)*(-1/(a^15*c^9))^(1/8) + 2*x
))) - 7*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*log(2*a^4*c^2*
(-1/(a^15*c^9))^(1/4) + 2*sqrt(2)*a^2*c*sqrt(x)*(-1/(a^15*c^9))^(1/8) + 2*x) + 7
*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*log(2*a^4*c^2*(-1/(a^
15*c^9))^(1/4) - 2*sqrt(2)*a^2*c*sqrt(x)*(-1/(a^15*c^9))^(1/8) + 2*x) - 8*sqrt(2
)*(c*x^4 - 7*a)*sqrt(x))/(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*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**(7/2)/(c*x**4+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.335732, size = 662, normalized size = 1.99 \[ \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{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^{2} c} + \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{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^{2} c} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{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^{2} c} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{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^{2} c} + \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{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^{2} c} - \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{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^{2} c} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{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^{2} c} - \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{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^{2} c} + \frac{c x^{\frac{9}{2}} - 7 \, a \sqrt{x}}{64 \,{\left (c x^{4} + a\right )}^{2} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

7/512*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*s
qrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*c) + 7/512*sqrt(sqrt(2) + 2)*(a/c)
^(1/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(
a/c)^(1/8)))/(a^2*c) + 7/512*sqrt(-sqrt(2) + 2)*(a/c)^(1/8)*arctan((sqrt(sqrt(2)
+ 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*c) + 7/512
*sqrt(-sqrt(2) + 2)*(a/c)^(1/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(
x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*c) + 7/1024*sqrt(sqrt(2) + 2)*(a/c)^(
1/8)*ln(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*c) - 7/102
4*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*ln(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x +
(a/c)^(1/4))/(a^2*c) + 7/1024*sqrt(-sqrt(2) + 2)*(a/c)^(1/8)*ln(sqrt(x)*sqrt(-sq
rt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*c) - 7/1024*sqrt(-sqrt(2) + 2)*(a
/c)^(1/8)*ln(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*c)
+ 1/64*(c*x^(9/2) - 7*a*sqrt(x))/((c*x^4 + a)^2*a*c)