3.197 \(\int \frac{x^{15/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=310 \[ -\frac{\sqrt [4]{b} (9 b B-5 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{\sqrt [4]{b} (9 b B-5 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}-\frac{\sqrt [4]{b} (9 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}+\frac{\sqrt [4]{b} (9 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{13/4}}-\frac{\sqrt{x} (9 b B-5 A c)}{2 c^3}+\frac{x^{5/2} (9 b B-5 A c)}{10 b c^2}-\frac{x^{9/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]

[Out]

-((9*b*B - 5*A*c)*Sqrt[x])/(2*c^3) + ((9*b*B - 5*A*c)*x^(5/2))/(10*b*c^2) - ((b*
B - A*c)*x^(9/2))/(2*b*c*(b + c*x^2)) - (b^(1/4)*(9*b*B - 5*A*c)*ArcTan[1 - (Sqr
t[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(13/4)) + (b^(1/4)*(9*b*B - 5*A*c)*
ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(13/4)) - (b^(1/4)*(
9*b*B - 5*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sq
rt[2]*c^(13/4)) + (b^(1/4)*(9*b*B - 5*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)
*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*c^(13/4))

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Rubi [A]  time = 0.525189, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{\sqrt [4]{b} (9 b B-5 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{\sqrt [4]{b} (9 b B-5 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}-\frac{\sqrt [4]{b} (9 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}+\frac{\sqrt [4]{b} (9 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{13/4}}-\frac{\sqrt{x} (9 b B-5 A c)}{2 c^3}+\frac{x^{5/2} (9 b B-5 A c)}{10 b c^2}-\frac{x^{9/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

-((9*b*B - 5*A*c)*Sqrt[x])/(2*c^3) + ((9*b*B - 5*A*c)*x^(5/2))/(10*b*c^2) - ((b*
B - A*c)*x^(9/2))/(2*b*c*(b + c*x^2)) - (b^(1/4)*(9*b*B - 5*A*c)*ArcTan[1 - (Sqr
t[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(13/4)) + (b^(1/4)*(9*b*B - 5*A*c)*
ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(13/4)) - (b^(1/4)*(
9*b*B - 5*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sq
rt[2]*c^(13/4)) + (b^(1/4)*(9*b*B - 5*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)
*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*c^(13/4))

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Rubi in Sympy [A]  time = 82.9099, size = 289, normalized size = 0.93 \[ \frac{\sqrt{2} \sqrt [4]{b} \left (5 A c - 9 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \left (5 A c - 9 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{13}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \left (5 A c - 9 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \left (5 A c - 9 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{13}{4}}} + \frac{\sqrt{x} \left (5 A c - 9 B b\right )}{2 c^{3}} + \frac{x^{\frac{9}{2}} \left (A c - B b\right )}{2 b c \left (b + c x^{2}\right )} - \frac{x^{\frac{5}{2}} \left (5 A c - 9 B b\right )}{10 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(15/2)*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

sqrt(2)*b**(1/4)*(5*A*c - 9*B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b
) + sqrt(c)*x)/(16*c**(13/4)) - sqrt(2)*b**(1/4)*(5*A*c - 9*B*b)*log(sqrt(2)*b**
(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*c**(13/4)) + sqrt(2)*b**(1/4)*
(5*A*c - 9*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*c**(13/4)) - sqrt
(2)*b**(1/4)*(5*A*c - 9*B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*c**(
13/4)) + sqrt(x)*(5*A*c - 9*B*b)/(2*c**3) + x**(9/2)*(A*c - B*b)/(2*b*c*(b + c*x
**2)) - x**(5/2)*(5*A*c - 9*B*b)/(10*b*c**2)

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Mathematica [A]  time = 0.418302, size = 277, normalized size = 0.89 \[ \frac{\frac{40 b \sqrt [4]{c} \sqrt{x} (A c-b B)}{b+c x^2}+160 \sqrt [4]{c} \sqrt{x} (A c-2 b B)-5 \sqrt{2} \sqrt [4]{b} (9 b B-5 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+5 \sqrt{2} \sqrt [4]{b} (9 b B-5 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-10 \sqrt{2} \sqrt [4]{b} (9 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+10 \sqrt{2} \sqrt [4]{b} (9 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+32 B c^{5/4} x^{5/2}}{80 c^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

(160*c^(1/4)*(-2*b*B + A*c)*Sqrt[x] + 32*B*c^(5/4)*x^(5/2) + (40*b*c^(1/4)*(-(b*
B) + A*c)*Sqrt[x])/(b + c*x^2) - 10*Sqrt[2]*b^(1/4)*(9*b*B - 5*A*c)*ArcTan[1 - (
Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 10*Sqrt[2]*b^(1/4)*(9*b*B - 5*A*c)*ArcTan[1
+ (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 5*Sqrt[2]*b^(1/4)*(9*b*B - 5*A*c)*Log[Sqr
t[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 5*Sqrt[2]*b^(1/4)*(9*b*B -
5*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(80*c^(13/4)
)

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Maple [A]  time = 0.021, size = 339, normalized size = 1.1 \[{\frac{2\,B}{5\,{c}^{2}}{x}^{{\frac{5}{2}}}}+2\,{\frac{A\sqrt{x}}{{c}^{2}}}-4\,{\frac{\sqrt{x}Bb}{{c}^{3}}}+{\frac{Ab}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }\sqrt{x}}-{\frac{{b}^{2}B}{2\,{c}^{3} \left ( c{x}^{2}+b \right ) }\sqrt{x}}-{\frac{5\,\sqrt{2}A}{8\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}A}{16\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{5\,\sqrt{2}A}{8\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{9\,b\sqrt{2}B}{8\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{9\,b\sqrt{2}B}{16\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{9\,b\sqrt{2}B}{8\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(15/2)*(B*x^2+A)/(c*x^4+b*x^2)^2,x)

[Out]

2/5/c^2*B*x^(5/2)+2/c^2*A*x^(1/2)-4/c^3*x^(1/2)*B*b+1/2*b/c^2*x^(1/2)/(c*x^2+b)*
A-1/2*b^2/c^3*x^(1/2)/(c*x^2+b)*B-5/8/c^2*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(
b/c)^(1/4)*x^(1/2)-1)-5/16/c^2*(b/c)^(1/4)*2^(1/2)*A*ln((x+(b/c)^(1/4)*x^(1/2)*2
^(1/2)+(b/c)^(1/2))/(x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))-5/8/c^2*(b/c)^(
1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+9/8*b/c^3*(b/c)^(1/4)*2^(1/
2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+9/16*b/c^3*(b/c)^(1/4)*2^(1/2)*B*ln((
x+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2))/(x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^
(1/2)))+9/8*b/c^3*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(15/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240282, size = 851, normalized size = 2.75 \[ \frac{20 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{6561 \, B^{4} b^{5} - 14580 \, A B^{3} b^{4} c + 12150 \, A^{2} B^{2} b^{3} c^{2} - 4500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{13}}\right )^{\frac{1}{4}} \arctan \left (-\frac{c^{3} \left (-\frac{6561 \, B^{4} b^{5} - 14580 \, A B^{3} b^{4} c + 12150 \, A^{2} B^{2} b^{3} c^{2} - 4500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{13}}\right )^{\frac{1}{4}}}{{\left (9 \, B b - 5 \, A c\right )} \sqrt{x} - \sqrt{c^{6} \sqrt{-\frac{6561 \, B^{4} b^{5} - 14580 \, A B^{3} b^{4} c + 12150 \, A^{2} B^{2} b^{3} c^{2} - 4500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{13}}} +{\left (81 \, B^{2} b^{2} - 90 \, A B b c + 25 \, A^{2} c^{2}\right )} x}}\right ) - 5 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{6561 \, B^{4} b^{5} - 14580 \, A B^{3} b^{4} c + 12150 \, A^{2} B^{2} b^{3} c^{2} - 4500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{13}}\right )^{\frac{1}{4}} \log \left (c^{3} \left (-\frac{6561 \, B^{4} b^{5} - 14580 \, A B^{3} b^{4} c + 12150 \, A^{2} B^{2} b^{3} c^{2} - 4500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{13}}\right )^{\frac{1}{4}} -{\left (9 \, B b - 5 \, A c\right )} \sqrt{x}\right ) + 5 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{6561 \, B^{4} b^{5} - 14580 \, A B^{3} b^{4} c + 12150 \, A^{2} B^{2} b^{3} c^{2} - 4500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{13}}\right )^{\frac{1}{4}} \log \left (-c^{3} \left (-\frac{6561 \, B^{4} b^{5} - 14580 \, A B^{3} b^{4} c + 12150 \, A^{2} B^{2} b^{3} c^{2} - 4500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{13}}\right )^{\frac{1}{4}} -{\left (9 \, B b - 5 \, A c\right )} \sqrt{x}\right ) + 4 \,{\left (4 \, B c^{2} x^{4} - 45 \, B b^{2} + 25 \, A b c - 4 \,{\left (9 \, B b c - 5 \, A c^{2}\right )} x^{2}\right )} \sqrt{x}}{40 \,{\left (c^{4} x^{2} + b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(15/2)/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

1/40*(20*(c^4*x^2 + b*c^3)*(-(6561*B^4*b^5 - 14580*A*B^3*b^4*c + 12150*A^2*B^2*b
^3*c^2 - 4500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^13)^(1/4)*arctan(-c^3*(-(6561*B^4
*b^5 - 14580*A*B^3*b^4*c + 12150*A^2*B^2*b^3*c^2 - 4500*A^3*B*b^2*c^3 + 625*A^4*
b*c^4)/c^13)^(1/4)/((9*B*b - 5*A*c)*sqrt(x) - sqrt(c^6*sqrt(-(6561*B^4*b^5 - 145
80*A*B^3*b^4*c + 12150*A^2*B^2*b^3*c^2 - 4500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^1
3) + (81*B^2*b^2 - 90*A*B*b*c + 25*A^2*c^2)*x))) - 5*(c^4*x^2 + b*c^3)*(-(6561*B
^4*b^5 - 14580*A*B^3*b^4*c + 12150*A^2*B^2*b^3*c^2 - 4500*A^3*B*b^2*c^3 + 625*A^
4*b*c^4)/c^13)^(1/4)*log(c^3*(-(6561*B^4*b^5 - 14580*A*B^3*b^4*c + 12150*A^2*B^2
*b^3*c^2 - 4500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^13)^(1/4) - (9*B*b - 5*A*c)*sqr
t(x)) + 5*(c^4*x^2 + b*c^3)*(-(6561*B^4*b^5 - 14580*A*B^3*b^4*c + 12150*A^2*B^2*
b^3*c^2 - 4500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^13)^(1/4)*log(-c^3*(-(6561*B^4*b
^5 - 14580*A*B^3*b^4*c + 12150*A^2*B^2*b^3*c^2 - 4500*A^3*B*b^2*c^3 + 625*A^4*b*
c^4)/c^13)^(1/4) - (9*B*b - 5*A*c)*sqrt(x)) + 4*(4*B*c^2*x^4 - 45*B*b^2 + 25*A*b
*c - 4*(9*B*b*c - 5*A*c^2)*x^2)*sqrt(x))/(c^4*x^2 + b*c^3)

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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**(15/2)*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220371, size = 402, normalized size = 1.3 \[ \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{4}} + \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{4}} + \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{4}} - \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{4}} - \frac{B b^{2} \sqrt{x} - A b c \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} c^{3}} + \frac{2 \,{\left (B c^{8} x^{\frac{5}{2}} - 10 \, B b c^{7} \sqrt{x} + 5 \, A c^{8} \sqrt{x}\right )}}{5 \, c^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(15/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

1/8*sqrt(2)*(9*(b*c^3)^(1/4)*B*b - 5*(b*c^3)^(1/4)*A*c)*arctan(1/2*sqrt(2)*(sqrt
(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/c^4 + 1/8*sqrt(2)*(9*(b*c^3)^(1/4)*B*b
- 5*(b*c^3)^(1/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b
/c)^(1/4))/c^4 + 1/16*sqrt(2)*(9*(b*c^3)^(1/4)*B*b - 5*(b*c^3)^(1/4)*A*c)*ln(sqr
t(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^4 - 1/16*sqrt(2)*(9*(b*c^3)^(1/4)*B*
b - 5*(b*c^3)^(1/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^4 -
1/2*(B*b^2*sqrt(x) - A*b*c*sqrt(x))/((c*x^2 + b)*c^3) + 2/5*(B*c^8*x^(5/2) - 10*
B*b*c^7*sqrt(x) + 5*A*c^8*sqrt(x))/c^10