3.206 \(\int \frac{A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=332 \[ \frac{c^{5/4} (9 b B-13 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}+\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{17/4}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )} \]

[Out]

(9*b*B - 13*A*c)/(18*b^2*c*x^(9/2)) - (9*b*B - 13*A*c)/(10*b^3*x^(5/2)) + (c*(9*
b*B - 13*A*c))/(2*b^4*Sqrt[x]) - (b*B - A*c)/(2*b*c*x^(9/2)*(b + c*x^2)) - (c^(5
/4)*(9*b*B - 13*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b
^(17/4)) + (c^(5/4)*(9*b*B - 13*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4
)])/(4*Sqrt[2]*b^(17/4)) + (c^(5/4)*(9*b*B - 13*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/
4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(17/4)) - (c^(5/4)*(9*b*B - 13*A*c
)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(17/4
))

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Rubi [A]  time = 0.609657, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{c^{5/4} (9 b B-13 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}+\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{17/4}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)^2),x]

[Out]

(9*b*B - 13*A*c)/(18*b^2*c*x^(9/2)) - (9*b*B - 13*A*c)/(10*b^3*x^(5/2)) + (c*(9*
b*B - 13*A*c))/(2*b^4*Sqrt[x]) - (b*B - A*c)/(2*b*c*x^(9/2)*(b + c*x^2)) - (c^(5
/4)*(9*b*B - 13*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b
^(17/4)) + (c^(5/4)*(9*b*B - 13*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4
)])/(4*Sqrt[2]*b^(17/4)) + (c^(5/4)*(9*b*B - 13*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/
4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(17/4)) - (c^(5/4)*(9*b*B - 13*A*c
)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(17/4
))

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Rubi in Sympy [A]  time = 92.8386, size = 311, normalized size = 0.94 \[ \frac{A c - B b}{2 b c x^{\frac{9}{2}} \left (b + c x^{2}\right )} - \frac{13 A c - 9 B b}{18 b^{2} c x^{\frac{9}{2}}} + \frac{13 A c - 9 B b}{10 b^{3} x^{\frac{5}{2}}} - \frac{c \left (13 A c - 9 B b\right )}{2 b^{4} \sqrt{x}} - \frac{\sqrt{2} c^{\frac{5}{4}} \left (13 A c - 9 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{17}{4}}} + \frac{\sqrt{2} c^{\frac{5}{4}} \left (13 A c - 9 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{17}{4}}} + \frac{\sqrt{2} c^{\frac{5}{4}} \left (13 A c - 9 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{17}{4}}} - \frac{\sqrt{2} c^{\frac{5}{4}} \left (13 A c - 9 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{17}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**(3/2)/(c*x**4+b*x**2)**2,x)

[Out]

(A*c - B*b)/(2*b*c*x**(9/2)*(b + c*x**2)) - (13*A*c - 9*B*b)/(18*b**2*c*x**(9/2)
) + (13*A*c - 9*B*b)/(10*b**3*x**(5/2)) - c*(13*A*c - 9*B*b)/(2*b**4*sqrt(x)) -
sqrt(2)*c**(5/4)*(13*A*c - 9*B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(
b) + sqrt(c)*x)/(16*b**(17/4)) + sqrt(2)*c**(5/4)*(13*A*c - 9*B*b)*log(sqrt(2)*b
**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(17/4)) + sqrt(2)*c**(5/4
)*(13*A*c - 9*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(17/4)) - s
qrt(2)*c**(5/4)*(13*A*c - 9*B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*
b**(17/4))

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Mathematica [A]  time = 1.05756, size = 301, normalized size = 0.91 \[ \frac{-\frac{288 b^{5/4} (b B-2 A c)}{x^{5/2}}-\frac{160 A b^{9/4}}{x^{9/2}}+45 \sqrt{2} c^{5/4} (9 b B-13 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+45 \sqrt{2} c^{5/4} (13 A c-9 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+90 \sqrt{2} c^{5/4} (13 A c-9 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+90 \sqrt{2} c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+\frac{360 \sqrt [4]{b} c^2 x^{3/2} (b B-A c)}{b+c x^2}+\frac{1440 \sqrt [4]{b} c (2 b B-3 A c)}{\sqrt{x}}}{720 b^{17/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)^2),x]

[Out]

((-160*A*b^(9/4))/x^(9/2) - (288*b^(5/4)*(b*B - 2*A*c))/x^(5/2) + (1440*b^(1/4)*
c*(2*b*B - 3*A*c))/Sqrt[x] + (360*b^(1/4)*c^2*(b*B - A*c)*x^(3/2))/(b + c*x^2) +
90*Sqrt[2]*c^(5/4)*(-9*b*B + 13*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/
4)] + 90*Sqrt[2]*c^(5/4)*(9*b*B - 13*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b
^(1/4)] + 45*Sqrt[2]*c^(5/4)*(9*b*B - 13*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1
/4)*Sqrt[x] + Sqrt[c]*x] + 45*Sqrt[2]*c^(5/4)*(-9*b*B + 13*A*c)*Log[Sqrt[b] + Sq
rt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(720*b^(17/4))

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Maple [A]  time = 0.03, size = 372, normalized size = 1.1 \[ -{\frac{2\,A}{9\,{b}^{2}}{x}^{-{\frac{9}{2}}}}+{\frac{4\,Ac}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}-6\,{\frac{A{c}^{2}}{{b}^{4}\sqrt{x}}}+4\,{\frac{Bc}{{b}^{3}\sqrt{x}}}-{\frac{A{c}^{3}}{2\,{b}^{4} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}+{\frac{B{c}^{2}}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}-{\frac{13\,{c}^{2}\sqrt{2}A}{16\,{b}^{4}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{13\,{c}^{2}\sqrt{2}A}{8\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{13\,{c}^{2}\sqrt{2}A}{8\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{9\,c\sqrt{2}B}{16\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{9\,c\sqrt{2}B}{8\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{9\,c\sqrt{2}B}{8\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(3/2)/(c*x^4+b*x^2)^2,x)

[Out]

-2/9*A/b^2/x^(9/2)+4/5/b^3/x^(5/2)*A*c-2/5/b^2/x^(5/2)*B-6*c^2/b^4/x^(1/2)*A+4*c
/b^3/x^(1/2)*B-1/2/b^4*c^3*x^(3/2)/(c*x^2+b)*A+1/2/b^3*c^2*x^(3/2)/(c*x^2+b)*B-1
3/16/b^4*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)))-13/8/b^4*c^2/(b/c)^(1/4)*2^(1/2)*
A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-13/8/b^4*c^2/(b/c)^(1/4)*2^(1/2)*A*arcta
n(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+9/16/b^3*c/(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^3*c/(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+9/8/b^3*c/(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)/((c*x^4 + b*x^2)^2*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249892, size = 1226, normalized size = 3.69 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/360*(180*(9*B*b*c^2 - 13*A*c^3)*x^6 + 144*(9*B*b^2*c - 13*A*b*c^2)*x^4 - 80*A*
b^3 - 16*(9*B*b^3 - 13*A*b^2*c)*x^2 - 180*(b^4*c*x^6 + b^5*x^4)*sqrt(x)*(-(6561*
B^4*b^4*c^5 - 37908*A*B^3*b^3*c^6 + 82134*A^2*B^2*b^2*c^7 - 79092*A^3*B*b*c^8 +
28561*A^4*c^9)/b^17)^(1/4)*arctan(-b^13*(-(6561*B^4*b^4*c^5 - 37908*A*B^3*b^3*c^
6 + 82134*A^2*B^2*b^2*c^7 - 79092*A^3*B*b*c^8 + 28561*A^4*c^9)/b^17)^(3/4)/((729
*B^3*b^3*c^4 - 3159*A*B^2*b^2*c^5 + 4563*A^2*B*b*c^6 - 2197*A^3*c^7)*sqrt(x) - s
qrt((531441*B^6*b^6*c^8 - 4605822*A*B^5*b^5*c^9 + 16632135*A^2*B^4*b^4*c^10 - 32
032260*A^3*B^3*b^3*c^11 + 34701615*A^4*B^2*b^2*c^12 - 20049822*A^5*B*b*c^13 + 48
26809*A^6*c^14)*x - (6561*B^4*b^13*c^5 - 37908*A*B^3*b^12*c^6 + 82134*A^2*B^2*b^
11*c^7 - 79092*A^3*B*b^10*c^8 + 28561*A^4*b^9*c^9)*sqrt(-(6561*B^4*b^4*c^5 - 379
08*A*B^3*b^3*c^6 + 82134*A^2*B^2*b^2*c^7 - 79092*A^3*B*b*c^8 + 28561*A^4*c^9)/b^
17)))) - 45*(b^4*c*x^6 + b^5*x^4)*sqrt(x)*(-(6561*B^4*b^4*c^5 - 37908*A*B^3*b^3*
c^6 + 82134*A^2*B^2*b^2*c^7 - 79092*A^3*B*b*c^8 + 28561*A^4*c^9)/b^17)^(1/4)*log
(b^13*(-(6561*B^4*b^4*c^5 - 37908*A*B^3*b^3*c^6 + 82134*A^2*B^2*b^2*c^7 - 79092*
A^3*B*b*c^8 + 28561*A^4*c^9)/b^17)^(3/4) - (729*B^3*b^3*c^4 - 3159*A*B^2*b^2*c^5
+ 4563*A^2*B*b*c^6 - 2197*A^3*c^7)*sqrt(x)) + 45*(b^4*c*x^6 + b^5*x^4)*sqrt(x)*
(-(6561*B^4*b^4*c^5 - 37908*A*B^3*b^3*c^6 + 82134*A^2*B^2*b^2*c^7 - 79092*A^3*B*
b*c^8 + 28561*A^4*c^9)/b^17)^(1/4)*log(-b^13*(-(6561*B^4*b^4*c^5 - 37908*A*B^3*b
^3*c^6 + 82134*A^2*B^2*b^2*c^7 - 79092*A^3*B*b*c^8 + 28561*A^4*c^9)/b^17)^(3/4)
- (729*B^3*b^3*c^4 - 3159*A*B^2*b^2*c^5 + 4563*A^2*B*b*c^6 - 2197*A^3*c^7)*sqrt(
x)))/((b^4*c*x^6 + b^5*x^4)*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((B*x**2+A)/x**(3/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227252, size = 443, normalized size = 1.33 \[ \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{5} c} + \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{5} c} - \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{5} c} + \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{5} c} + \frac{B b c^{2} x^{\frac{3}{2}} - A c^{3} x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} b^{4}} + \frac{2 \,{\left (90 \, B b c x^{4} - 135 \, A c^{2} x^{4} - 9 \, B b^{2} x^{2} + 18 \, A b c x^{2} - 5 \, A b^{2}\right )}}{45 \, b^{4} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*(9*(b*c^3)^(3/4)*B*b - 13*(b*c^3)^(3/4)*A*c)*arctan(1/2*sqrt(2)*(sqr
t(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/(b^5*c) + 1/8*sqrt(2)*(9*(b*c^3)^(3/4
)*B*b - 13*(b*c^3)^(3/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(
x))/(b/c)^(1/4))/(b^5*c) - 1/16*sqrt(2)*(9*(b*c^3)^(3/4)*B*b - 13*(b*c^3)^(3/4)*
A*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^5*c) + 1/16*sqrt(2)*(9*(
b*c^3)^(3/4)*B*b - 13*(b*c^3)^(3/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + s
qrt(b/c))/(b^5*c) + 1/2*(B*b*c^2*x^(3/2) - A*c^3*x^(3/2))/((c*x^2 + b)*b^4) + 2/
45*(90*B*b*c*x^4 - 135*A*c^2*x^4 - 9*B*b^2*x^2 + 18*A*b*c*x^2 - 5*A*b^2)/(b^4*x^
(9/2))