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

Optimal. Leaf size=289 \[ -\frac{(3 b B-7 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{(3 b B-7 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}-\frac{(3 b B-7 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{(3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )} \]

[Out]

(3*b*B - 7*A*c)/(6*b^2*c*x^(3/2)) - (b*B - A*c)/(2*b*c*x^(3/2)*(b + c*x^2)) - ((
3*b*B - 7*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(11/4
)*c^(1/4)) + ((3*b*B - 7*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*
Sqrt[2]*b^(11/4)*c^(1/4)) - ((3*b*B - 7*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/
4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(11/4)*c^(1/4)) + ((3*b*B - 7*A*c)*Log[Sqr
t[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(11/4)*c^(1/4)
)

_______________________________________________________________________________________

Rubi [A]  time = 0.468259, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{(3 b B-7 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{(3 b B-7 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}-\frac{(3 b B-7 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{(3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(3*b*B - 7*A*c)/(6*b^2*c*x^(3/2)) - (b*B - A*c)/(2*b*c*x^(3/2)*(b + c*x^2)) - ((
3*b*B - 7*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(11/4
)*c^(1/4)) + ((3*b*B - 7*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*
Sqrt[2]*b^(11/4)*c^(1/4)) - ((3*b*B - 7*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/
4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(11/4)*c^(1/4)) + ((3*b*B - 7*A*c)*Log[Sqr
t[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(11/4)*c^(1/4)
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 75.1617, size = 269, normalized size = 0.93 \[ \frac{A c - B b}{2 b c x^{\frac{3}{2}} \left (b + c x^{2}\right )} - \frac{7 A c - 3 B b}{6 b^{2} c x^{\frac{3}{2}}} + \frac{\sqrt{2} \left (7 A c - 3 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{11}{4}} \sqrt [4]{c}} - \frac{\sqrt{2} \left (7 A c - 3 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{11}{4}} \sqrt [4]{c}} + \frac{\sqrt{2} \left (7 A c - 3 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{11}{4}} \sqrt [4]{c}} - \frac{\sqrt{2} \left (7 A c - 3 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{11}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*c - B*b)/(2*b*c*x**(3/2)*(b + c*x**2)) - (7*A*c - 3*B*b)/(6*b**2*c*x**(3/2))
+ sqrt(2)*(7*A*c - 3*B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqr
t(c)*x)/(16*b**(11/4)*c**(1/4)) - sqrt(2)*(7*A*c - 3*B*b)*log(sqrt(2)*b**(1/4)*c
**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(11/4)*c**(1/4)) + sqrt(2)*(7*A*c
- 3*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(11/4)*c**(1/4)) - sq
rt(2)*(7*A*c - 3*B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(11/4)*c
**(1/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.48044, size = 256, normalized size = 0.89 \[ \frac{\frac{24 b^{3/4} \sqrt{x} (b B-A c)}{b+c x^2}-\frac{32 A b^{3/4}}{x^{3/2}}+\frac{3 \sqrt{2} (7 A c-3 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{3 \sqrt{2} (3 b B-7 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} (7 A c-3 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} (3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt [4]{c}}}{48 b^{11/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*A*b^(3/4))/x^(3/2) + (24*b^(3/4)*(b*B - A*c)*Sqrt[x])/(b + c*x^2) + (6*Sqr
t[2]*(-3*b*B + 7*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4) + (
6*Sqrt[2]*(3*b*B - 7*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/c^(1/4)
+ (3*Sqrt[2]*(-3*b*B + 7*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + S
qrt[c]*x])/c^(1/4) + (3*Sqrt[2]*(3*b*B - 7*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^
(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4))/(48*b^(11/4))

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 317, normalized size = 1.1 \[ -{\frac{2\,A}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{Ac}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{B}{2\,b \left ( c{x}^{2}+b \right ) }\sqrt{x}}-{\frac{7\,\sqrt{2}Ac}{8\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{7\,\sqrt{2}Ac}{8\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{7\,\sqrt{2}Ac}{16\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}B}{8\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}B}{8\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}B}{16\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*A/b^2/x^(3/2)-1/2/b^2*x^(1/2)/(c*x^2+b)*A*c+1/2/b*x^(1/2)/(c*x^2+b)*B-7/8/b
^3*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)*c-7/8/b^3*(b/c)^(
1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)*c-7/16/b^3*(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)))*c+3/8/b^2*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(
1/2)+1)+3/8/b^2*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+3/16
/b^2*(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)))

_______________________________________________________________________________________

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^(3/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.246541, size = 838, normalized size = 2.9 \[ \frac{4 \,{\left (3 \, B b - 7 \, A c\right )} x^{2} + 12 \,{\left (b^{2} c x^{3} + b^{3} x\right )} \sqrt{x} \left (-\frac{81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{3} \left (-\frac{81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac{1}{4}}}{{\left (3 \, B b - 7 \, A c\right )} \sqrt{x} - \sqrt{b^{6} \sqrt{-\frac{81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}} +{\left (9 \, B^{2} b^{2} - 42 \, A B b c + 49 \, A^{2} c^{2}\right )} x}}\right ) - 3 \,{\left (b^{2} c x^{3} + b^{3} x\right )} \sqrt{x} \left (-\frac{81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac{1}{4}} \log \left (b^{3} \left (-\frac{81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac{1}{4}} -{\left (3 \, B b - 7 \, A c\right )} \sqrt{x}\right ) + 3 \,{\left (b^{2} c x^{3} + b^{3} x\right )} \sqrt{x} \left (-\frac{81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac{1}{4}} \log \left (-b^{3} \left (-\frac{81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac{1}{4}} -{\left (3 \, B b - 7 \, A c\right )} \sqrt{x}\right ) - 16 \, A b}{24 \,{\left (b^{2} c x^{3} + b^{3} x\right )} \sqrt{x}} \]

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, algorithm="fricas")

[Out]

1/24*(4*(3*B*b - 7*A*c)*x^2 + 12*(b^2*c*x^3 + b^3*x)*sqrt(x)*(-(81*B^4*b^4 - 756
*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))
^(1/4)*arctan(-b^3*(-(81*B^4*b^4 - 756*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116
*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))^(1/4)/((3*B*b - 7*A*c)*sqrt(x) - sqrt(b^6
*sqrt(-(81*B^4*b^4 - 756*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 +
2401*A^4*c^4)/(b^11*c)) + (9*B^2*b^2 - 42*A*B*b*c + 49*A^2*c^2)*x))) - 3*(b^2*c
*x^3 + b^3*x)*sqrt(x)*(-(81*B^4*b^4 - 756*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4
116*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))^(1/4)*log(b^3*(-(81*B^4*b^4 - 756*A*B^
3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))^(1/4
) - (3*B*b - 7*A*c)*sqrt(x)) + 3*(b^2*c*x^3 + b^3*x)*sqrt(x)*(-(81*B^4*b^4 - 756
*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))
^(1/4)*log(-b^3*(-(81*B^4*b^4 - 756*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^
3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))^(1/4) - (3*B*b - 7*A*c)*sqrt(x)) - 16*A*b)/(
(b^2*c*x^3 + b^3*x)*sqrt(x))

_______________________________________________________________________________________

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**(3/2)*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223308, size = 382, normalized size = 1.32 \[ \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 7 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3} c} + \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 7 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3} c} + \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 7 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{3} c} - \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 7 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{3} c} + \frac{B b \sqrt{x} - A c \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} b^{2}} - \frac{2 \, A}{3 \, b^{2} x^{\frac{3}{2}}} \]

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, algorithm="giac")

[Out]

1/8*sqrt(2)*(3*(b*c^3)^(1/4)*B*b - 7*(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))/(b^3*c) + 1/8*sqrt(2)*(3*(b*c^3)^(1/4)
*B*b - 7*(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))/(b^3*c) + 1/16*sqrt(2)*(3*(b*c^3)^(1/4)*B*b - 7*(b*c^3)^(1/4)*A*c
)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^3*c) - 1/16*sqrt(2)*(3*(b*c
^3)^(1/4)*B*b - 7*(b*c^3)^(1/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(
b/c))/(b^3*c) + 1/2*(B*b*sqrt(x) - A*c*sqrt(x))/((c*x^2 + b)*b^2) - 2/3*A/(b^2*x
^(3/2))