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

Optimal. Leaf size=343 \[ \frac{11 c^{3/4} (7 b B-15 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}-\frac{11 c^{3/4} (7 b B-15 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}+\frac{11 c^{3/4} (7 b B-15 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{19/4}}-\frac{11 c^{3/4} (7 b B-15 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{19/4}}-\frac{11 (7 b B-15 A c)}{48 b^4 x^{3/2}}+\frac{11 (7 b B-15 A c)}{112 b^3 c x^{7/2}}-\frac{7 b B-15 A c}{16 b^2 c x^{7/2} \left (b+c x^2\right )}-\frac{b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2} \]

[Out]

(11*(7*b*B - 15*A*c))/(112*b^3*c*x^(7/2)) - (11*(7*b*B - 15*A*c))/(48*b^4*x^(3/2
)) - (b*B - A*c)/(4*b*c*x^(7/2)*(b + c*x^2)^2) - (7*b*B - 15*A*c)/(16*b^2*c*x^(7
/2)*(b + c*x^2)) + (11*c^(3/4)*(7*b*B - 15*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt
[x])/b^(1/4)])/(32*Sqrt[2]*b^(19/4)) - (11*c^(3/4)*(7*b*B - 15*A*c)*ArcTan[1 + (
Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(19/4)) + (11*c^(3/4)*(7*b*B -
15*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*
b^(19/4)) - (11*c^(3/4)*(7*b*B - 15*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*S
qrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(19/4))

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Rubi [A]  time = 0.612287, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423 \[ \frac{11 c^{3/4} (7 b B-15 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}-\frac{11 c^{3/4} (7 b B-15 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}+\frac{11 c^{3/4} (7 b B-15 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{19/4}}-\frac{11 c^{3/4} (7 b B-15 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{19/4}}-\frac{11 (7 b B-15 A c)}{48 b^4 x^{3/2}}+\frac{11 (7 b B-15 A c)}{112 b^3 c x^{7/2}}-\frac{7 b B-15 A c}{16 b^2 c x^{7/2} \left (b+c x^2\right )}-\frac{b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(11*(7*b*B - 15*A*c))/(112*b^3*c*x^(7/2)) - (11*(7*b*B - 15*A*c))/(48*b^4*x^(3/2
)) - (b*B - A*c)/(4*b*c*x^(7/2)*(b + c*x^2)^2) - (7*b*B - 15*A*c)/(16*b^2*c*x^(7
/2)*(b + c*x^2)) + (11*c^(3/4)*(7*b*B - 15*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt
[x])/b^(1/4)])/(32*Sqrt[2]*b^(19/4)) - (11*c^(3/4)*(7*b*B - 15*A*c)*ArcTan[1 + (
Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(19/4)) + (11*c^(3/4)*(7*b*B -
15*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*
b^(19/4)) - (11*c^(3/4)*(7*b*B - 15*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*S
qrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(19/4))

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Rubi in Sympy [A]  time = 91.9226, size = 326, normalized size = 0.95 \[ \frac{A c - B b}{4 b c x^{\frac{7}{2}} \left (b + c x^{2}\right )^{2}} + \frac{15 A c - 7 B b}{16 b^{2} c x^{\frac{7}{2}} \left (b + c x^{2}\right )} - \frac{11 \left (15 A c - 7 B b\right )}{112 b^{3} c x^{\frac{7}{2}}} + \frac{11 \left (15 A c - 7 B b\right )}{48 b^{4} x^{\frac{3}{2}}} - \frac{11 \sqrt{2} c^{\frac{3}{4}} \left (15 A c - 7 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{19}{4}}} + \frac{11 \sqrt{2} c^{\frac{3}{4}} \left (15 A c - 7 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{19}{4}}} - \frac{11 \sqrt{2} c^{\frac{3}{4}} \left (15 A c - 7 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{19}{4}}} + \frac{11 \sqrt{2} c^{\frac{3}{4}} \left (15 A c - 7 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{19}{4}}} \]

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)**3,x)

[Out]

(A*c - B*b)/(4*b*c*x**(7/2)*(b + c*x**2)**2) + (15*A*c - 7*B*b)/(16*b**2*c*x**(7
/2)*(b + c*x**2)) - 11*(15*A*c - 7*B*b)/(112*b**3*c*x**(7/2)) + 11*(15*A*c - 7*B
*b)/(48*b**4*x**(3/2)) - 11*sqrt(2)*c**(3/4)*(15*A*c - 7*B*b)*log(-sqrt(2)*b**(1
/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(19/4)) + 11*sqrt(2)*c**(3/4
)*(15*A*c - 7*B*b)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/
(128*b**(19/4)) - 11*sqrt(2)*c**(3/4)*(15*A*c - 7*B*b)*atan(1 - sqrt(2)*c**(1/4)
*sqrt(x)/b**(1/4))/(64*b**(19/4)) + 11*sqrt(2)*c**(3/4)*(15*A*c - 7*B*b)*atan(1
+ sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(19/4))

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Mathematica [A]  time = 0.528684, size = 308, normalized size = 0.9 \[ \frac{-\frac{1792 b^{3/4} (b B-3 A c)}{x^{3/2}}-\frac{672 b^{7/4} c \sqrt{x} (b B-A c)}{\left (b+c x^2\right )^2}-\frac{168 b^{3/4} c \sqrt{x} (15 b B-23 A c)}{b+c x^2}-\frac{768 A b^{7/4}}{x^{7/2}}+231 \sqrt{2} c^{3/4} (7 b B-15 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+231 \sqrt{2} c^{3/4} (15 A c-7 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-462 \sqrt{2} c^{3/4} (15 A c-7 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+462 \sqrt{2} c^{3/4} (15 A c-7 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{2688 b^{19/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-768*A*b^(7/4))/x^(7/2) - (1792*b^(3/4)*(b*B - 3*A*c))/x^(3/2) - (672*b^(7/4)*
c*(b*B - A*c)*Sqrt[x])/(b + c*x^2)^2 - (168*b^(3/4)*c*(15*b*B - 23*A*c)*Sqrt[x])
/(b + c*x^2) - 462*Sqrt[2]*c^(3/4)*(-7*b*B + 15*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)
*Sqrt[x])/b^(1/4)] + 462*Sqrt[2]*c^(3/4)*(-7*b*B + 15*A*c)*ArcTan[1 + (Sqrt[2]*c
^(1/4)*Sqrt[x])/b^(1/4)] + 231*Sqrt[2]*c^(3/4)*(7*b*B - 15*A*c)*Log[Sqrt[b] - Sq
rt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 231*Sqrt[2]*c^(3/4)*(-7*b*B + 15*A*
c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2688*b^(19/4))

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Maple [A]  time = 0.031, size = 390, normalized size = 1.1 \[ -{\frac{2\,A}{7\,{b}^{3}}{x}^{-{\frac{7}{2}}}}+2\,{\frac{Ac}{{x}^{3/2}{b}^{4}}}-{\frac{2\,B}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}+{\frac{23\,A{c}^{3}}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{15\,B{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{27\,A{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{19\,Bc}{16\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}+{\frac{165\,{c}^{2}\sqrt{2}A}{64\,{b}^{5}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{165\,{c}^{2}\sqrt{2}A}{64\,{b}^{5}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{165\,{c}^{2}\sqrt{2}A}{128\,{b}^{5}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{77\,c\sqrt{2}B}{64\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{77\,c\sqrt{2}B}{64\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{77\,c\sqrt{2}B}{128\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) } \]

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)^3,x)

[Out]

-2/7*A/b^3/x^(7/2)+2/x^(3/2)/b^4*A*c-2/3/x^(3/2)/b^3*B+23/16/b^4*c^3/(c*x^2+b)^2
*x^(5/2)*A-15/16/b^3*c^2/(c*x^2+b)^2*x^(5/2)*B+27/16/b^3*c^2/(c*x^2+b)^2*A*x^(1/
2)-19/16/b^2*c/(c*x^2+b)^2*B*x^(1/2)+165/64/b^5*c^2*(b/c)^(1/4)*2^(1/2)*A*arctan
(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+165/64/b^5*c^2*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1
/2)/(b/c)^(1/4)*x^(1/2)-1)+165/128/b^5*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)))-77/
64/b^4*c*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-77/64/b^4*c
*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)-77/128/b^4*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)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2536, size = 1010, normalized size = 2.94 \[ \text{result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

-1/1344*(308*(7*B*b*c^2 - 15*A*c^3)*x^6 + 484*(7*B*b^2*c - 15*A*b*c^2)*x^4 + 384
*A*b^3 + 128*(7*B*b^3 - 15*A*b^2*c)*x^2 + 924*(b^4*c^2*x^7 + 2*b^5*c*x^5 + b^6*x
^3)*sqrt(x)*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 -
94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4)*arctan(-b^5*(-(2401*B^4*b^4*c^3 -
20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7
)/b^19)^(1/4)/((7*B*b*c - 15*A*c^2)*sqrt(x) - sqrt(b^10*sqrt(-(2401*B^4*b^4*c^3
- 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^
7)/b^19) + (49*B^2*b^2*c^2 - 210*A*B*b*c^3 + 225*A^2*c^4)*x))) - 231*(b^4*c^2*x^
7 + 2*b^5*c*x^5 + b^6*x^3)*sqrt(x)*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 6
6150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4)*log(11*b^5
*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B
*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4) - 11*(7*B*b*c - 15*A*c^2)*sqrt(x)) + 231*(b^
4*c^2*x^7 + 2*b^5*c*x^5 + b^6*x^3)*sqrt(x)*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3
*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4)*lo
g(-11*b^5*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94
500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4) - 11*(7*B*b*c - 15*A*c^2)*sqrt(x)))
/((b^4*c^2*x^7 + 2*b^5*c*x^5 + b^6*x^3)*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(x**(3/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22532, size = 425, normalized size = 1.24 \[ -\frac{11 \, \sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{5}} - \frac{11 \, \sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{5}} - \frac{11 \, \sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{5}} + \frac{11 \, \sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{5}} - \frac{15 \, B b c^{2} x^{\frac{5}{2}} - 23 \, A c^{3} x^{\frac{5}{2}} + 19 \, B b^{2} c \sqrt{x} - 27 \, A b c^{2} \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{4}} - \frac{2 \,{\left (7 \, B b x^{2} - 21 \, A c x^{2} + 3 \, A b\right )}}{21 \, b^{4} x^{\frac{7}{2}}} \]

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)^3,x, algorithm="giac")

[Out]

-11/64*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 15*(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^5 - 11/64*sqrt(2)*(7*(b*c^3)^(1/
4)*B*b - 15*(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^5 - 11/128*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 15*(b*c^3)^(1/4)*A
*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^5 + 11/128*sqrt(2)*(7*(b*c
^3)^(1/4)*B*b - 15*(b*c^3)^(1/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt
(b/c))/b^5 - 1/16*(15*B*b*c^2*x^(5/2) - 23*A*c^3*x^(5/2) + 19*B*b^2*c*sqrt(x) -
27*A*b*c^2*sqrt(x))/((c*x^2 + b)^2*b^4) - 2/21*(7*B*b*x^2 - 21*A*c*x^2 + 3*A*b)/
(b^4*x^(7/2))