3.207 \(\int \frac{x^{23/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=343 \[ -\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}-\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} c^{17/4}}-\frac{9 \sqrt{x} (13 b B-5 A c)}{16 c^4}+\frac{9 x^{5/2} (13 b B-5 A c)}{80 b c^3}-\frac{x^{9/2} (13 b B-5 A c)}{16 b c^2 \left (b+c x^2\right )}-\frac{x^{13/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

[Out]

(-9*(13*b*B - 5*A*c)*Sqrt[x])/(16*c^4) + (9*(13*b*B - 5*A*c)*x^(5/2))/(80*b*c^3)
- ((b*B - A*c)*x^(13/2))/(4*b*c*(b + c*x^2)^2) - ((13*b*B - 5*A*c)*x^(9/2))/(16
*b*c^2*(b + c*x^2)) - (9*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(32*Sqrt[2]*c^(17/4)) + (9*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 +
(Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*c^(17/4)) - (9*b^(1/4)*(13*b*B -
5*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*
c^(17/4)) + (9*b^(1/4)*(13*b*B - 5*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sq
rt[x] + Sqrt[c]*x])/(64*Sqrt[2]*c^(17/4))

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Rubi [A]  time = 0.589671, 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 \[ -\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}-\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} c^{17/4}}-\frac{9 \sqrt{x} (13 b B-5 A c)}{16 c^4}+\frac{9 x^{5/2} (13 b B-5 A c)}{80 b c^3}-\frac{x^{9/2} (13 b B-5 A c)}{16 b c^2 \left (b+c x^2\right )}-\frac{x^{13/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(23/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(-9*(13*b*B - 5*A*c)*Sqrt[x])/(16*c^4) + (9*(13*b*B - 5*A*c)*x^(5/2))/(80*b*c^3)
- ((b*B - A*c)*x^(13/2))/(4*b*c*(b + c*x^2)^2) - ((13*b*B - 5*A*c)*x^(9/2))/(16
*b*c^2*(b + c*x^2)) - (9*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(32*Sqrt[2]*c^(17/4)) + (9*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 +
(Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*c^(17/4)) - (9*b^(1/4)*(13*b*B -
5*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*
c^(17/4)) + (9*b^(1/4)*(13*b*B - 5*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sq
rt[x] + Sqrt[c]*x])/(64*Sqrt[2]*c^(17/4))

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Rubi in Sympy [A]  time = 91.4798, size = 330, normalized size = 0.96 \[ \frac{9 \sqrt{2} \sqrt [4]{b} \left (5 A c - 13 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 c^{\frac{17}{4}}} - \frac{9 \sqrt{2} \sqrt [4]{b} \left (5 A c - 13 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 c^{\frac{17}{4}}} + \frac{9 \sqrt{2} \sqrt [4]{b} \left (5 A c - 13 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 c^{\frac{17}{4}}} - \frac{9 \sqrt{2} \sqrt [4]{b} \left (5 A c - 13 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 c^{\frac{17}{4}}} + \frac{9 \sqrt{x} \left (5 A c - 13 B b\right )}{16 c^{4}} + \frac{x^{\frac{13}{2}} \left (A c - B b\right )}{4 b c \left (b + c x^{2}\right )^{2}} + \frac{x^{\frac{9}{2}} \left (5 A c - 13 B b\right )}{16 b c^{2} \left (b + c x^{2}\right )} - \frac{9 x^{\frac{5}{2}} \left (5 A c - 13 B b\right )}{80 b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(23/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

9*sqrt(2)*b**(1/4)*(5*A*c - 13*B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqr
t(b) + sqrt(c)*x)/(128*c**(17/4)) - 9*sqrt(2)*b**(1/4)*(5*A*c - 13*B*b)*log(sqrt
(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*c**(17/4)) + 9*sqrt(2)
*b**(1/4)*(5*A*c - 13*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*c**(1
7/4)) - 9*sqrt(2)*b**(1/4)*(5*A*c - 13*B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b*
*(1/4))/(64*c**(17/4)) + 9*sqrt(x)*(5*A*c - 13*B*b)/(16*c**4) + x**(13/2)*(A*c -
B*b)/(4*b*c*(b + c*x**2)**2) + x**(9/2)*(5*A*c - 13*B*b)/(16*b*c**2*(b + c*x**2
)) - 9*x**(5/2)*(5*A*c - 13*B*b)/(80*b*c**3)

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Mathematica [A]  time = 0.463628, size = 310, normalized size = 0.9 \[ \frac{\frac{160 b^2 \sqrt [4]{c} \sqrt{x} (b B-A c)}{\left (b+c x^2\right )^2}+\frac{40 b \sqrt [4]{c} \sqrt{x} (17 A c-25 b B)}{b+c x^2}+1280 \sqrt [4]{c} \sqrt{x} (A c-3 b B)-45 \sqrt{2} \sqrt [4]{b} (13 b B-5 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+45 \sqrt{2} \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-90 \sqrt{2} \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+90 \sqrt{2} \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+256 B c^{5/4} x^{5/2}}{640 c^{17/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(23/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(1280*c^(1/4)*(-3*b*B + A*c)*Sqrt[x] + 256*B*c^(5/4)*x^(5/2) + (160*b^2*c^(1/4)*
(b*B - A*c)*Sqrt[x])/(b + c*x^2)^2 + (40*b*c^(1/4)*(-25*b*B + 17*A*c)*Sqrt[x])/(
b + c*x^2) - 90*Sqrt[2]*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqr
t[x])/b^(1/4)] + 90*Sqrt[2]*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)
*Sqrt[x])/b^(1/4)] - 45*Sqrt[2]*b^(1/4)*(13*b*B - 5*A*c)*Log[Sqrt[b] - Sqrt[2]*b
^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 45*Sqrt[2]*b^(1/4)*(13*b*B - 5*A*c)*Log[Sq
rt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(640*c^(17/4))

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Maple [A]  time = 0.029, size = 381, normalized size = 1.1 \[{\frac{2\,B}{5\,{c}^{3}}{x}^{{\frac{5}{2}}}}+2\,{\frac{A\sqrt{x}}{{c}^{3}}}-6\,{\frac{\sqrt{x}Bb}{{c}^{4}}}+{\frac{17\,Ab}{16\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{25\,{b}^{2}B}{16\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{13\,{b}^{2}A}{16\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{21\,B{b}^{3}}{16\,{c}^{4} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{45\,\sqrt{2}A}{128\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{45\,\sqrt{2}A}{64\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{45\,\sqrt{2}A}{64\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{117\,b\sqrt{2}B}{128\,{c}^{4}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{117\,b\sqrt{2}B}{64\,{c}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{117\,b\sqrt{2}B}{64\,{c}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(23/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

2/5/c^3*B*x^(5/2)+2/c^3*A*x^(1/2)-6/c^4*x^(1/2)*B*b+17/16*b/c^2/(c*x^2+b)^2*x^(5
/2)*A-25/16*b^2/c^3/(c*x^2+b)^2*x^(5/2)*B+13/16*b^2/c^3/(c*x^2+b)^2*A*x^(1/2)-21
/16*b^3/c^4/(c*x^2+b)^2*B*x^(1/2)-45/128/c^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)))-4
5/64/c^3*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-45/64/c^3*(
b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+117/128*b/c^4*(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)))+117/64*b/c^4*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/
c)^(1/4)*x^(1/2)+1)+117/64*b/c^4*(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^(23/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24594, size = 944, normalized size = 2.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(23/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

1/320*(180*(c^6*x^4 + 2*b*c^5*x^2 + b^2*c^4)*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*
c + 25350*A^2*B^2*b^3*c^2 - 6500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(1/4)*arct
an(-c^4*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500*A^3*
B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(1/4)/((13*B*b - 5*A*c)*sqrt(x) - sqrt(c^8*sqrt
(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500*A^3*B*b^2*c^
3 + 625*A^4*b*c^4)/c^17) + (169*B^2*b^2 - 130*A*B*b*c + 25*A^2*c^2)*x))) - 45*(c
^6*x^4 + 2*b*c^5*x^2 + b^2*c^4)*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2
*B^2*b^3*c^2 - 6500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(1/4)*log(9*c^4*(-(2856
1*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500*A^3*B*b^2*c^3 + 625
*A^4*b*c^4)/c^17)^(1/4) - 9*(13*B*b - 5*A*c)*sqrt(x)) + 45*(c^6*x^4 + 2*b*c^5*x^
2 + b^2*c^4)*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500
*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(1/4)*log(-9*c^4*(-(28561*B^4*b^5 - 43940*
A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^
(1/4) - 9*(13*B*b - 5*A*c)*sqrt(x)) + 4*(32*B*c^3*x^6 - 32*(13*B*b*c^2 - 5*A*c^3
)*x^4 - 585*B*b^3 + 225*A*b^2*c - 81*(13*B*b^2*c - 5*A*b*c^2)*x^2)*sqrt(x))/(c^6
*x^4 + 2*b*c^5*x^2 + b^2*c^4)

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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**(23/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222285, size = 433, normalized size = 1.26 \[ \frac{9 \, \sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, c^{5}} + \frac{9 \, \sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, c^{5}} + \frac{9 \, \sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, c^{5}} - \frac{9 \, \sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, c^{5}} - \frac{25 \, B b^{2} c x^{\frac{5}{2}} - 17 \, A b c^{2} x^{\frac{5}{2}} + 21 \, B b^{3} \sqrt{x} - 13 \, A b^{2} c \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} c^{4}} + \frac{2 \,{\left (B c^{12} x^{\frac{5}{2}} - 15 \, B b c^{11} \sqrt{x} + 5 \, A c^{12} \sqrt{x}\right )}}{5 \, c^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(23/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

9/64*sqrt(2)*(13*(b*c^3)^(1/4)*B*b - 5*(b*c^3)^(1/4)*A*c)*arctan(1/2*sqrt(2)*(sq
rt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/c^5 + 9/64*sqrt(2)*(13*(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^5 + 9/128*sqrt(2)*(13*(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^5 - 9/128*sqrt(2)*(13*(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^5 - 1/16*(25*B*b^2*c*x^(5/2) - 17*A*b*c^2*x^(5/2) + 21*B*b^3*sqrt(x) - 13*A*
b^2*c*sqrt(x))/((c*x^2 + b)^2*c^4) + 2/5*(B*c^12*x^(5/2) - 15*B*b*c^11*sqrt(x) +
5*A*c^12*sqrt(x))/c^15