3.211 \(\int \frac{x^{15/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=298 \[ -\frac{(3 A c+5 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{9/4}}+\frac{(3 A c+5 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{9/4}}-\frac{(3 A c+5 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{9/4}}+\frac{(3 A c+5 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{7/4} c^{9/4}}-\frac{\sqrt{x} (3 A c+5 b B)}{16 b c^2 \left (b+c x^2\right )}-\frac{x^{5/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

[Out]

-((b*B - A*c)*x^(5/2))/(4*b*c*(b + c*x^2)^2) - ((5*b*B + 3*A*c)*Sqrt[x])/(16*b*c
^2*(b + c*x^2)) - ((5*b*B + 3*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]
)/(32*Sqrt[2]*b^(7/4)*c^(9/4)) + ((5*b*B + 3*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(32*Sqrt[2]*b^(7/4)*c^(9/4)) - ((5*b*B + 3*A*c)*Log[Sqrt[b] - S
qrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(7/4)*c^(9/4)) + ((5*
b*B + 3*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqr
t[2]*b^(7/4)*c^(9/4))

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Rubi [A]  time = 0.496616, antiderivative size = 298, 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 \[ -\frac{(3 A c+5 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{9/4}}+\frac{(3 A c+5 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{9/4}}-\frac{(3 A c+5 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{9/4}}+\frac{(3 A c+5 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{7/4} c^{9/4}}-\frac{\sqrt{x} (3 A c+5 b B)}{16 b c^2 \left (b+c x^2\right )}-\frac{x^{5/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

-((b*B - A*c)*x^(5/2))/(4*b*c*(b + c*x^2)^2) - ((5*b*B + 3*A*c)*Sqrt[x])/(16*b*c
^2*(b + c*x^2)) - ((5*b*B + 3*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]
)/(32*Sqrt[2]*b^(7/4)*c^(9/4)) + ((5*b*B + 3*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(32*Sqrt[2]*b^(7/4)*c^(9/4)) - ((5*b*B + 3*A*c)*Log[Sqrt[b] - S
qrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(7/4)*c^(9/4)) + ((5*
b*B + 3*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqr
t[2]*b^(7/4)*c^(9/4))

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Rubi in Sympy [A]  time = 75.0331, size = 277, normalized size = 0.93 \[ \frac{x^{\frac{5}{2}} \left (A c - B b\right )}{4 b c \left (b + c x^{2}\right )^{2}} - \frac{\sqrt{x} \left (3 A c + 5 B b\right )}{16 b c^{2} \left (b + c x^{2}\right )} - \frac{\sqrt{2} \left (3 A c + 5 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{7}{4}} c^{\frac{9}{4}}} + \frac{\sqrt{2} \left (3 A c + 5 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{7}{4}} c^{\frac{9}{4}}} - \frac{\sqrt{2} \left (3 A c + 5 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{7}{4}} c^{\frac{9}{4}}} + \frac{\sqrt{2} \left (3 A c + 5 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{7}{4}} c^{\frac{9}{4}}} \]

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)**3,x)

[Out]

x**(5/2)*(A*c - B*b)/(4*b*c*(b + c*x**2)**2) - sqrt(x)*(3*A*c + 5*B*b)/(16*b*c**
2*(b + c*x**2)) - sqrt(2)*(3*A*c + 5*B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x)
+ sqrt(b) + sqrt(c)*x)/(128*b**(7/4)*c**(9/4)) + sqrt(2)*(3*A*c + 5*B*b)*log(sq
rt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(7/4)*c**(9/4)) -
sqrt(2)*(3*A*c + 5*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(7/4
)*c**(9/4)) + sqrt(2)*(3*A*c + 5*B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4)
)/(64*b**(7/4)*c**(9/4))

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Mathematica [A]  time = 0.528269, size = 274, normalized size = 0.92 \[ \frac{-\frac{\sqrt{2} (3 A c+5 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{7/4}}+\frac{\sqrt{2} (3 A c+5 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{7/4}}-\frac{2 \sqrt{2} (3 A c+5 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{b^{7/4}}+\frac{2 \sqrt{2} (3 A c+5 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{b^{7/4}}+\frac{8 \sqrt [4]{c} \sqrt{x} (A c-9 b B)}{b \left (b+c x^2\right )}-\frac{32 \sqrt [4]{c} \sqrt{x} (A c-b B)}{\left (b+c x^2\right )^2}}{128 c^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

((-32*c^(1/4)*(-(b*B) + A*c)*Sqrt[x])/(b + c*x^2)^2 + (8*c^(1/4)*(-9*b*B + A*c)*
Sqrt[x])/(b*(b + c*x^2)) - (2*Sqrt[2]*(5*b*B + 3*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4
)*Sqrt[x])/b^(1/4)])/b^(7/4) + (2*Sqrt[2]*(5*b*B + 3*A*c)*ArcTan[1 + (Sqrt[2]*c^
(1/4)*Sqrt[x])/b^(1/4)])/b^(7/4) - (Sqrt[2]*(5*b*B + 3*A*c)*Log[Sqrt[b] - Sqrt[2
]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(7/4) + (Sqrt[2]*(5*b*B + 3*A*c)*Log[S
qrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(7/4))/(128*c^(9/4))

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Maple [A]  time = 0.024, size = 334, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( Ac-9\,Bb \right ){x}^{5/2}}{bc}}-1/32\,{\frac{ \left ( 3\,Ac+5\,Bb \right ) \sqrt{x}}{{c}^{2}}} \right ) }+{\frac{3\,\sqrt{2}A}{64\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}A}{64\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}A}{128\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{5\,\sqrt{2}B}{64\,b{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}B}{64\,b{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{5\,\sqrt{2}B}{128\,b{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) } \]

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)^3,x)

[Out]

2*(1/32*(A*c-9*B*b)/b/c*x^(5/2)-1/32*(3*A*c+5*B*b)/c^2*x^(1/2))/(c*x^2+b)^2+3/64
/c/b^2*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+3/64/c/b^2*(b
/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+3/128/c/b^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)))+5/64/c^2/b*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1
/4)*x^(1/2)+1)+5/64/c^2/b*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/
2)-1)+5/128/c^2/b*(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^(15/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255513, size = 921, normalized size = 3.09 \[ -\frac{4 \,{\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \left (-\frac{625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} c^{2} \left (-\frac{625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac{1}{4}}}{{\left (5 \, B b + 3 \, A c\right )} \sqrt{x} + \sqrt{b^{4} c^{4} \sqrt{-\frac{625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}} +{\left (25 \, B^{2} b^{2} + 30 \, A B b c + 9 \, A^{2} c^{2}\right )} x}}\right ) -{\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \left (-\frac{625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac{1}{4}} \log \left (b^{2} c^{2} \left (-\frac{625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac{1}{4}} +{\left (5 \, B b + 3 \, A c\right )} \sqrt{x}\right ) +{\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \left (-\frac{625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac{1}{4}} \log \left (-b^{2} c^{2} \left (-\frac{625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac{1}{4}} +{\left (5 \, B b + 3 \, A c\right )} \sqrt{x}\right ) + 4 \,{\left (5 \, B b^{2} + 3 \, A b c +{\left (9 \, B b c - A c^{2}\right )} x^{2}\right )} \sqrt{x}}{64 \,{\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

-1/64*(4*(b*c^4*x^4 + 2*b^2*c^3*x^2 + b^3*c^2)*(-(625*B^4*b^4 + 1500*A*B^3*b^3*c
+ 1350*A^2*B^2*b^2*c^2 + 540*A^3*B*b*c^3 + 81*A^4*c^4)/(b^7*c^9))^(1/4)*arctan(
b^2*c^2*(-(625*B^4*b^4 + 1500*A*B^3*b^3*c + 1350*A^2*B^2*b^2*c^2 + 540*A^3*B*b*c
^3 + 81*A^4*c^4)/(b^7*c^9))^(1/4)/((5*B*b + 3*A*c)*sqrt(x) + sqrt(b^4*c^4*sqrt(-
(625*B^4*b^4 + 1500*A*B^3*b^3*c + 1350*A^2*B^2*b^2*c^2 + 540*A^3*B*b*c^3 + 81*A^
4*c^4)/(b^7*c^9)) + (25*B^2*b^2 + 30*A*B*b*c + 9*A^2*c^2)*x))) - (b*c^4*x^4 + 2*
b^2*c^3*x^2 + b^3*c^2)*(-(625*B^4*b^4 + 1500*A*B^3*b^3*c + 1350*A^2*B^2*b^2*c^2
+ 540*A^3*B*b*c^3 + 81*A^4*c^4)/(b^7*c^9))^(1/4)*log(b^2*c^2*(-(625*B^4*b^4 + 15
00*A*B^3*b^3*c + 1350*A^2*B^2*b^2*c^2 + 540*A^3*B*b*c^3 + 81*A^4*c^4)/(b^7*c^9))
^(1/4) + (5*B*b + 3*A*c)*sqrt(x)) + (b*c^4*x^4 + 2*b^2*c^3*x^2 + b^3*c^2)*(-(625
*B^4*b^4 + 1500*A*B^3*b^3*c + 1350*A^2*B^2*b^2*c^2 + 540*A^3*B*b*c^3 + 81*A^4*c^
4)/(b^7*c^9))^(1/4)*log(-b^2*c^2*(-(625*B^4*b^4 + 1500*A*B^3*b^3*c + 1350*A^2*B^
2*b^2*c^2 + 540*A^3*B*b*c^3 + 81*A^4*c^4)/(b^7*c^9))^(1/4) + (5*B*b + 3*A*c)*sqr
t(x)) + 4*(5*B*b^2 + 3*A*b*c + (9*B*b*c - A*c^2)*x^2)*sqrt(x))/(b*c^4*x^4 + 2*b^
2*c^3*x^2 + b^3*c^2)

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221233, size = 402, normalized size = 1.35 \[ \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{1}{4}} B b + 3 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{3}} + \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{1}{4}} B b + 3 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{3}} + \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{1}{4}} B b + 3 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{3}} - \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{1}{4}} B b + 3 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{3}} - \frac{9 \, B b c x^{\frac{5}{2}} - A c^{2} x^{\frac{5}{2}} + 5 \, B b^{2} \sqrt{x} + 3 \, A b c \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} b c^{2}} \]

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)^3,x, algorithm="giac")

[Out]

1/64*sqrt(2)*(5*(b*c^3)^(1/4)*B*b + 3*(b*c^3)^(1/4)*A*c)*arctan(1/2*sqrt(2)*(sqr
t(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/(b^2*c^3) + 1/64*sqrt(2)*(5*(b*c^3)^(
1/4)*B*b + 3*(b*c^3)^(1/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqr
t(x))/(b/c)^(1/4))/(b^2*c^3) + 1/128*sqrt(2)*(5*(b*c^3)^(1/4)*B*b + 3*(b*c^3)^(1
/4)*A*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^2*c^3) - 1/128*sqrt(
2)*(5*(b*c^3)^(1/4)*B*b + 3*(b*c^3)^(1/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) +
x + sqrt(b/c))/(b^2*c^3) - 1/16*(9*B*b*c*x^(5/2) - A*c^2*x^(5/2) + 5*B*b^2*sqrt
(x) + 3*A*b*c*sqrt(x))/((c*x^2 + b)^2*b*c^2)