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

Optimal. Leaf size=237 \[ -\frac{(b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{(b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4} \sqrt [4]{c}}-\frac{(b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{7/4} \sqrt [4]{c}}-\frac{2 A}{3 b x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.385983, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ -\frac{(b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{(b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4} \sqrt [4]{c}}-\frac{(b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{7/4} \sqrt [4]{c}}-\frac{2 A}{3 b x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 64.4065, size = 221, normalized size = 0.93 \[ - \frac{2 A}{3 b x^{\frac{3}{2}}} + \frac{\sqrt{2} \left (A c - B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{7}{4}} \sqrt [4]{c}} - \frac{\sqrt{2} \left (A c - B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{7}{4}} \sqrt [4]{c}} + \frac{\sqrt{2} \left (A c - B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{7}{4}} \sqrt [4]{c}} - \frac{\sqrt{2} \left (A c - B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{7}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/(c*x**4+b*x**2)/x**(1/2),x)

[Out]

-2*A/(3*b*x**(3/2)) + sqrt(2)*(A*c - B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x)
+ sqrt(b) + sqrt(c)*x)/(4*b**(7/4)*c**(1/4)) - sqrt(2)*(A*c - B*b)*log(sqrt(2)*
b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(4*b**(7/4)*c**(1/4)) + sqrt(2)
*(A*c - B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b**(7/4)*c**(1/4)) -
sqrt(2)*(A*c - B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b**(7/4)*c**
(1/4))

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Mathematica [A]  time = 0.328013, size = 223, normalized size = 0.94 \[ \frac{-\frac{8 A b^{3/4}}{x^{3/2}}+\frac{3 \sqrt{2} (A c-b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{3 \sqrt{2} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} (A c-b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt [4]{c}}}{12 b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-8*A*b^(3/4))/x^(3/2) + (6*Sqrt[2]*(-(b*B) + A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*
Sqrt[x])/b^(1/4)])/c^(1/4) + (6*Sqrt[2]*(b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*
Sqrt[x])/b^(1/4)])/c^(1/4) + (3*Sqrt[2]*(-(b*B) + A*c)*Log[Sqrt[b] - Sqrt[2]*b^(
1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) + (3*Sqrt[2]*(b*B - A*c)*Log[Sqrt[b]
+ Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4))/(12*b^(7/4))

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Maple [A]  time = 0.014, size = 280, normalized size = 1.2 \[ -{\frac{2\,A}{3\,b}{x}^{-{\frac{3}{2}}}}-{\frac{\sqrt{2}Ac}{2\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{\sqrt{2}Ac}{4\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}Ac}{2\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{\sqrt{2}B}{2\,b}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{\sqrt{2}B}{4\,b}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}B}{2\,b}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(c*x^4+b*x^2)/x^(1/2),x)

[Out]

-2/3*A/b/x^(3/2)-1/2/b^2*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2
)-1)*c-1/4/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)))*c-1/2/b^2*(b/c)^(1/4)*2^(1/2)*A
*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)*c+1/2/b*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1
/2)/(b/c)^(1/4)*x^(1/2)-1)+1/4/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)))+1/2/b*(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)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243756, size = 722, normalized size = 3.05 \[ \frac{12 \, b x^{\frac{3}{2}} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{2} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac{1}{4}}}{{\left (B b - A c\right )} \sqrt{x} - \sqrt{b^{4} \sqrt{-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}} +{\left (B^{2} b^{2} - 2 \, A B b c + A^{2} c^{2}\right )} x}}\right ) - 3 \, b x^{\frac{3}{2}} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac{1}{4}} \log \left (b^{2} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac{1}{4}} -{\left (B b - A c\right )} \sqrt{x}\right ) + 3 \, b x^{\frac{3}{2}} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac{1}{4}} \log \left (-b^{2} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac{1}{4}} -{\left (B b - A c\right )} \sqrt{x}\right ) - 4 \, A}{6 \, b x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(12*b*x^(3/2)*(-(B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3
+ A^4*c^4)/(b^7*c))^(1/4)*arctan(-b^2*(-(B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*B^2*b^
2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b^7*c))^(1/4)/((B*b - A*c)*sqrt(x) - sqrt(b^4*
sqrt(-(B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b
^7*c)) + (B^2*b^2 - 2*A*B*b*c + A^2*c^2)*x))) - 3*b*x^(3/2)*(-(B^4*b^4 - 4*A*B^3
*b^3*c + 6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b^7*c))^(1/4)*log(b^2*(-(
B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b^7*c))^
(1/4) - (B*b - A*c)*sqrt(x)) + 3*b*x^(3/2)*(-(B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*B^
2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b^7*c))^(1/4)*log(-b^2*(-(B^4*b^4 - 4*A*B^
3*b^3*c + 6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b^7*c))^(1/4) - (B*b - A
*c)*sqrt(x)) - 4*A)/(b*x^(3/2))

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Sympy [A]  time = 86.1173, size = 379, normalized size = 1.6 \[ \begin{cases} \tilde{\infty } \left (- \frac{2 A}{7 x^{\frac{7}{2}}} - \frac{2 B}{3 x^{\frac{3}{2}}}\right ) & \text{for}\: b = 0 \wedge c = 0 \\\frac{- \frac{2 A}{7 x^{\frac{7}{2}}} - \frac{2 B}{3 x^{\frac{3}{2}}}}{c} & \text{for}\: b = 0 \\\frac{- \frac{2 A}{3 x^{\frac{3}{2}}} + 2 B \sqrt{x}}{b} & \text{for}\: c = 0 \\- \frac{2 A}{3 b x^{\frac{3}{2}}} + \frac{\sqrt [4]{-1} A c^{6} \left (\frac{1}{c}\right )^{\frac{21}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{7}{4}}} - \frac{\sqrt [4]{-1} A c^{6} \left (\frac{1}{c}\right )^{\frac{21}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{7}{4}}} + \frac{\sqrt [4]{-1} A c^{6} \left (\frac{1}{c}\right )^{\frac{21}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{b^{\frac{7}{4}}} - \frac{\sqrt [4]{-1} B c^{5} \left (\frac{1}{c}\right )^{\frac{21}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{3}{4}}} + \frac{\sqrt [4]{-1} B c^{5} \left (\frac{1}{c}\right )^{\frac{21}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{3}{4}}} - \frac{\sqrt [4]{-1} B c^{5} \left (\frac{1}{c}\right )^{\frac{21}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{b^{\frac{3}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/(c*x**4+b*x**2)/x**(1/2),x)

[Out]

Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/(3*x**(3/2))), Eq(b, 0) & Eq(c, 0)), ((-
2*A/(7*x**(7/2)) - 2*B/(3*x**(3/2)))/c, Eq(b, 0)), ((-2*A/(3*x**(3/2)) + 2*B*sqr
t(x))/b, Eq(c, 0)), (-2*A/(3*b*x**(3/2)) + (-1)**(1/4)*A*c**6*(1/c)**(21/4)*log(
-(-1)**(1/4)*b**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*b**(7/4)) - (-1)**(1/4)*A*c**6*
(1/c)**(21/4)*log((-1)**(1/4)*b**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*b**(7/4)) + (-
1)**(1/4)*A*c**6*(1/c)**(21/4)*atan((-1)**(3/4)*sqrt(x)/(b**(1/4)*(1/c)**(1/4)))
/b**(7/4) - (-1)**(1/4)*B*c**5*(1/c)**(21/4)*log(-(-1)**(1/4)*b**(1/4)*(1/c)**(1
/4) + sqrt(x))/(2*b**(3/4)) + (-1)**(1/4)*B*c**5*(1/c)**(21/4)*log((-1)**(1/4)*b
**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*b**(3/4)) - (-1)**(1/4)*B*c**5*(1/c)**(21/4)*
atan((-1)**(3/4)*sqrt(x)/(b**(1/4)*(1/c)**(1/4)))/b**(3/4), True))

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GIAC/XCAS [A]  time = 0.218781, size = 339, normalized size = 1.43 \[ \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2} c} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2} c} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2} c} - \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2} c} - \frac{2 \, A}{3 \, b x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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