3.371 \(\int \frac{A+B x^2}{\sqrt{x} \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=235 \[ -\frac{(A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{(A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}-\frac{(A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} b^{5/4}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} b^{5/4}}+\frac{2 B \sqrt{x}}{b} \]

[Out]

(2*B*Sqrt[x])/b - ((A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(S
qrt[2]*a^(3/4)*b^(5/4)) + ((A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1
/4)])/(Sqrt[2]*a^(3/4)*b^(5/4)) - ((A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*b^(5/4)) + ((A*b - a*B)*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*b^(5/4))

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Rubi [A]  time = 0.374576, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{(A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{(A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}-\frac{(A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} b^{5/4}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} b^{5/4}}+\frac{2 B \sqrt{x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*Sqrt[x])/b - ((A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(S
qrt[2]*a^(3/4)*b^(5/4)) + ((A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1
/4)])/(Sqrt[2]*a^(3/4)*b^(5/4)) - ((A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*b^(5/4)) + ((A*b - a*B)*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*b^(5/4))

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Rubi in Sympy [A]  time = 71.4617, size = 219, normalized size = 0.93 \[ \frac{2 B \sqrt{x}}{b} - \frac{\sqrt{2} \left (A b - B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{3}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (A b - B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{3}{4}} b^{\frac{5}{4}}} - \frac{\sqrt{2} \left (A b - B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (A b - B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(x)/b - sqrt(2)*(A*b - B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqr
t(a) + sqrt(b)*x)/(4*a**(3/4)*b**(5/4)) + sqrt(2)*(A*b - B*a)*log(sqrt(2)*a**(1/
4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(3/4)*b**(5/4)) - sqrt(2)*(A*b
- B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(3/4)*b**(5/4)) + sqrt(
2)*(A*b - B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(3/4)*b**(5/4))

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Mathematica [A]  time = 0.203323, size = 212, normalized size = 0.9 \[ \frac{8 a^{3/4} \sqrt [4]{b} B \sqrt{x}-\sqrt{2} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+\sqrt{2} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-2 \sqrt{2} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+2 \sqrt{2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 a^{3/4} b^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 277, normalized size = 1.2 \[ 2\,{\frac{B\sqrt{x}}{b}}+{\frac{\sqrt{2}A}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{\sqrt{2}A}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}A}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{\sqrt{2}B}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{\sqrt{2}B}{4\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}B}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25076, size = 709, normalized size = 3.02 \[ -\frac{4 \, b \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a b \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac{1}{4}}}{{\left (B a - A b\right )} \sqrt{x} - \sqrt{a^{2} b^{2} \sqrt{-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}} +{\left (B^{2} a^{2} - 2 \, A B a b + A^{2} b^{2}\right )} x}}\right ) - b \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac{1}{4}} \log \left (a b \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac{1}{4}} -{\left (B a - A b\right )} \sqrt{x}\right ) + b \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac{1}{4}} \log \left (-a b \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac{1}{4}} -{\left (B a - A b\right )} \sqrt{x}\right ) - 4 \, B \sqrt{x}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b
^4)/(a^3*b^5))^(1/4)*arctan(-a*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2
- 4*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^5))^(1/4)/((B*a - A*b)*sqrt(x) - sqrt(a^2*b^2*
sqrt(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a
^3*b^5)) + (B^2*a^2 - 2*A*B*a*b + A^2*b^2)*x))) - b*(-(B^4*a^4 - 4*A*B^3*a^3*b +
6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^5))^(1/4)*log(a*b*(-(B^4*a^
4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^5))^(1/4
) - (B*a - A*b)*sqrt(x)) + b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*
A^3*B*a*b^3 + A^4*b^4)/(a^3*b^5))^(1/4)*log(-a*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*
A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^5))^(1/4) - (B*a - A*b)*sqrt(x
)) - 4*B*sqrt(x))/b

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Sympy [A]  time = 41.0706, size = 371, normalized size = 1.58 \[ \begin{cases} \tilde{\infty } \left (- \frac{2 A}{3 x^{\frac{3}{2}}} + 2 B \sqrt{x}\right ) & \text{for}\: a = 0 \wedge b = 0 \\\frac{- \frac{2 A}{3 x^{\frac{3}{2}}} + 2 B \sqrt{x}}{b} & \text{for}\: a = 0 \\\frac{2 A \sqrt{x} + \frac{2 B x^{\frac{5}{2}}}{5}}{a} & \text{for}\: b = 0 \\- \frac{\sqrt [4]{-1} A \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{3}{4}} b^{4} \left (\frac{1}{b}\right )^{\frac{15}{4}}} + \frac{\sqrt [4]{-1} A \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{3}{4}} b^{4} \left (\frac{1}{b}\right )^{\frac{15}{4}}} - \frac{\sqrt [4]{-1} A \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{a^{\frac{3}{4}} b^{4} \left (\frac{1}{b}\right )^{\frac{15}{4}}} + \frac{\sqrt [4]{-1} B \sqrt [4]{a} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 b^{5} \left (\frac{1}{b}\right )^{\frac{15}{4}}} - \frac{\sqrt [4]{-1} B \sqrt [4]{a} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 b^{5} \left (\frac{1}{b}\right )^{\frac{15}{4}}} + \frac{\sqrt [4]{-1} B \sqrt [4]{a} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{b^{5} \left (\frac{1}{b}\right )^{\frac{15}{4}}} + \frac{2 B \sqrt{x}}{b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo*(-2*A/(3*x**(3/2)) + 2*B*sqrt(x)), Eq(a, 0) & Eq(b, 0)), ((-2*A/(
3*x**(3/2)) + 2*B*sqrt(x))/b, Eq(a, 0)), ((2*A*sqrt(x) + 2*B*x**(5/2)/5)/a, Eq(b
, 0)), (-(-1)**(1/4)*A*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(
3/4)*b**4*(1/b)**(15/4)) + (-1)**(1/4)*A*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) +
sqrt(x))/(2*a**(3/4)*b**4*(1/b)**(15/4)) - (-1)**(1/4)*A*atan((-1)**(3/4)*sqrt(
x)/(a**(1/4)*(1/b)**(1/4)))/(a**(3/4)*b**4*(1/b)**(15/4)) + (-1)**(1/4)*B*a**(1/
4)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*b**5*(1/b)**(15/4)) - (-
1)**(1/4)*B*a**(1/4)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*b**5*(1
/b)**(15/4)) + (-1)**(1/4)*B*a**(1/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**
(1/4)))/(b**5*(1/b)**(15/4)) + 2*B*sqrt(x)/b, True))

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GIAC/XCAS [A]  time = 0.295454, size = 339, normalized size = 1.44 \[ \frac{2 \, B \sqrt{x}}{b} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{2}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{2}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a b^{2}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(x)/b - 1/2*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(1/2*s
qrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^2) - 1/2*sqrt(2)*((a*
b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2
*sqrt(x))/(a/b)^(1/4))/(a*b^2) - 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*
A*b)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^2) + 1/4*sqrt(2)*((a*b
^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/
b))/(a*b^2)