\(\int \frac {\sqrt {x} (A+B x)}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [398]

Optimal result

Integrand size = 29, antiderivative size = 125 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {(A b-a B) \sqrt {x}}{3 b^2 (a+b x)^3}+\frac {(A b-7 a B) \sqrt {x}}{12 a b^2 (a+b x)^2}+\frac {(A b+a B) \sqrt {x}}{8 a^2 b^2 (a+b x)}+\frac {(A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2}} \] Output:

-1/3*(A*b-B*a)*x^(1/2)/b^2/(b*x+a)^3+1/12*(A*b-7*B*a)*x^(1/2)/a/b^2/(b*x+a 
)^2+1/8*(A*b+B*a)*x^(1/2)/a^2/b^2/(b*x+a)+1/8*(A*b+B*a)*arctan(b^(1/2)*x^( 
1/2)/a^(1/2))/a^(5/2)/b^(5/2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {x} \left (-3 a^3 B+3 A b^3 x^2+a b^2 x (8 A+3 B x)-a^2 b (3 A+8 B x)\right )}{24 a^2 b^2 (a+b x)^3}+\frac {(A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2}} \] Input:

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

Output:

(Sqrt[x]*(-3*a^3*B + 3*A*b^3*x^2 + a*b^2*x*(8*A + 3*B*x) - a^2*b*(3*A + 8* 
B*x)))/(24*a^2*b^2*(a + b*x)^3) + ((A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sq 
rt[a]])/(8*a^(5/2)*b^(5/2))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1184, 27, 87, 51, 52, 73, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1184

\(\displaystyle b^4 \int \frac {\sqrt {x} (A+B x)}{b^4 (a+b x)^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {\sqrt {x} (A+B x)}{(a+b x)^4}dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(a B+A b) \int \frac {\sqrt {x}}{(a+b x)^3}dx}{2 a b}+\frac {x^{3/2} (A b-a B)}{3 a b (a+b x)^3}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(a B+A b) \left (\frac {\int \frac {1}{\sqrt {x} (a+b x)^2}dx}{4 b}-\frac {\sqrt {x}}{2 b (a+b x)^2}\right )}{2 a b}+\frac {x^{3/2} (A b-a B)}{3 a b (a+b x)^3}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a B+A b) \left (\frac {\frac {\int \frac {1}{\sqrt {x} (a+b x)}dx}{2 a}+\frac {\sqrt {x}}{a (a+b x)}}{4 b}-\frac {\sqrt {x}}{2 b (a+b x)^2}\right )}{2 a b}+\frac {x^{3/2} (A b-a B)}{3 a b (a+b x)^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(a B+A b) \left (\frac {\frac {\int \frac {1}{a+b x}d\sqrt {x}}{a}+\frac {\sqrt {x}}{a (a+b x)}}{4 b}-\frac {\sqrt {x}}{2 b (a+b x)^2}\right )}{2 a b}+\frac {x^{3/2} (A b-a B)}{3 a b (a+b x)^3}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {(a B+A b) \left (\frac {\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)}}{4 b}-\frac {\sqrt {x}}{2 b (a+b x)^2}\right )}{2 a b}+\frac {x^{3/2} (A b-a B)}{3 a b (a+b x)^3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {\frac {\left (A b +B a \right ) x^{\frac {5}{2}}}{8 a^{2}}+\frac {\left (A b -B a \right ) x^{\frac {3}{2}}}{3 a b}-\frac {\left (A b +B a \right ) \sqrt {x}}{8 b^{2}}}{\left (b x +a \right )^{3}}+\frac {\left (A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 a^{2} b^{2} \sqrt {a b}}\)	\(92\)
default	\(\frac {\frac {\left (A b +B a \right ) x^{\frac {5}{2}}}{8 a^{2}}+\frac {\left (A b -B a \right ) x^{\frac {3}{2}}}{3 a b}-\frac {\left (A b +B a \right ) \sqrt {x}}{8 b^{2}}}{\left (b x +a \right )^{3}}+\frac {\left (A b +B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 a^{2} b^{2} \sqrt {a b}}\)	\(92\)

Input:

int(x^(1/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
 

Output:

2*(1/16*(A*b+B*a)/a^2*x^(5/2)+1/6*(A*b-B*a)/a/b*x^(3/2)-1/16*(A*b+B*a)/b^2 
*x^(1/2))/(b*x+a)^3+1/8*(A*b+B*a)/a^2/b^2/(a*b)^(1/2)*arctan(b*x^(1/2)/(a* 
b)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 399, normalized size of antiderivative = 3.19 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [-\frac {3 \, {\left (B a^{4} + A a^{3} b + {\left (B a b^{3} + A b^{4}\right )} x^{3} + 3 \, {\left (B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \, {\left (B a^{3} b + A a^{2} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (3 \, B a^{4} b + 3 \, A a^{3} b^{2} - 3 \, {\left (B a^{2} b^{3} + A a b^{4}\right )} x^{2} + 8 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x\right )} \sqrt {x}}{48 \, {\left (a^{3} b^{6} x^{3} + 3 \, a^{4} b^{5} x^{2} + 3 \, a^{5} b^{4} x + a^{6} b^{3}\right )}}, -\frac {3 \, {\left (B a^{4} + A a^{3} b + {\left (B a b^{3} + A b^{4}\right )} x^{3} + 3 \, {\left (B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \, {\left (B a^{3} b + A a^{2} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (3 \, B a^{4} b + 3 \, A a^{3} b^{2} - 3 \, {\left (B a^{2} b^{3} + A a b^{4}\right )} x^{2} + 8 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x\right )} \sqrt {x}}{24 \, {\left (a^{3} b^{6} x^{3} + 3 \, a^{4} b^{5} x^{2} + 3 \, a^{5} b^{4} x + a^{6} b^{3}\right )}}\right ] \] Input:

integrate(x^(1/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
 

Output:

[-1/48*(3*(B*a^4 + A*a^3*b + (B*a*b^3 + A*b^4)*x^3 + 3*(B*a^2*b^2 + A*a*b^ 
3)*x^2 + 3*(B*a^3*b + A*a^2*b^2)*x)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b) 
*sqrt(x))/(b*x + a)) + 2*(3*B*a^4*b + 3*A*a^3*b^2 - 3*(B*a^2*b^3 + A*a*b^4 
)*x^2 + 8*(B*a^3*b^2 - A*a^2*b^3)*x)*sqrt(x))/(a^3*b^6*x^3 + 3*a^4*b^5*x^2 
 + 3*a^5*b^4*x + a^6*b^3), -1/24*(3*(B*a^4 + A*a^3*b + (B*a*b^3 + A*b^4)*x 
^3 + 3*(B*a^2*b^2 + A*a*b^3)*x^2 + 3*(B*a^3*b + A*a^2*b^2)*x)*sqrt(a*b)*ar 
ctan(sqrt(a*b)/(b*sqrt(x))) + (3*B*a^4*b + 3*A*a^3*b^2 - 3*(B*a^2*b^3 + A* 
a*b^4)*x^2 + 8*(B*a^3*b^2 - A*a^2*b^3)*x)*sqrt(x))/(a^3*b^6*x^3 + 3*a^4*b^ 
5*x^2 + 3*a^5*b^4*x + a^6*b^3)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2302 vs. \(2 (117) = 234\).

Time = 22.53 (sec) , antiderivative size = 2302, normalized size of antiderivative = 18.42 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((zoo*(-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(b, 0) 
), ((2*A*x**(3/2)/3 + 2*B*x**(5/2)/5)/a**4, Eq(b, 0)), ((-2*A/(5*x**(5/2)) 
 - 2*B/(3*x**(3/2)))/b**4, Eq(a, 0)), (3*A*a**3*b*log(sqrt(x) - sqrt(-a/b) 
)/(48*a**5*b**3*sqrt(-a/b) + 144*a**4*b**4*x*sqrt(-a/b) + 144*a**3*b**5*x* 
*2*sqrt(-a/b) + 48*a**2*b**6*x**3*sqrt(-a/b)) - 3*A*a**3*b*log(sqrt(x) + s 
qrt(-a/b))/(48*a**5*b**3*sqrt(-a/b) + 144*a**4*b**4*x*sqrt(-a/b) + 144*a** 
3*b**5*x**2*sqrt(-a/b) + 48*a**2*b**6*x**3*sqrt(-a/b)) - 6*A*a**2*b**2*sqr 
t(x)*sqrt(-a/b)/(48*a**5*b**3*sqrt(-a/b) + 144*a**4*b**4*x*sqrt(-a/b) + 14 
4*a**3*b**5*x**2*sqrt(-a/b) + 48*a**2*b**6*x**3*sqrt(-a/b)) + 9*A*a**2*b** 
2*x*log(sqrt(x) - sqrt(-a/b))/(48*a**5*b**3*sqrt(-a/b) + 144*a**4*b**4*x*s 
qrt(-a/b) + 144*a**3*b**5*x**2*sqrt(-a/b) + 48*a**2*b**6*x**3*sqrt(-a/b)) 
- 9*A*a**2*b**2*x*log(sqrt(x) + sqrt(-a/b))/(48*a**5*b**3*sqrt(-a/b) + 144 
*a**4*b**4*x*sqrt(-a/b) + 144*a**3*b**5*x**2*sqrt(-a/b) + 48*a**2*b**6*x** 
3*sqrt(-a/b)) + 16*A*a*b**3*x**(3/2)*sqrt(-a/b)/(48*a**5*b**3*sqrt(-a/b) + 
 144*a**4*b**4*x*sqrt(-a/b) + 144*a**3*b**5*x**2*sqrt(-a/b) + 48*a**2*b**6 
*x**3*sqrt(-a/b)) + 9*A*a*b**3*x**2*log(sqrt(x) - sqrt(-a/b))/(48*a**5*b** 
3*sqrt(-a/b) + 144*a**4*b**4*x*sqrt(-a/b) + 144*a**3*b**5*x**2*sqrt(-a/b) 
+ 48*a**2*b**6*x**3*sqrt(-a/b)) - 9*A*a*b**3*x**2*log(sqrt(x) + sqrt(-a/b) 
)/(48*a**5*b**3*sqrt(-a/b) + 144*a**4*b**4*x*sqrt(-a/b) + 144*a**3*b**5*x* 
*2*sqrt(-a/b) + 48*a**2*b**6*x**3*sqrt(-a/b)) + 6*A*b**4*x**(5/2)*sqrt(...
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {3 \, {\left (B a b^{2} + A b^{3}\right )} x^{\frac {5}{2}} - 8 \, {\left (B a^{2} b - A a b^{2}\right )} x^{\frac {3}{2}} - 3 \, {\left (B a^{3} + A a^{2} b\right )} \sqrt {x}}{24 \, {\left (a^{2} b^{5} x^{3} + 3 \, a^{3} b^{4} x^{2} + 3 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} + \frac {{\left (B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{2}} \] Input:

integrate(x^(1/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
 

Output:

1/24*(3*(B*a*b^2 + A*b^3)*x^(5/2) - 8*(B*a^2*b - A*a*b^2)*x^(3/2) - 3*(B*a 
^3 + A*a^2*b)*sqrt(x))/(a^2*b^5*x^3 + 3*a^3*b^4*x^2 + 3*a^4*b^3*x + a^5*b^ 
2) + 1/8*(B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2*b^2)
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {{\left (B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{2}} + \frac {3 \, B a b^{2} x^{\frac {5}{2}} + 3 \, A b^{3} x^{\frac {5}{2}} - 8 \, B a^{2} b x^{\frac {3}{2}} + 8 \, A a b^{2} x^{\frac {3}{2}} - 3 \, B a^{3} \sqrt {x} - 3 \, A a^{2} b \sqrt {x}}{24 \, {\left (b x + a\right )}^{3} a^{2} b^{2}} \] Input:

integrate(x^(1/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
 

Output:

1/8*(B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2*b^2) + 1/24*(3* 
B*a*b^2*x^(5/2) + 3*A*b^3*x^(5/2) - 8*B*a^2*b*x^(3/2) + 8*A*a*b^2*x^(3/2) 
- 3*B*a^3*sqrt(x) - 3*A*a^2*b*sqrt(x))/((b*x + a)^3*a^2*b^2)
 

Mupad [B] (verification not implemented)

Time = 11.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {x^{5/2}\,\left (A\,b+B\,a\right )}{8\,a^2}-\frac {\sqrt {x}\,\left (A\,b+B\,a\right )}{8\,b^2}+\frac {x^{3/2}\,\left (A\,b-B\,a\right )}{3\,a\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b+B\,a\right )}{8\,a^{5/2}\,b^{5/2}} \] Input:

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

Output:

((x^(5/2)*(A*b + B*a))/(8*a^2) - (x^(1/2)*(A*b + B*a))/(8*b^2) + (x^(3/2)* 
(A*b - B*a))/(3*a*b))/(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x) + (atan((b 
^(1/2)*x^(1/2))/a^(1/2))*(A*b + B*a))/(8*a^(5/2)*b^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2}+2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a b x +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} x^{2}-\sqrt {x}\, a^{2} b +\sqrt {x}\, a \,b^{2} x}{4 a^{2} b^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:

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

Output:

(sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**2 + 2*sqrt(b)*sqrt 
(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a*b*x + sqrt(b)*sqrt(a)*atan((sqrt 
(x)*b)/(sqrt(b)*sqrt(a)))*b**2*x**2 - sqrt(x)*a**2*b + sqrt(x)*a*b**2*x)/( 
4*a**2*b**2*(a**2 + 2*a*b*x + b**2*x**2))