\(\int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^3} \, dx\) [301]

Optimal result

Integrand size = 33, antiderivative size = 136 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^3} \, dx=\frac {(3 f-2 g) \sqrt {3-2 x}}{2 \sqrt {6} g^2 (f+g x)^2}-\frac {(15 f+16 g) \sqrt {3-2 x}}{2 \sqrt {6} g^2 (2 f+3 g) (f+g x)}+\frac {(9 f+20 g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2 f+3 g}}\right )}{\sqrt {6} g^{5/2} (2 f+3 g)^{3/2}} \] Output:

1/12*(3*f-2*g)*(3-2*x)^(1/2)*6^(1/2)/g^2/(g*x+f)^2-1/12*(15*f+16*g)*(3-2*x 
)^(1/2)*6^(1/2)/g^2/(2*f+3*g)/(g*x+f)+1/6*(9*f+20*g)*arctanh(g^(1/2)*(3-2* 
x)^(1/2)/(2*f+3*g)^(1/2))*6^(1/2)/g^(5/2)/(2*f+3*g)^(3/2)
 

Mathematica [A] (verified)

Time = 2.68 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^3} \, dx=\frac {-\frac {\sqrt {g} \sqrt {6+5 x-6 x^2} \left (9 f^2+2 g^2 (3+8 x)+f g (11+15 x)\right )}{(2 f+3 g) \sqrt {2+3 x} (f+g x)^2}+\frac {2 (9 f+20 g) \text {arctanh}\left (\frac {\sqrt {2 f+3 g} \sqrt {6+5 x-6 x^2}}{\sqrt {g} (3-2 x) \sqrt {2+3 x}}\right )}{(2 f+3 g)^{3/2}}}{2 \sqrt {6} g^{5/2}} \] Input:

Integrate[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^3,x]
 

Output:

(-((Sqrt[g]*Sqrt[6 + 5*x - 6*x^2]*(9*f^2 + 2*g^2*(3 + 8*x) + f*g*(11 + 15* 
x)))/((2*f + 3*g)*Sqrt[2 + 3*x]*(f + g*x)^2)) + (2*(9*f + 20*g)*ArcTanh[(S 
qrt[2*f + 3*g]*Sqrt[6 + 5*x - 6*x^2])/(Sqrt[g]*(3 - 2*x)*Sqrt[2 + 3*x])])/ 
(2*f + 3*g)^(3/2))/(2*Sqrt[6]*g^(5/2))
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1245, 87, 27, 51, 73, 221}

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.

\(\displaystyle \int \frac {\sqrt {3 x+2} \sqrt {-x^2+\frac {5 x}{6}+1}}{(f+g x)^3} \, dx\)

\(\Big \downarrow \) 1245

\(\displaystyle \int \frac {\sqrt {\frac {1}{2}-\frac {x}{3}} (3 x+2)}{(f+g x)^3}dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(9 f+20 g) \int \frac {\sqrt {3-2 x}}{\sqrt {6} (f+g x)^2}dx}{2 g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{2 \sqrt {6} g (2 f+3 g) (f+g x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(9 f+20 g) \int \frac {\sqrt {3-2 x}}{(f+g x)^2}dx}{2 \sqrt {6} g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{2 \sqrt {6} g (2 f+3 g) (f+g x)^2}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(9 f+20 g) \left (-\frac {\int \frac {1}{\sqrt {3-2 x} (f+g x)}dx}{g}-\frac {\sqrt {3-2 x}}{g (f+g x)}\right )}{2 \sqrt {6} g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{2 \sqrt {6} g (2 f+3 g) (f+g x)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(9 f+20 g) \left (\frac {\int \frac {1}{\frac {1}{2} (2 f+3 g)-\frac {1}{2} g (3-2 x)}d\sqrt {3-2 x}}{g}-\frac {\sqrt {3-2 x}}{g (f+g x)}\right )}{2 \sqrt {6} g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{2 \sqrt {6} g (2 f+3 g) (f+g x)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(9 f+20 g) \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2 f+3 g}}\right )}{g^{3/2} \sqrt {2 f+3 g}}-\frac {\sqrt {3-2 x}}{g (f+g x)}\right )}{2 \sqrt {6} g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{2 \sqrt {6} g (2 f+3 g) (f+g x)^2}\)

Input:

Int[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^3,x]
 

Output:

((3*f - 2*g)*(3 - 2*x)^(3/2))/(2*Sqrt[6]*g*(2*f + 3*g)*(f + g*x)^2) + ((9* 
f + 20*g)*(-(Sqrt[3 - 2*x]/(g*(f + g*x))) + (2*ArcTanh[(Sqrt[g]*Sqrt[3 - 2 
*x])/Sqrt[2*f + 3*g]])/(g^(3/2)*Sqrt[2*f + 3*g])))/(2*Sqrt[6]*g*(2*f + 3*g 
))
 

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 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 221

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

rule 1245

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(113)=226\).

Time = 1.41 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.72

method	result	size
default	\(\frac {\sqrt {-6 x^{2}+5 x +6}\, \sqrt {6}\, \left (18 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) f \,g^{2} x^{2}+40 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) g^{3} x^{2}+36 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) f^{2} g x +80 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) f \,g^{2} x -15 \sqrt {g \left (2 f +3 g \right )}\, \sqrt {3-2 x}\, f g x -16 \sqrt {g \left (2 f +3 g \right )}\, \sqrt {3-2 x}\, g^{2} x +18 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) f^{3}+40 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) f^{2} g -9 \sqrt {g \left (2 f +3 g \right )}\, \sqrt {3-2 x}\, f^{2}-11 \sqrt {g \left (2 f +3 g \right )}\, \sqrt {3-2 x}\, f g -6 \sqrt {g \left (2 f +3 g \right )}\, \sqrt {3-2 x}\, g^{2}\right )}{12 \sqrt {3 x +2}\, \sqrt {3-2 x}\, \sqrt {g \left (2 f +3 g \right )}\, g^{2} \left (2 f +3 g \right ) \left (g x +f \right )^{2}}\)	\(370\)

Input:

int(1/6*(3*x+2)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^3,x,method=_RETURNVE 
RBOSE)
 

Output:

1/12*(-6*x^2+5*x+6)^(1/2)*6^(1/2)*(18*arctanh(1/6*g*(-12*x+18)^(1/2)*6^(1/ 
2)/(g*(2*f+3*g))^(1/2))*f*g^2*x^2+40*arctanh(1/6*g*(-12*x+18)^(1/2)*6^(1/2 
)/(g*(2*f+3*g))^(1/2))*g^3*x^2+36*arctanh(1/6*g*(-12*x+18)^(1/2)*6^(1/2)/( 
g*(2*f+3*g))^(1/2))*f^2*g*x+80*arctanh(1/6*g*(-12*x+18)^(1/2)*6^(1/2)/(g*( 
2*f+3*g))^(1/2))*f*g^2*x-15*(g*(2*f+3*g))^(1/2)*(3-2*x)^(1/2)*f*g*x-16*(g* 
(2*f+3*g))^(1/2)*(3-2*x)^(1/2)*g^2*x+18*arctanh(1/6*g*(-12*x+18)^(1/2)*6^( 
1/2)/(g*(2*f+3*g))^(1/2))*f^3+40*arctanh(1/6*g*(-12*x+18)^(1/2)*6^(1/2)/(g 
*(2*f+3*g))^(1/2))*f^2*g-9*(g*(2*f+3*g))^(1/2)*(3-2*x)^(1/2)*f^2-11*(g*(2* 
f+3*g))^(1/2)*(3-2*x)^(1/2)*f*g-6*(g*(2*f+3*g))^(1/2)*(3-2*x)^(1/2)*g^2)/( 
3*x+2)^(1/2)/(3-2*x)^(1/2)/(g*(2*f+3*g))^(1/2)/g^2/(2*f+3*g)/(g*x+f)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (113) = 226\).

Time = 0.10 (sec) , antiderivative size = 684, normalized size of antiderivative = 5.03 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^3} \, dx =\text {Too large to display} \] Input:

integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^3,x, algorithm 
="fricas")
 

Output:

[1/12*((3*(9*f*g^2 + 20*g^3)*x^3 + 18*f^3 + 40*f^2*g + 2*(27*f^2*g + 69*f* 
g^2 + 20*g^3)*x^2 + (27*f^3 + 96*f^2*g + 80*f*g^2)*x)*sqrt(12*f*g + 18*g^2 
)*log(-(18*g*x^2 - 6*(3*f + 7*g)*x - sqrt(12*f*g + 18*g^2)*sqrt(-36*x^2 + 
30*x + 36)*sqrt(3*x + 2) - 12*f - 36*g)/(3*g*x^2 + (3*f + 2*g)*x + 2*f)) - 
 (18*f^3*g + 49*f^2*g^2 + 45*f*g^3 + 18*g^4 + (30*f^2*g^2 + 77*f*g^3 + 48* 
g^4)*x)*sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2))/(8*f^4*g^3 + 24*f^3*g^4 + 
 18*f^2*g^5 + 3*(4*f^2*g^5 + 12*f*g^6 + 9*g^7)*x^3 + 2*(12*f^3*g^4 + 40*f^ 
2*g^5 + 39*f*g^6 + 9*g^7)*x^2 + (12*f^4*g^3 + 52*f^3*g^4 + 75*f^2*g^5 + 36 
*f*g^6)*x), -1/12*(2*(3*(9*f*g^2 + 20*g^3)*x^3 + 18*f^3 + 40*f^2*g + 2*(27 
*f^2*g + 69*f*g^2 + 20*g^3)*x^2 + (27*f^3 + 96*f^2*g + 80*f*g^2)*x)*sqrt(- 
12*f*g - 18*g^2)*arctan(1/6*sqrt(-12*f*g - 18*g^2)*sqrt(-36*x^2 + 30*x + 3 
6)*sqrt(3*x + 2)/(3*(2*f + 3*g)*x + 4*f + 6*g)) + (18*f^3*g + 49*f^2*g^2 + 
 45*f*g^3 + 18*g^4 + (30*f^2*g^2 + 77*f*g^3 + 48*g^4)*x)*sqrt(-36*x^2 + 30 
*x + 36)*sqrt(3*x + 2))/(8*f^4*g^3 + 24*f^3*g^4 + 18*f^2*g^5 + 3*(4*f^2*g^ 
5 + 12*f*g^6 + 9*g^7)*x^3 + 2*(12*f^3*g^4 + 40*f^2*g^5 + 39*f*g^6 + 9*g^7) 
*x^2 + (12*f^4*g^3 + 52*f^3*g^4 + 75*f^2*g^5 + 36*f*g^6)*x)]
 

Sympy [F]

\[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^3} \, dx=\frac {\sqrt {6} \int \frac {\sqrt {3 x + 2} \sqrt {- 6 x^{2} + 5 x + 6}}{f^{3} + 3 f^{2} g x + 3 f g^{2} x^{2} + g^{3} x^{3}}\, dx}{6} \] Input:

integrate(1/6*(2+3*x)**(1/2)*(-36*x**2+30*x+36)**(1/2)/(g*x+f)**3,x)
 

Output:

sqrt(6)*Integral(sqrt(3*x + 2)*sqrt(-6*x**2 + 5*x + 6)/(f**3 + 3*f**2*g*x 
+ 3*f*g**2*x**2 + g**3*x**3), x)/6
 

Maxima [F]

\[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^3} \, dx=\int { \frac {\sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{6 \, {\left (g x + f\right )}^{3}} \,d x } \] Input:

integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^3,x, algorithm 
="maxima")
 

Output:

1/6*integrate(sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2)/(g*x + f)^3, x)
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^3} \, dx=-\frac {1}{6} \, \sqrt {6} {\left (\frac {{\left (9 \, f + 20 \, g\right )} \arctan \left (\frac {g \sqrt {-2 \, x + 3}}{\sqrt {-2 \, f g - 3 \, g^{2}}}\right )}{{\left (2 \, f g^{2} + 3 \, g^{3}\right )} \sqrt {-2 \, f g - 3 \, g^{2}}} - \frac {15 \, f g {\left (-2 \, x + 3\right )}^{\frac {3}{2}} + 16 \, g^{2} {\left (-2 \, x + 3\right )}^{\frac {3}{2}} - 18 \, f^{2} \sqrt {-2 \, x + 3} - 67 \, f g \sqrt {-2 \, x + 3} - 60 \, g^{2} \sqrt {-2 \, x + 3}}{{\left (2 \, f g^{2} + 3 \, g^{3}\right )} {\left (g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g\right )}^{2}}\right )} \] Input:

integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^3,x, algorithm 
="giac")
 

Output:

-1/6*sqrt(6)*((9*f + 20*g)*arctan(g*sqrt(-2*x + 3)/sqrt(-2*f*g - 3*g^2))/( 
(2*f*g^2 + 3*g^3)*sqrt(-2*f*g - 3*g^2)) - (15*f*g*(-2*x + 3)^(3/2) + 16*g^ 
2*(-2*x + 3)^(3/2) - 18*f^2*sqrt(-2*x + 3) - 67*f*g*sqrt(-2*x + 3) - 60*g^ 
2*sqrt(-2*x + 3))/((2*f*g^2 + 3*g^3)*(g*(2*x - 3) + 2*f + 3*g)^2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^3} \, dx=\int \frac {\sqrt {3\,x+2}\,\sqrt {-36\,x^2+30\,x+36}}{6\,{\left (f+g\,x\right )}^3} \,d x \] Input:

int(((3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/(6*(f + g*x)^3),x)
 

Output:

int(((3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/(6*(f + g*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 410, normalized size of antiderivative = 3.01 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^3} \, dx=\frac {\sqrt {6}\, \left (18 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) f^{3}+36 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) f^{2} g x +40 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) f^{2} g +18 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) f \,g^{2} x^{2}+80 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) f \,g^{2} x +40 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) g^{3} x^{2}-18 \sqrt {-2 x +3}\, f^{3} g -30 \sqrt {-2 x +3}\, f^{2} g^{2} x -49 \sqrt {-2 x +3}\, f^{2} g^{2}-77 \sqrt {-2 x +3}\, f \,g^{3} x -45 \sqrt {-2 x +3}\, f \,g^{3}-48 \sqrt {-2 x +3}\, g^{4} x -18 \sqrt {-2 x +3}\, g^{4}\right )}{12 g^{3} \left (4 f^{2} g^{2} x^{2}+12 f \,g^{3} x^{2}+9 g^{4} x^{2}+8 f^{3} g x +24 f^{2} g^{2} x +18 f \,g^{3} x +4 f^{4}+12 f^{3} g +9 f^{2} g^{2}\right )} \] Input:

int(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^3,x)
 

Output:

(sqrt(6)*(18*sqrt(g)*sqrt( - 2*f - 3*g)*atan((sqrt( - 2*x + 3)*g)/(sqrt(g) 
*sqrt( - 2*f - 3*g)))*f**3 + 36*sqrt(g)*sqrt( - 2*f - 3*g)*atan((sqrt( - 2 
*x + 3)*g)/(sqrt(g)*sqrt( - 2*f - 3*g)))*f**2*g*x + 40*sqrt(g)*sqrt( - 2*f 
 - 3*g)*atan((sqrt( - 2*x + 3)*g)/(sqrt(g)*sqrt( - 2*f - 3*g)))*f**2*g + 1 
8*sqrt(g)*sqrt( - 2*f - 3*g)*atan((sqrt( - 2*x + 3)*g)/(sqrt(g)*sqrt( - 2* 
f - 3*g)))*f*g**2*x**2 + 80*sqrt(g)*sqrt( - 2*f - 3*g)*atan((sqrt( - 2*x + 
 3)*g)/(sqrt(g)*sqrt( - 2*f - 3*g)))*f*g**2*x + 40*sqrt(g)*sqrt( - 2*f - 3 
*g)*atan((sqrt( - 2*x + 3)*g)/(sqrt(g)*sqrt( - 2*f - 3*g)))*g**3*x**2 - 18 
*sqrt( - 2*x + 3)*f**3*g - 30*sqrt( - 2*x + 3)*f**2*g**2*x - 49*sqrt( - 2* 
x + 3)*f**2*g**2 - 77*sqrt( - 2*x + 3)*f*g**3*x - 45*sqrt( - 2*x + 3)*f*g* 
*3 - 48*sqrt( - 2*x + 3)*g**4*x - 18*sqrt( - 2*x + 3)*g**4))/(12*g**3*(4*f 
**4 + 8*f**3*g*x + 12*f**3*g + 4*f**2*g**2*x**2 + 24*f**2*g**2*x + 9*f**2* 
g**2 + 12*f*g**3*x**2 + 18*f*g**3*x + 9*g**4*x**2))