\(\int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [133]

Optimal result

Integrand size = 46, antiderivative size = 169 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {\left (2 a e^2 g-c d (e f+d g)\right ) \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{g^{3/2} (c d f-a e g)^{3/2}} \] Output:

-(-d*g+e*f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(-a*e*g+c*d*f)/(e*x+ 
d)^(1/2)/(g*x+f)+(2*a*e^2*g-c*d*(d*g+e*f))*arctan(1/g^(1/2)/(a*d*e+(a*e^2+ 
c*d^2)*x+c*d*e*x^2)^(1/2)*(-a*e*g+c*d*f)^(1/2)*(e*x+d)^(1/2))/g^(3/2)/(-a* 
e*g+c*d*f)^(3/2)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {d+e x} \left (-\frac {\sqrt {g} (-e f+d g) (a e+c d x)}{(-c d f+a e g) (f+g x)}+\frac {\left (-2 a e^2 g+c d (e f+d g)\right ) \sqrt {a e+c d x} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{3/2}}\right )}{g^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(d + e*x)^(3/2)/((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2]),x]
 

Output:

(Sqrt[d + e*x]*(-((Sqrt[g]*(-(e*f) + d*g)*(a*e + c*d*x))/((-(c*d*f) + a*e* 
g)*(f + g*x))) + ((-2*a*e^2*g + c*d*(e*f + d*g))*Sqrt[a*e + c*d*x]*ArcTan[ 
(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(c*d*f - a*e*g)^(3/2)))/ 
(g^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1257, 1255, 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.

\(\displaystyle \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\)

\(\Big \downarrow \) 1257

\(\displaystyle -\frac {\left (2 a e^2 g-c d (d g+e f)\right ) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 g (c d f-a e g)}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x} (f+g x) (c d f-a e g)}\)

\(\Big \downarrow \) 1255

\(\displaystyle -\frac {e^2 \left (2 a e^2 g-c d (d g+e f)\right ) \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{g (c d f-a e g)}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x} (f+g x) (c d f-a e g)}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\left (2 a e^2 g-c d (d g+e f)\right ) \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{3/2} (c d f-a e g)^{3/2}}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x} (f+g x) (c d f-a e g)}\)

Input:

Int[(d + e*x)^(3/2)/((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 
2]),x]
 

Output:

-(((e*f - d*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g*(c*d*f - a* 
e*g)*Sqrt[d + e*x]*(f + g*x))) - ((2*a*e^2*g - c*d*(e*f + d*g))*ArcTan[(Sq 
rt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sq 
rt[d + e*x])])/(g^(3/2)*(c*d*f - a*e*g)^(3/2))
 

Defintions of rubi rules used

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 1255

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + 
 (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2   Subst[Int[1/(c*(e*f + d*g) - b*e 
*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{ 
a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

rule 1257

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e^2*(e*f - d*g)*(d + e*x)^(m - 2)*( 
f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/(g*(n + 1)*(c*e*f + c*d*g - b*e 
*g))), x] - Simp[e*((b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/( 
g*(n + 1)*(c*e*f + c*d*g - b*e*g)))   Int[(d + e*x)^(m - 1)*(f + g*x)^(n + 
1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && 
EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p - 1, 0] && LtQ[n, -1] && Integer 
Q[2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(336\) vs. \(2(153)=306\).

Time = 3.05 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.99

method	result	size
default	\(\frac {\left (-2 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) a \,e^{2} g^{2} x +\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c \,d^{2} g^{2} x +\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c d e f g x -2 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) a \,e^{2} f g +\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c \,d^{2} f g +\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c d e \,f^{2}-\sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, d g +\sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, e f \right ) \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}}{\sqrt {e x +d}\, \sqrt {c d x +a e}\, g \left (a e g -d f c \right ) \left (g x +f \right ) \sqrt {\left (a e g -d f c \right ) g}}\)	\(337\)

Input:

int((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,meth 
od=_RETURNVERBOSE)
 

Output:

(-2*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*e^2*g^2*x+arcta 
nh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c*d^2*g^2*x+arctanh(g*(c*d 
*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c*d*e*f*g*x-2*arctanh(g*(c*d*x+a*e) 
^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*e^2*f*g+arctanh(g*(c*d*x+a*e)^(1/2)/((a* 
e*g-c*d*f)*g)^(1/2))*c*d^2*f*g+arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)* 
g)^(1/2))*c*d*e*f^2-((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*d*g+((a*e*g- 
c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*e*f)/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e)) 
^(1/2)/(c*d*x+a*e)^(1/2)/g/(a*e*g-c*d*f)/(g*x+f)/((a*e*g-c*d*f)*g)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (153) = 306\).

Time = 0.13 (sec) , antiderivative size = 897, normalized size of antiderivative = 5.31 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), 
x, algorithm="fricas")
 

Output:

[-1/2*((c*d^2*e*f^2 + (c*d^3 - 2*a*d*e^2)*f*g + (c*d*e^2*f*g + (c*d^2*e - 
2*a*e^3)*g^2)*x^2 + (c*d*e^2*f^2 + 2*(c*d^2*e - a*e^3)*f*g + (c*d^3 - 2*a* 
d*e^2)*g^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a* 
d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x - 2*sqrt(c*d*e*x^2 + a*d*e + (c* 
d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + ( 
e*f + d*g)*x)) + 2*(c*d*e*f^2*g + a*d*e*g^3 - (c*d^2 + a*e^2)*f*g^2)*sqrt( 
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^2*d^3*f^3*g^2 - 2 
*a*c*d^2*e*f^2*g^3 + a^2*d*e^2*f*g^4 + (c^2*d^2*e*f^2*g^3 - 2*a*c*d*e^2*f* 
g^4 + a^2*e^3*g^5)*x^2 + (c^2*d^2*e*f^3*g^2 + a^2*d*e^2*g^5 + (c^2*d^3 - 2 
*a*c*d*e^2)*f^2*g^3 - (2*a*c*d^2*e - a^2*e^3)*f*g^4)*x), -((c*d^2*e*f^2 + 
(c*d^3 - 2*a*d*e^2)*f*g + (c*d*e^2*f*g + (c*d^2*e - 2*a*e^3)*g^2)*x^2 + (c 
*d*e^2*f^2 + 2*(c*d^2*e - a*e^3)*f*g + (c*d^3 - 2*a*d*e^2)*g^2)*x)*sqrt(c* 
d*f*g - a*e*g^2)*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt( 
c*d*f*g - a*e*g^2)*sqrt(e*x + d)/(c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)* 
x)) + (c*d*e*f^2*g + a*d*e*g^3 - (c*d^2 + a*e^2)*f*g^2)*sqrt(c*d*e*x^2 + a 
*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^2*d^3*f^3*g^2 - 2*a*c*d^2*e*f^ 
2*g^3 + a^2*d*e^2*f*g^4 + (c^2*d^2*e*f^2*g^3 - 2*a*c*d*e^2*f*g^4 + a^2*e^3 
*g^5)*x^2 + (c^2*d^2*e*f^3*g^2 + a^2*d*e^2*g^5 + (c^2*d^3 - 2*a*c*d*e^2)*f 
^2*g^3 - (2*a*c*d^2*e - a^2*e^3)*f*g^4)*x)]
 

Sympy [F]

\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{2}}\, dx \] Input:

integrate((e*x+d)**(3/2)/(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)** 
(1/2),x)
 

Output:

Integral((d + e*x)**(3/2)/(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)**2), x)
 

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{2}} \,d x } \] Input:

integrate((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), 
x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(3/2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g* 
x + f)^2), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.50 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {e {\left (\frac {{\left (c^{2} d^{2} e^{2} f + c^{2} d^{3} e g - 2 \, a c d e^{3} g\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{{\left (c d f g {\left | e \right |} - a e g^{2} {\left | e \right |}\right )} \sqrt {c d f g - a e g^{2}} e} - \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{2} e^{2} f - \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{3} e g}{{\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )} {\left (c d f g {\left | e \right |} - a e g^{2} {\left | e \right |}\right )}}\right )}}{c d} \] Input:

integrate((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), 
x, algorithm="giac")
 

Output:

e*((c^2*d^2*e^2*f + c^2*d^3*e*g - 2*a*c*d*e^3*g)*arctan(sqrt((e*x + d)*c*d 
*e - c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e))/((c*d*f*g*abs(e) - a* 
e*g^2*abs(e))*sqrt(c*d*f*g - a*e*g^2)*e) - (sqrt((e*x + d)*c*d*e - c*d^2*e 
 + a*e^3)*c^2*d^2*e^2*f - sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^2*d^3* 
e*g)/((c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g)*(c*d*f 
*g*abs(e) - a*e*g^2*abs(e))))/(c*d)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \] Input:

int((d + e*x)^(3/2)/((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^( 
1/2)),x)
 

Output:

int((d + e*x)^(3/2)/((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^( 
1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.63 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-2 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) a \,e^{2} f g -2 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) a \,e^{2} g^{2} x +\sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c \,d^{2} f g +\sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c \,d^{2} g^{2} x +\sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c d e \,f^{2}+\sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c d e f g x -\sqrt {c d x +a e}\, a d e \,g^{3}+\sqrt {c d x +a e}\, a \,e^{2} f \,g^{2}+\sqrt {c d x +a e}\, c \,d^{2} f \,g^{2}-\sqrt {c d x +a e}\, c d e \,f^{2} g}{g^{2} \left (a^{2} e^{2} g^{3} x -2 a c d e f \,g^{2} x +c^{2} d^{2} f^{2} g x +a^{2} e^{2} f \,g^{2}-2 a c d e \,f^{2} g +c^{2} d^{2} f^{3}\right )} \] Input:

int((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
 

Output:

( - 2*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*s 
qrt( - a*e*g + c*d*f)))*a*e**2*f*g - 2*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan 
((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*a*e**2*g**2*x + s 
qrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - 
a*e*g + c*d*f)))*c*d**2*f*g + sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a* 
e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c*d**2*g**2*x + sqrt(g)*sq 
rt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c 
*d*f)))*c*d*e*f**2 + sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x 
)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c*d*e*f*g*x - sqrt(a*e + c*d*x)*a*d 
*e*g**3 + sqrt(a*e + c*d*x)*a*e**2*f*g**2 + sqrt(a*e + c*d*x)*c*d**2*f*g** 
2 - sqrt(a*e + c*d*x)*c*d*e*f**2*g)/(g**2*(a**2*e**2*f*g**2 + a**2*e**2*g* 
*3*x - 2*a*c*d*e*f**2*g - 2*a*c*d*e*f*g**2*x + c**2*d**2*f**3 + c**2*d**2* 
f**2*g*x))