\(\int \frac {(a+b x^2)^{3/2}}{x^2 (c+d x)^2} \, dx\) [1091]

Optimal result

Integrand size = 22, antiderivative size = 182 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^2 (c+d x)^2} \, dx=-\frac {a \sqrt {a+b x^2}}{c^2 x}-\frac {\left (b+\frac {a d^2}{c^2}\right ) \sqrt {a+b x^2}}{d (c+d x)}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^2}+\frac {\left (b c^2-2 a d^2\right ) \sqrt {b c^2+a d^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^3 d^2}+\frac {2 a^{3/2} d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^3} \] Output:

-a*(b*x^2+a)^(1/2)/c^2/x-(b+a*d^2/c^2)*(b*x^2+a)^(1/2)/d/(d*x+c)+b^(3/2)*a 
rctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/d^2+(-2*a*d^2+b*c^2)*(a*d^2+b*c^2)^(1/2) 
*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/c^3/d^2+2*a^(3/ 
2)*d*arctanh((b*x^2+a)^(1/2)/a^(1/2))/c^3
 

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^2 (c+d x)^2} \, dx=-\frac {\frac {d \sqrt {a+b x^2} \left (b c^2 x+a d (c+2 d x)\right )}{c^2 x (c+d x)}+\frac {2 \left (b c^2-2 a d^2\right ) \sqrt {-b c^2-a d^2} \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{c^3}+\frac {4 a^{3/2} d^3 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^3}+b^{3/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{d^2} \] Input:

Integrate[(a + b*x^2)^(3/2)/(x^2*(c + d*x)^2),x]
 

Output:

-(((d*Sqrt[a + b*x^2]*(b*c^2*x + a*d*(c + 2*d*x)))/(c^2*x*(c + d*x)) + (2* 
(b*c^2 - 2*a*d^2)*Sqrt[-(b*c^2) - a*d^2]*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqr 
t[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/c^3 + (4*a^(3/2)*d^3*ArcTanh[(Sqrt[ 
b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/c^3 + b^(3/2)*Log[-(Sqrt[b]*x) + Sqrt[a 
+ b*x^2]])/d^2)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(423\) vs. \(2(182)=364\).

Time = 1.20 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.32, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 2009}

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.

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

\(\Big \downarrow \) 617

\(\displaystyle \int \left (\frac {2 d^2 \left (a+b x^2\right )^{3/2}}{c^3 (c+d x)}-\frac {2 d \left (a+b x^2\right )^{3/2}}{c^3 x}+\frac {d^2 \left (a+b x^2\right )^{3/2}}{c^2 (c+d x)^2}+\frac {\left (a+b x^2\right )^{3/2}}{c^2 x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 a^{3/2} d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^3}+\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+2 b c^2\right )}{2 c^2 d^2}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a d^2+2 b c^2\right )}{c^2 d^2}+\frac {3 b \sqrt {a d^2+b c^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c d^2}+\frac {3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 c^2}-\frac {2 \left (a d^2+b c^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 d^2}-\frac {2 a d \sqrt {a+b x^2}}{c^3}-\frac {d \left (a+b x^2\right )^{3/2}}{c^2 (c+d x)}-\frac {3 b \sqrt {a+b x^2} (2 c-d x)}{2 c^2 d}-\frac {\left (a+b x^2\right )^{3/2}}{c^2 x}+\frac {3 b x \sqrt {a+b x^2}}{2 c^2}+\frac {\sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )-b c d x\right )}{c^3 d}\)

Input:

Int[(a + b*x^2)^(3/2)/(x^2*(c + d*x)^2),x]
 

Output:

(-2*a*d*Sqrt[a + b*x^2])/c^3 + (3*b*x*Sqrt[a + b*x^2])/(2*c^2) - (3*b*(2*c 
 - d*x)*Sqrt[a + b*x^2])/(2*c^2*d) + ((2*(b*c^2 + a*d^2) - b*c*d*x)*Sqrt[a 
 + b*x^2])/(c^3*d) - (a + b*x^2)^(3/2)/(c^2*x) - (d*(a + b*x^2)^(3/2))/(c^ 
2*(c + d*x)) + (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*c^2) 
+ (3*Sqrt[b]*(2*b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*c^ 
2*d^2) - (Sqrt[b]*(2*b*c^2 + 3*a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]] 
)/(c^2*d^2) + (3*b*Sqrt[b*c^2 + a*d^2]*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + 
 a*d^2]*Sqrt[a + b*x^2])])/(c*d^2) - (2*(b*c^2 + a*d^2)^(3/2)*ArcTanh[(a*d 
 - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c^3*d^2) + (2*a^(3/2)*d 
*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/c^3
 

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(466\) vs. \(2(160)=320\).

Time = 0.47 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.57

method	result	size
risch	\(-\frac {a \sqrt {b \,x^{2}+a}}{c^{2} x}+\frac {\frac {b^{\frac {3}{2}} c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{2}}+\frac {\left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}-\frac {2 \left (a^{2} d^{4}-b^{2} c^{4}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{3} c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 a^{\frac {3}{2}} d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c}}{c^{2}}\)	\(467\)
default	\(\text {Expression too large to display}\)	\(1507\)

Input:

int((b*x^2+a)^(3/2)/x^2/(d*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

-a*(b*x^2+a)^(1/2)/c^2/x+1/c^2*(b^(3/2)*c^2/d^2*ln(b^(1/2)*x+(b*x^2+a)^(1/ 
2))+1/d^4*(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b 
*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^2)/(( 
a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+ 
b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/( 
x+c/d)))-2/d^3*(a^2*d^4-b^2*c^4)/c/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+ 
b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/ 
d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d))+2*a^(3/2)*d/c*ln((2*a+2*a^(1/ 
2)*(b*x^2+a)^(1/2))/x))
 

Fricas [A] (verification not implemented)

Time = 33.84 (sec) , antiderivative size = 2238, normalized size of antiderivative = 12.30 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^2 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)/x^2/(d*x+c)^2,x, algorithm="fricas")
 

Output:

[1/2*((b*c^3*d*x^2 + b*c^4*x)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqr 
t(b)*x - a) - sqrt(b*c^2 + a*d^2)*((b*c^2*d - 2*a*d^3)*x^2 + (b*c^3 - 2*a* 
c*d^2)*x)*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x 
^2 - 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d 
*x + c^2)) + 2*(a*d^4*x^2 + a*c*d^3*x)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 
+ a)*sqrt(a) + 2*a)/x^2) - 2*(a*c^2*d^2 + (b*c^3*d + 2*a*c*d^3)*x)*sqrt(b* 
x^2 + a))/(c^3*d^3*x^2 + c^4*d^2*x), -1/2*(2*(b*c^3*d*x^2 + b*c^4*x)*sqrt( 
-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + sqrt(b*c^2 + a*d^2)*((b*c^2*d - 2 
*a*d^3)*x^2 + (b*c^3 - 2*a*c*d^2)*x)*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^ 
2 - (2*b^2*c^2 + a*b*d^2)*x^2 - 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b 
*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - 2*(a*d^4*x^2 + a*c*d^3*x)*sqrt(a)* 
log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(a*c^2*d^2 + (b*c^ 
3*d + 2*a*c*d^3)*x)*sqrt(b*x^2 + a))/(c^3*d^3*x^2 + c^4*d^2*x), 1/2*(2*sqr 
t(-b*c^2 - a*d^2)*((b*c^2*d - 2*a*d^3)*x^2 + (b*c^3 - 2*a*c*d^2)*x)*arctan 
(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + ( 
b^2*c^2 + a*b*d^2)*x^2)) + (b*c^3*d*x^2 + b*c^4*x)*sqrt(b)*log(-2*b*x^2 - 
2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(a*d^4*x^2 + a*c*d^3*x)*sqrt(a)*log(- 
(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a*c^2*d^2 + (b*c^3*d + 
 2*a*c*d^3)*x)*sqrt(b*x^2 + a))/(c^3*d^3*x^2 + c^4*d^2*x), (sqrt(-b*c^2 - 
a*d^2)*((b*c^2*d - 2*a*d^3)*x^2 + (b*c^3 - 2*a*c*d^2)*x)*arctan(sqrt(-b...
 

Sympy [F]

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

integrate((b*x**2+a)**(3/2)/x**2/(d*x+c)**2,x)
 

Output:

Integral((a + b*x**2)**(3/2)/(x**2*(c + d*x)**2), x)
 

Maxima [F]

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

integrate((b*x^2+a)^(3/2)/x^2/(d*x+c)^2,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x + c)^2*x^2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^2 (c+d x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a)^(3/2)/x^2/(d*x+c)^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((a + b*x^2)^(3/2)/(x^2*(c + d*x)^2),x)
 

Output:

int((a + b*x^2)^(3/2)/(x^2*(c + d*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 572, normalized size of antiderivative = 3.14 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^2 (c+d x)^2} \, dx=\frac {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a c \,d^{2} x +4 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a \,d^{3} x^{2}-2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) b \,c^{3} x -2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) b \,c^{2} d \,x^{2}-4 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a c \,d^{2} x -4 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a \,d^{3} x^{2}+2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) b \,c^{3} x +2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) b \,c^{2} d \,x^{2}-2 \sqrt {b \,x^{2}+a}\, a \,c^{2} d^{2}-4 \sqrt {b \,x^{2}+a}\, a c \,d^{3} x -2 \sqrt {b \,x^{2}+a}\, b \,c^{3} d x -2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {a}\right ) a c \,d^{3} x -2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {a}\right ) a \,d^{4} x^{2}+2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {a}\right ) a c \,d^{3} x +2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {a}\right ) a \,d^{4} x^{2}-\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {b}\, x \right ) b \,c^{4} x -\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {b}\, x \right ) b \,c^{3} d \,x^{2}+\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {b}\, x \right ) b \,c^{4} x +\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {b}\, x \right ) b \,c^{3} d \,x^{2}}{2 c^{3} d^{2} x \left (d x +c \right )} \] Input:

int((b*x^2+a)^(3/2)/x^2/(d*x+c)^2,x)
 

Output:

(4*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
+ b*c*x)*a*c*d**2*x + 4*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a* 
d**2 + b*c**2) - a*d + b*c*x)*a*d**3*x**2 - 2*sqrt(a*d**2 + b*c**2)*log(sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b*c**3*x - 2*sqrt(a*d* 
*2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b*c 
**2*d*x**2 - 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*c*d**2*x - 4*sqrt(a*d* 
*2 + b*c**2)*log(c + d*x)*a*d**3*x**2 + 2*sqrt(a*d**2 + b*c**2)*log(c + d* 
x)*b*c**3*x + 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b*c**2*d*x**2 - 2*sqrt( 
a + b*x**2)*a*c**2*d**2 - 4*sqrt(a + b*x**2)*a*c*d**3*x - 2*sqrt(a + b*x** 
2)*b*c**3*d*x - 2*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a*c*d**3*x - 2*s 
qrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a*d**4*x**2 + 2*sqrt(a)*log(sqrt(a 
+ b*x**2) + sqrt(a))*a*c*d**3*x + 2*sqrt(a)*log(sqrt(a + b*x**2) + sqrt(a) 
)*a*d**4*x**2 - sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b*c**4*x - sqrt( 
b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b*c**3*d*x**2 + sqrt(b)*log(sqrt(a + 
b*x**2) + sqrt(b)*x)*b*c**4*x + sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)* 
b*c**3*d*x**2)/(2*c**3*d**2*x*(c + d*x))