\(\int \frac {1}{x (a+c x^2)^{3/2} (d+e x+f x^2)} \, dx\) [42]

Optimal result

Integrand size = 27, antiderivative size = 526 \[ \int \frac {1}{x \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\frac {1}{a d \sqrt {a+c x^2}}-\frac {a \left (a f^2+c \left (e^2-d f\right )\right )+c^2 d e x}{a d \left (a c e^2+(c d-a f)^2\right ) \sqrt {a+c x^2}}+\frac {f \left (2 e \left (a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}-\frac {f \left (2 e \left (a f^2+c \left (e^2-2 d f\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d} \] Output:

1/a/d/(c*x^2+a)^(1/2)-(a*(a*f^2+c*(-d*f+e^2))+c^2*d*e*x)/a/d/(a*c*e^2+(-a* 
f+c*d)^2)/(c*x^2+a)^(1/2)+1/2*f*(2*e*(a*f^2+c*(-2*d*f+e^2))-(e-(-4*d*f+e^2 
)^(1/2))*(a*f^2+c*(-d*f+e^2)))*arctanh(1/2*(2*a*f-c*(e-(-4*d*f+e^2)^(1/2)) 
*x)*2^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)/(c*x^2+a)^( 
1/2))*2^(1/2)/d/(-4*d*f+e^2)^(1/2)/(a*c*e^2+(-a*f+c*d)^2)/(2*a*f^2+c*(e^2- 
2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)-1/2*f*(2*e*(a*f^2+c*(-2*d*f+e^2))-(e+(- 
4*d*f+e^2)^(1/2))*(a*f^2+c*(-d*f+e^2)))*arctanh(1/2*(2*a*f-c*(e+(-4*d*f+e^ 
2)^(1/2))*x)*2^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)/(c 
*x^2+a)^(1/2))*2^(1/2)/d/(-4*d*f+e^2)^(1/2)/(a*c*e^2+(-a*f+c*d)^2)/(2*a*f^ 
2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)-arctanh((c*x^2+a)^(1/2)/a^(1/2 
))/a^(3/2)/d
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.25 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=-\frac {c (-c d+a f+c e x)}{a \left (c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )\right ) \sqrt {a+c x^2}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {\text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {a c e^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-a c d f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+a^2 f^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 c^{3/2} e^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 c^{3/2} d e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a \sqrt {c} e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c e^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+c d f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a f^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{d \left (c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )\right )} \] Input:

Integrate[1/(x*(a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]
 

Output:

-((c*(-(c*d) + a*f + c*e*x))/(a*(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))*Sq 
rt[a + c*x^2])) + (2*ArcTanh[(Sqrt[c]*x - Sqrt[a + c*x^2])/Sqrt[a]])/(a^(3 
/2)*d) + RootSum[a^2*f + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 - 2*a*f*#1^2 - 2*Sq 
rt[c]*e*#1^3 + f*#1^4 & , (a*c*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - 
#1] - a*c*d*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1] + a^2*f^3*Log[-(S 
qrt[c]*x) + Sqrt[a + c*x^2] - #1] + 2*c^(3/2)*e^3*Log[-(Sqrt[c]*x) + Sqrt[ 
a + c*x^2] - #1]*#1 - 4*c^(3/2)*d*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - 
 #1]*#1 + 2*a*Sqrt[c]*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 - 
c*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 + c*d*f^2*Log[-(Sqrt 
[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 - a*f^3*Log[-(Sqrt[c]*x) + Sqrt[a + c* 
x^2] - #1]*#1^2)/(a*Sqrt[c]*e + 4*c*d*#1 - 2*a*f*#1 - 3*Sqrt[c]*e*#1^2 + 2 
*f*#1^3) & ]/(d*(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)))
 

Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7279, 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.

\(\displaystyle \int \frac {1}{x \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {-e-f x}{d \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+c x^2\right )^{3/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {f \left (2 e \left (a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {f \left (2 e \left (a f^2+c \left (e^2-2 d f\right )\right )-\left (\sqrt {e^2-4 d f}+e\right ) \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {a \left (a f^2+c \left (e^2-d f\right )\right )+c^2 d e x}{a d \sqrt {a+c x^2} \left ((c d-a f)^2+a c e^2\right )}+\frac {1}{a d \sqrt {a+c x^2}}\)

Input:

Int[1/(x*(a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]
 

Output:

1/(a*d*Sqrt[a + c*x^2]) - (a*(a*f^2 + c*(e^2 - d*f)) + c^2*d*e*x)/(a*d*(a* 
c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2]) + (f*(2*e*(a*f^2 + c*(e^2 - 2*d*f) 
) - (e - Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - d*f)))*ArcTanh[(2*a*f - c*(e 
 - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e 
^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + ( 
c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - (f* 
(2*e*(a*f^2 + c*(e^2 - 2*d*f)) - (e + Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - 
 d*f)))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^ 
2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d*S 
qrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + 
 e*Sqrt[e^2 - 4*d*f])]) - ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]/(a^(3/2)*d)
 

Defintions of rubi rules used

rule 2009

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

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1564\) vs. \(2(479)=958\).

Time = 2.16 (sec) , antiderivative size = 1565, normalized size of antiderivative = 2.98

method	result	size
default	\(\text {Expression too large to display}\)	\(1565\)

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{x \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {1}{x \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{x \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e x + d\right )} x} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionNot implemented, e.g. for multivariate mod/approx polynomi 
alsError:
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {1}{x\,{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\frac {2 \sqrt {c \,x^{2}+a}\, a +\sqrt {a}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}-\sqrt {a}\right ) a +\sqrt {a}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}-\sqrt {a}\right ) c \,x^{2}-\sqrt {a}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}+\sqrt {a}\right ) a -\sqrt {a}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}+\sqrt {a}\right ) c \,x^{2}-2 \left (\int \frac {\sqrt {c \,x^{2}+a}}{c^{2} f \,x^{6}+c^{2} e \,x^{5}+2 a c f \,x^{4}+c^{2} d \,x^{4}+2 a c e \,x^{3}+a^{2} f \,x^{2}+2 a c d \,x^{2}+a^{2} e x +a^{2} d}d x \right ) a^{3} e -2 \left (\int \frac {\sqrt {c \,x^{2}+a}}{c^{2} f \,x^{6}+c^{2} e \,x^{5}+2 a c f \,x^{4}+c^{2} d \,x^{4}+2 a c e \,x^{3}+a^{2} f \,x^{2}+2 a c d \,x^{2}+a^{2} e x +a^{2} d}d x \right ) a^{2} c e \,x^{2}-2 \left (\int \frac {\sqrt {c \,x^{2}+a}\, x}{c^{2} f \,x^{6}+c^{2} e \,x^{5}+2 a c f \,x^{4}+c^{2} d \,x^{4}+2 a c e \,x^{3}+a^{2} f \,x^{2}+2 a c d \,x^{2}+a^{2} e x +a^{2} d}d x \right ) a^{3} f -2 \left (\int \frac {\sqrt {c \,x^{2}+a}\, x}{c^{2} f \,x^{6}+c^{2} e \,x^{5}+2 a c f \,x^{4}+c^{2} d \,x^{4}+2 a c e \,x^{3}+a^{2} f \,x^{2}+2 a c d \,x^{2}+a^{2} e x +a^{2} d}d x \right ) a^{2} c f \,x^{2}}{2 a^{2} d \left (c \,x^{2}+a \right )} \] Input:

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

Output:

(2*sqrt(a + c*x**2)*a + sqrt(a)*log(sqrt(a + c*x**2) - sqrt(a))*a + sqrt(a 
)*log(sqrt(a + c*x**2) - sqrt(a))*c*x**2 - sqrt(a)*log(sqrt(a + c*x**2) + 
sqrt(a))*a - sqrt(a)*log(sqrt(a + c*x**2) + sqrt(a))*c*x**2 - 2*int(sqrt(a 
 + c*x**2)/(a**2*d + a**2*e*x + a**2*f*x**2 + 2*a*c*d*x**2 + 2*a*c*e*x**3 
+ 2*a*c*f*x**4 + c**2*d*x**4 + c**2*e*x**5 + c**2*f*x**6),x)*a**3*e - 2*in 
t(sqrt(a + c*x**2)/(a**2*d + a**2*e*x + a**2*f*x**2 + 2*a*c*d*x**2 + 2*a*c 
*e*x**3 + 2*a*c*f*x**4 + c**2*d*x**4 + c**2*e*x**5 + c**2*f*x**6),x)*a**2* 
c*e*x**2 - 2*int((sqrt(a + c*x**2)*x)/(a**2*d + a**2*e*x + a**2*f*x**2 + 2 
*a*c*d*x**2 + 2*a*c*e*x**3 + 2*a*c*f*x**4 + c**2*d*x**4 + c**2*e*x**5 + c* 
*2*f*x**6),x)*a**3*f - 2*int((sqrt(a + c*x**2)*x)/(a**2*d + a**2*e*x + a** 
2*f*x**2 + 2*a*c*d*x**2 + 2*a*c*e*x**3 + 2*a*c*f*x**4 + c**2*d*x**4 + c**2 
*e*x**5 + c**2*f*x**6),x)*a**2*c*f*x**2)/(2*a**2*d*(a + c*x**2))