\(\int \frac {(d-c^2 d x^2)^3 (a+b \text {arccosh}(c x))}{x^2} \, dx\) [25]

Optimal result

Integrand size = 25, antiderivative size = 190 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {61}{25} b c d^3 \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{25} b c^3 d^3 x^2 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{25} b c^5 d^3 x^4 \sqrt {-1+c x} \sqrt {1+c x}-\frac {d^3 (a+b \text {arccosh}(c x))}{x}-3 c^2 d^3 x (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+b c d^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \] Output:

61/25*b*c*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)-7/25*b*c^3*d^3*x^2*(c*x-1)^(1/2) 
*(c*x+1)^(1/2)+1/25*b*c^5*d^3*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)-d^3*(a+b*arc 
cosh(c*x))/x-3*c^2*d^3*x*(a+b*arccosh(c*x))+c^4*d^3*x^3*(a+b*arccosh(c*x)) 
-1/5*c^6*d^3*x^5*(a+b*arccosh(c*x))+b*c*d^3*arctan((c*x-1)^(1/2)*(c*x+1)^( 
1/2))
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{25} d^3 \left (-\frac {25 a}{x}-75 a c^2 x+25 a c^4 x^3-5 a c^6 x^5+b c \sqrt {-1+c x} \sqrt {1+c x} \left (61-7 c^2 x^2+c^4 x^4\right )-\frac {5 b \left (5+15 c^2 x^2-5 c^4 x^4+c^6 x^6\right ) \text {arccosh}(c x)}{x}-25 b c \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right ) \] Input:

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

Output:

(d^3*((-25*a)/x - 75*a*c^2*x + 25*a*c^4*x^3 - 5*a*c^6*x^5 + b*c*Sqrt[-1 + 
c*x]*Sqrt[1 + c*x]*(61 - 7*c^2*x^2 + c^4*x^4) - (5*b*(5 + 15*c^2*x^2 - 5*c 
^4*x^4 + c^6*x^6)*ArcCosh[c*x])/x - 25*b*c*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 
 + c*x])]))/25
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6336, 27, 2113, 2331, 2123, 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.

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

\(\Big \downarrow \) 6336

\(\displaystyle -b c \int -\frac {d^3 \left (c^6 x^6-5 c^4 x^4+15 c^2 x^2+5\right )}{5 x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} b c d^3 \int \frac {c^6 x^6-5 c^4 x^4+15 c^2 x^2+5}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\)

\(\Big \downarrow \) 2113

\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {c^6 x^6-5 c^4 x^4+15 c^2 x^2+5}{x \sqrt {c^2 x^2-1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\)

\(\Big \downarrow \) 2331

\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {c^6 x^6-5 c^4 x^4+15 c^2 x^2+5}{x^2 \sqrt {c^2 x^2-1}}dx^2}{10 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\)

\(\Big \downarrow \) 2123

\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \int \left (\left (c^2 x^2-1\right )^{3/2} c^2-3 \sqrt {c^2 x^2-1} c^2+\frac {11 c^2}{\sqrt {c^2 x^2-1}}+\frac {5}{x^2 \sqrt {c^2 x^2-1}}\right )dx^2}{10 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{5} c^6 d^3 x^5 (a+b \text {arccosh}(c x))+c^4 d^3 x^3 (a+b \text {arccosh}(c x))-3 c^2 d^3 x (a+b \text {arccosh}(c x))-\frac {d^3 (a+b \text {arccosh}(c x))}{x}+\frac {b c d^3 \sqrt {c^2 x^2-1} \left (10 \arctan \left (\sqrt {c^2 x^2-1}\right )+\frac {2}{5} \left (c^2 x^2-1\right )^{5/2}-2 \left (c^2 x^2-1\right )^{3/2}+22 \sqrt {c^2 x^2-1}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-((d^3*(a + b*ArcCosh[c*x]))/x) - 3*c^2*d^3*x*(a + b*ArcCosh[c*x]) + c^4*d 
^3*x^3*(a + b*ArcCosh[c*x]) - (c^6*d^3*x^5*(a + b*ArcCosh[c*x]))/5 + (b*c* 
d^3*Sqrt[-1 + c^2*x^2]*(22*Sqrt[-1 + c^2*x^2] - 2*(-1 + c^2*x^2)^(3/2) + ( 
2*(-1 + c^2*x^2)^(5/2))/5 + 10*ArcTan[Sqrt[-1 + c^2*x^2]]))/(10*Sqrt[-1 + 
c*x]*Sqrt[1 + c*x])
 

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 2009

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

rule 2113

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. 
)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ 
m]/(a*c + b*d*x^2)^FracPart[m])   Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a 
*d, 0] && EqQ[m, n] &&  !IntegerQ[m]
 

rule 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 6336

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcCosh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && E 
qQ[c^2*d + e, 0] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.94

method	result	size
parts	\(-d^{3} a \left (\frac {c^{6} x^{5}}{5}-c^{4} x^{3}+3 c^{2} x +\frac {1}{x}\right )-d^{3} b c \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+7 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+25 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-61 \sqrt {c^{2} x^{2}-1}\right )}{25 \sqrt {c^{2} x^{2}-1}}\right )\)	\(178\)
derivativedivides	\(c \left (-d^{3} a \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+7 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+25 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-61 \sqrt {c^{2} x^{2}-1}\right )}{25 \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(181\)
default	\(c \left (-d^{3} a \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+7 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+25 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-61 \sqrt {c^{2} x^{2}-1}\right )}{25 \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(181\)

Input:

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

Output:

-d^3*a*(1/5*c^6*x^5-c^4*x^3+3*c^2*x+1/x)-d^3*b*c*(1/5*arccosh(c*x)*c^5*x^5 
-c^3*x^3*arccosh(c*x)+3*c*x*arccosh(c*x)+arccosh(c*x)/c/x+1/25*(c*x-1)^(1/ 
2)*(c*x+1)^(1/2)*(-c^4*x^4*(c^2*x^2-1)^(1/2)+7*c^2*x^2*(c^2*x^2-1)^(1/2)+2 
5*arctan(1/(c^2*x^2-1)^(1/2))-61*(c^2*x^2-1)^(1/2))/(c^2*x^2-1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.31 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {5 \, a c^{6} d^{3} x^{6} - 25 \, a c^{4} d^{3} x^{4} + 75 \, a c^{2} d^{3} x^{2} - 50 \, b c d^{3} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 5 \, {\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 25 \, a d^{3} + 5 \, {\left (b c^{6} d^{3} x^{6} - 5 \, b c^{4} d^{3} x^{4} + 15 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x + 5 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{5} d^{3} x^{5} - 7 \, b c^{3} d^{3} x^{3} + 61 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{25 \, x} \] Input:

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

Output:

-1/25*(5*a*c^6*d^3*x^6 - 25*a*c^4*d^3*x^4 + 75*a*c^2*d^3*x^2 - 50*b*c*d^3* 
x*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 5*(b*c^6 - 5*b*c^4 + 15*b*c^2 + 5*b)* 
d^3*x*log(-c*x + sqrt(c^2*x^2 - 1)) + 25*a*d^3 + 5*(b*c^6*d^3*x^6 - 5*b*c^ 
4*d^3*x^4 + 15*b*c^2*d^3*x^2 - (b*c^6 - 5*b*c^4 + 15*b*c^2 + 5*b)*d^3*x + 
5*b*d^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^5*d^3*x^5 - 7*b*c^3*d^3*x^3 + 
 61*b*c*d^3*x)*sqrt(c^2*x^2 - 1))/x
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=- d^{3} \left (\int 3 a c^{2}\, dx + \int \left (- \frac {a}{x^{2}}\right )\, dx + \int \left (- 3 a c^{4} x^{2}\right )\, dx + \int a c^{6} x^{4}\, dx + \int 3 b c^{2} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\right )\, dx + \int \left (- 3 b c^{4} x^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{4} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

-d**3*(Integral(3*a*c**2, x) + Integral(-a/x**2, x) + Integral(-3*a*c**4*x 
**2, x) + Integral(a*c**6*x**4, x) + Integral(3*b*c**2*acosh(c*x), x) + In 
tegral(-b*acosh(c*x)/x**2, x) + Integral(-3*b*c**4*x**2*acosh(c*x), x) + I 
ntegral(b*c**6*x**4*acosh(c*x), x))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {1}{5} \, a c^{6} d^{3} x^{5} - \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{6} d^{3} + a c^{4} d^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{4} d^{3} - 3 \, a c^{2} d^{3} x - 3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c d^{3} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{3} - \frac {a d^{3}}{x} \] Input:

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

Output:

-1/5*a*c^6*d^3*x^5 - 1/75*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/ 
c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*c^6*d^3 
+ a*c^4*d^3*x^3 + 1/3*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 
 2*sqrt(c^2*x^2 - 1)/c^4))*b*c^4*d^3 - 3*a*c^2*d^3*x - 3*(c*x*arccosh(c*x) 
 - sqrt(c^2*x^2 - 1))*b*c*d^3 - (c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x)* 
b*d^3 - a*d^3/x
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {d^{3} \left (-5 \mathit {acosh} \left (c x \right ) b \,c^{6} x^{6}+25 \mathit {acosh} \left (c x \right ) b \,c^{4} x^{4}-75 \mathit {acosh} \left (c x \right ) b \,c^{2} x^{2}-25 \mathit {acosh} \left (c x \right ) b -50 \mathit {atan} \left (\sqrt {c^{2} x^{2}-1}+c x \right ) b c x +\sqrt {c^{2} x^{2}-1}\, b \,c^{5} x^{5}-7 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} x^{3}-14 \sqrt {c^{2} x^{2}-1}\, b c x +75 \sqrt {c x +1}\, \sqrt {c x -1}\, b c x -5 a \,c^{6} x^{6}+25 a \,c^{4} x^{4}-75 a \,c^{2} x^{2}-25 a \right )}{25 x} \] Input:

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

Output:

(d**3*( - 5*acosh(c*x)*b*c**6*x**6 + 25*acosh(c*x)*b*c**4*x**4 - 75*acosh( 
c*x)*b*c**2*x**2 - 25*acosh(c*x)*b - 50*atan(sqrt(c**2*x**2 - 1) + c*x)*b* 
c*x + sqrt(c**2*x**2 - 1)*b*c**5*x**5 - 7*sqrt(c**2*x**2 - 1)*b*c**3*x**3 
- 14*sqrt(c**2*x**2 - 1)*b*c*x + 75*sqrt(c*x + 1)*sqrt(c*x - 1)*b*c*x - 5* 
a*c**6*x**6 + 25*a*c**4*x**4 - 75*a*c**2*x**2 - 25*a))/(25*x)