Integrand size = 23, antiderivative size = 286 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x^4}+\frac {223 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{24 \sqrt {1-\frac {1}{a x}} x^3}-\frac {1115 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{96 \sqrt {1-\frac {1}{a x}} x^2}+\frac {1115 a^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{64 \sqrt {1-\frac {1}{a x}} x}-\frac {1115 a^4 \sqrt {\frac {1}{a x}} \sqrt {c-a c x} \text {arcsinh}\left (\sqrt {\frac {1}{a x}}\right )}{64 \sqrt {1-\frac {1}{a x}}} \] Output:
-8*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)/x^4-1/4*(1+1/a/x)^(1/2 )*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/x^4+223/24*a*(1+1/a/x)^(1/2)*(-a*c*x+c) ^(1/2)/(1-1/a/x)^(1/2)/x^3-1115/96*a^2*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/(1 -1/a/x)^(1/2)/x^2+1115/64*a^3*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/(1-1/a/x)^( 1/2)/x-1115/64*a^4*(1/a/x)^(1/2)*(-a*c*x+c)^(1/2)*arcsinh((1/a/x)^(1/2))/( 1-1/a/x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.37 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {\sqrt {c-a c x} \left (48-200 a x+446 a^2 x^2-1115 a^3 x^3-3345 a^4 x^4+\frac {3345 a^{9/2} \sqrt {1+\frac {1}{a x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{9/2}}\right )}{192 a \sqrt {1-\frac {1}{a^2 x^2}} x^5} \] Input:
Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^5),x]
Output:
-1/192*(Sqrt[c - a*c*x]*(48 - 200*a*x + 446*a^2*x^2 - 1115*a^3*x^3 - 3345* a^4*x^4 + (3345*a^(9/2)*Sqrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/( x^(-1))^(9/2)))/(a*Sqrt[1 - 1/(a^2*x^2)]*x^5)
Time = 0.62 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.71, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6730, 27, 100, 27, 90, 60, 60, 60, 63, 222}
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.
\(\displaystyle \int \frac {\sqrt {c-a c x} e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2 \left (\frac {1}{x}\right )^{5/2}}{a^2 \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2 \left (\frac {1}{x}\right )^{5/2}}{\left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{7/2}}{\sqrt {\frac {1}{a x}+1}}-2 a^2 \int \frac {\left (27 a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{5/2}}{2 a \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{7/2}}{\sqrt {\frac {1}{a x}+1}}-a \int \frac {\left (27 a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{5/2}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{7/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {223}{8} a \int \frac {\left (\frac {1}{x}\right )^{5/2}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {1}{4} a \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{7/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {223}{8} a \left (\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}-\frac {5}{6} a \int \frac {\left (\frac {1}{x}\right )^{3/2}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )-\frac {1}{4} a \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{7/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {223}{8} a \left (\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}-\frac {5}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \int \frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )\right )-\frac {1}{4} a \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{7/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {223}{8} a \left (\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}-\frac {5}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \left (a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}-\frac {1}{2} a \int \frac {1}{\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )\right )\right )-\frac {1}{4} a \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 63 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{7/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {223}{8} a \left (\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}-\frac {5}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \left (a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}-a \int \frac {1}{\sqrt {1+\frac {1}{x^2 a}}}d\sqrt {\frac {1}{x}}\right )\right )\right )-\frac {1}{4} a \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{7/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {223}{8} a \left (\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}-\frac {5}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \left (a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}-a^{3/2} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )\right )\right )-\frac {1}{4} a \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}\right )\right ) \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
Input:
Int[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^5),x]
Output:
-((Sqrt[x^(-1)]*Sqrt[c - a*c*x]*((8*a^2*(x^(-1))^(7/2))/Sqrt[1 + 1/(a*x)] - a*(-1/4*(a*Sqrt[1 + 1/(a*x)]*(x^(-1))^(7/2)) + (223*a*((a*Sqrt[1 + 1/(a* x)]*(x^(-1))^(5/2))/3 - (5*a*((a*Sqrt[1 + 1/(a*x)]*(x^(-1))^(3/2))/2 - (3* a*(a*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)] - a^(3/2)*ArcSinh[Sqrt[x^(-1)]/Sqrt[a] ]))/4))/6))/8)))/(a^2*Sqrt[1 - 1/(a*x)]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p ) Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d ^2, 0] && !IntegerQ[p]
Time = 0.25 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.44
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (3345 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{4} x^{4} \sqrt {-c \left (a x +1\right )}+3345 a^{4} x^{4} \sqrt {c}+1115 a^{3} x^{3} \sqrt {c}-446 \sqrt {c}\, a^{2} x^{2}+200 \sqrt {c}\, a x -48 \sqrt {c}\right )}{192 \left (a x -1\right )^{2} \sqrt {c}\, x^{4}}\) | \(125\) |
risch | \(-\frac {\left (1809 a^{4} x^{4}+1115 a^{3} x^{3}-446 a^{2} x^{2}+200 a x -48\right ) c \sqrt {\frac {a x -1}{a x +1}}}{192 x^{4} \sqrt {-c \left (a x -1\right )}}-\frac {\left (\frac {8 a^{4}}{\sqrt {-a c x -c}}+\frac {1115 a^{4} \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{64 \sqrt {c}}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}{\sqrt {-c \left (a x -1\right )}}\) | \(142\) |
Input:
int((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x,method=_RETURNVERBOSE)
Output:
1/192*((a*x-1)/(a*x+1))^(3/2)/(a*x-1)^2*(a*x+1)*(-c*(a*x-1))^(1/2)*(3345*a rctan((-c*(a*x+1))^(1/2)/c^(1/2))*a^4*x^4*(-c*(a*x+1))^(1/2)+3345*a^4*x^4* c^(1/2)+1115*a^3*x^3*c^(1/2)-446*c^(1/2)*a^2*x^2+200*c^(1/2)*a*x-48*c^(1/2 ))/c^(1/2)/x^4
Time = 0.09 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\left [\frac {3345 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{384 \, {\left (a x^{5} - x^{4}\right )}}, -\frac {3345 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{192 \, {\left (a x^{5} - x^{4}\right )}}\right ] \] Input:
integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="frica s")
Output:
[1/384*(3345*(a^5*x^5 - a^4*x^4)*sqrt(-c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt (-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 2*c)/(a*x^2 - x)) + 2*(3345*a^4*x^4 + 1115*a^3*x^3 - 446*a^2*x^2 + 200*a*x - 48)*sqrt(-a *c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^5 - x^4), -1/192*(3345*(a^5*x^5 - a^4*x^4)*sqrt(c)*arctan(sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)*sqrt((a*x - 1 )/(a*x + 1))/(a*c*x - c)) - (3345*a^4*x^4 + 1115*a^3*x^3 - 446*a^2*x^2 + 2 00*a*x - 48)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^5 - x^4)]
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\text {Timed out} \] Input:
integrate((-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**5,x)
Output:
Timed out
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int { \frac {\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \] Input:
integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="maxim a")
Output:
integrate(sqrt(-a*c*x + c)*((a*x - 1)/(a*x + 1))^(3/2)/x^5, x)
Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,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
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^5} \,d x \] Input:
int(((c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^5,x)
Output:
int(((c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^5, x)
Time = 0.16 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.37 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c}\, i \left (-3345 \sqrt {a x +1}\, \mathrm {log}\left (\frac {2 \sqrt {a x +1}-2}{\sqrt {2}}\right ) a^{4} x^{4}+3345 \sqrt {a x +1}\, \mathrm {log}\left (\frac {2 \sqrt {a x +1}+2}{\sqrt {2}}\right ) a^{4} x^{4}-6690 a^{4} x^{4}-2230 a^{3} x^{3}+892 a^{2} x^{2}-400 a x +96\right )}{384 \sqrt {a x +1}\, x^{4}} \] Input:
int((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x)
Output:
(sqrt(c)*i*( - 3345*sqrt(a*x + 1)*log((2*sqrt(a*x + 1) - 2)/sqrt(2))*a**4* x**4 + 3345*sqrt(a*x + 1)*log((2*sqrt(a*x + 1) + 2)/sqrt(2))*a**4*x**4 - 6 690*a**4*x**4 - 2230*a**3*x**3 + 892*a**2*x**2 - 400*a*x + 96))/(384*sqrt( a*x + 1)*x**4)