\(\int e^{\coth ^{-1}(a x)} (c-\frac {c}{a x})^{7/2} \, dx\) [454]

Optimal result

Integrand size = 22, antiderivative size = 196 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {49 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{15 a \sqrt {c-\frac {c}{a x}}}+\frac {31 c^3 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}}{15 a}+\frac {7 c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 a}+c \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2} x-\frac {5 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \] Output:

49/15*c^4*(1-1/a^2/x^2)^(1/2)/a/(c-c/a/x)^(1/2)+31/15*c^3*(1-1/a^2/x^2)^(1 
/2)*(c-c/a/x)^(1/2)/a+7/5*c^2*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(3/2)/a+c*(1-1 
/a^2/x^2)^(1/2)*(c-c/a/x)^(5/2)*x-5*c^(7/2)*arctanh(c^(1/2)*(1-1/a^2/x^2)^ 
(1/2)/(c-c/a/x)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.52 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} \left (6-28 a x+56 a^2 x^2+15 a^3 x^3\right )-75 a^2 x^2 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{15 a^3 \sqrt {1-\frac {1}{a x}} x^2} \] Input:

Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^(7/2),x]
 

Output:

(c^3*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(6 - 28*a*x + 56*a^2*x^2 + 15*a^ 
3*x^3) - 75*a^2*x^2*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(15*a^3*Sqrt[1 - 1/(a*x)] 
*x^2)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.77, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6731, 585, 27, 108, 27, 170, 27, 164, 73, 221}

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.

\(\displaystyle \int \left (c-\frac {c}{a x}\right )^{7/2} e^{\coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6731

\(\displaystyle -c \int \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2} x^2d\frac {1}{x}\)

\(\Big \downarrow \) 585

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 108

\(\displaystyle -\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (\int -\frac {\left (a-\frac {1}{x}\right )^2 \left (5 a+\frac {7}{x}\right ) x}{2 a \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-x \left (a-\frac {1}{x}\right )^3 \sqrt {\frac {1}{a x}+1}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^3-\frac {\int \frac {\left (a-\frac {1}{x}\right )^2 \left (5 a+\frac {7}{x}\right ) x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 a}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 170

\(\displaystyle -\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^3-\frac {\frac {2}{5} a \int \frac {\left (a-\frac {1}{x}\right ) \left (25 a+\frac {31}{x}\right ) x}{2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {14}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2}{2 a}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^3-\frac {\frac {1}{5} a \int \frac {\left (a-\frac {1}{x}\right ) \left (25 a+\frac {31}{x}\right ) x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {14}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2}{2 a}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 164

\(\displaystyle -\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^3-\frac {\frac {1}{5} a \left (25 a^2 \int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {2}{3} a \left (80 a-\frac {31}{x}\right ) \sqrt {\frac {1}{a x}+1}\right )+\frac {14}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2}{2 a}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^3-\frac {\frac {1}{5} a \left (50 a^3 \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}+\frac {2}{3} a \left (80 a-\frac {31}{x}\right ) \sqrt {\frac {1}{a x}+1}\right )+\frac {14}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2}{2 a}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c^3 \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^3-\frac {\frac {1}{5} a \left (\frac {2}{3} a \left (80 a-\frac {31}{x}\right ) \sqrt {\frac {1}{a x}+1}-50 a^2 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )\right )+\frac {14}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2}{2 a}\right ) \sqrt {c-\frac {c}{a x}}}{a^3 \sqrt {1-\frac {1}{a x}}}\)

Input:

Int[E^ArcCoth[a*x]*(c - c/(a*x))^(7/2),x]
 

Output:

-((c^3*Sqrt[c - c/(a*x)]*(-((a - x^(-1))^3*Sqrt[1 + 1/(a*x)]*x) - ((14*a*( 
a - x^(-1))^2*Sqrt[1 + 1/(a*x)])/5 + (a*((2*a*(80*a - 31/x)*Sqrt[1 + 1/(a* 
x)])/3 - 50*a^2*ArcTanh[Sqrt[1 + 1/(a*x)]]))/5)/(2*a)))/(a^3*Sqrt[1 - 1/(a 
*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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 108

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

rule 164

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

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 585

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) 
, x_Symbol] :> Simp[a^p*c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^F 
racPart[n])   Int[(e*x)^m*(1 - d*(x/c))^p*(1 + d*(x/c))^(n + p), x], x] /; 
FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[a, 0]
 

rule 6731

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S 
imp[-c^n   Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ 
x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] 
&& IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.76

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (-30 a^{\frac {7}{2}} x^{3} \sqrt {x \left (a x +1\right )}+75 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} x^{3}-112 a^{\frac {5}{2}} x^{2} \sqrt {x \left (a x +1\right )}+56 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}-12 \sqrt {x \left (a x +1\right )}\, \sqrt {a}\right )}{30 \sqrt {\frac {a x -1}{a x +1}}\, x^{2} a^{\frac {7}{2}} \sqrt {x \left (a x +1\right )}}\)	\(149\)
risch	\(\frac {\left (15 a^{4} x^{4}+71 a^{3} x^{3}+28 a^{2} x^{2}-22 a x +6\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {5 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(176\)

Input:

int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/30/((a*x-1)/(a*x+1))^(1/2)*(c*(a*x-1)/a/x)^(1/2)/x^2*c^3/a^(7/2)*(-30*a 
^(7/2)*x^3*(x*(a*x+1))^(1/2)+75*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a*x+ 
1)/a^(1/2))*a^3*x^3-112*a^(5/2)*x^2*(x*(a*x+1))^(1/2)+56*a^(3/2)*x*(x*(a*x 
+1))^(1/2)-12*(x*(a*x+1))^(1/2)*a^(1/2))/(x*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.12 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\left [\frac {75 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (15 \, a^{4} c^{3} x^{4} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{60 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, \frac {75 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (15 \, a^{4} c^{3} x^{4} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{30 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}\right ] \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(7/2),x, algorithm="fricas")
 

Output:

[1/60*(75*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x 
- 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt(( 
a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(15*a^4*c^3*x^4 + 71*a^3*c^3*x^3 + 2 
8*a^2*c^3*x^2 - 22*a*c^3*x + 6*c^3)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x 
- c)/(a*x)))/(a^4*x^3 - a^3*x^2), 1/30*(75*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqr 
t(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a* 
c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(15*a^4*c^3*x^4 + 71*a^3*c^ 
3*x^3 + 28*a^2*c^3*x^2 - 22*a*c^3*x + 6*c^3)*sqrt((a*x - 1)/(a*x + 1))*sqr 
t((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Timed out} \] Input:

integrate(1/((a*x-1)/(a*x+1))**(1/2)*(c-c/a/x)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(7/2),x, algorithm="maxima")
 

Output:

integrate((c - c/(a*x))^(7/2)/sqrt((a*x - 1)/(a*x + 1)), x)
 

Giac [F(-2)]

Exception generated. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(7/2),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. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:

int((c - c/(a*x))^(7/2)/((a*x - 1)/(a*x + 1))^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int((c - c/(a*x))^(7/2)/((a*x - 1)/(a*x + 1))^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} \left (60 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} x^{3}+224 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-112 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +24 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}-300 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a^{3} x^{3}-197 a^{3} x^{3}\right )}{60 a^{4} x^{3}} \] Input:

int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(7/2),x)
 

Output:

(sqrt(c)*c**3*(60*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**3*x**3 + 224*sqrt(x)*sq 
rt(a)*sqrt(a*x + 1)*a**2*x**2 - 112*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x + 24 
*sqrt(x)*sqrt(a)*sqrt(a*x + 1) - 300*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))* 
a**3*x**3 - 197*a**3*x**3))/(60*a**4*x**3)