\(\int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx\) [173]

Optimal result

Integrand size = 16, antiderivative size = 133 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {47 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {58 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac {1}{x}\right )^3} \] Output:

1/9*a^5*(1-1/a^2/x^2)^(3/2)/c^5/(a-1/x)^6-8/21*a^4*(1-1/a^2/x^2)^(3/2)/c^5 
/(a-1/x)^5+47/105*a^3*(1-1/a^2/x^2)^(3/2)/c^5/(a-1/x)^4-58/315*a^2*(1-1/a^ 
2/x^2)^(3/2)/c^5/(a-1/x)^3
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.44 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-58-25 a x+21 a^2 x^2-10 a^3 x^3+2 a^4 x^4\right )}{315 c^5 (-1+a x)^5} \] Input:

Integrate[E^ArcCoth[a*x]/(c - a*c*x)^5,x]
 

Output:

-1/315*(Sqrt[1 - 1/(a^2*x^2)]*x*(-58 - 25*a*x + 21*a^2*x^2 - 10*a^3*x^3 + 
2*a^4*x^4))/(c^5*(-1 + a*x)^5)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.44, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6724, 27, 581, 25, 2170, 27, 671, 461, 461, 460}

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.

\(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx\)

\(\Big \downarrow \) 6724

\(\displaystyle a c \int \frac {\sqrt {1-\frac {1}{a^2 x^2}}}{c^6 \left (a-\frac {1}{x}\right )^6 x^3}d\frac {1}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \int \frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\left (a-\frac {1}{x}\right )^6 x^3}d\frac {1}{x}}{c^5}\)

\(\Big \downarrow \) 581

\(\displaystyle \frac {a \left (\int -\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (4 a^3-\frac {7 a^2}{x}+\frac {2 a}{x^2}\right )}{\left (a-\frac {1}{x}\right )^6}d\frac {1}{x}+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^4}\right )}{c^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a \left (\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^4}-\int \frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (4 a^3-\frac {7 a^2}{x}+\frac {2 a}{x^2}\right )}{\left (a-\frac {1}{x}\right )^6}d\frac {1}{x}\right )}{c^5}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {a \left (-\frac {1}{2} a^2 \int \frac {2 \sqrt {1-\frac {1}{a^2 x^2}} \left (9 a-\frac {10}{x}\right )}{\left (a-\frac {1}{x}\right )^6}d\frac {1}{x}+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^4}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^5}\right )}{c^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (-a^2 \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (9 a-\frac {10}{x}\right )}{\left (a-\frac {1}{x}\right )^6}d\frac {1}{x}+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^4}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^5}\right )}{c^5}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {a \left (-a^2 \left (\frac {29}{3} \int \frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\left (a-\frac {1}{x}\right )^5}d\frac {1}{x}-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 \left (a-\frac {1}{x}\right )^6}\right )+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^4}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^5}\right )}{c^5}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {a \left (-a^2 \left (\frac {29}{3} \left (\frac {2 \int \frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\left (a-\frac {1}{x}\right )^4}d\frac {1}{x}}{7 a}+\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{7 \left (a-\frac {1}{x}\right )^5}\right )-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 \left (a-\frac {1}{x}\right )^6}\right )+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^4}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^5}\right )}{c^5}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {a \left (-a^2 \left (\frac {29}{3} \left (\frac {2 \left (\frac {\int \frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\left (a-\frac {1}{x}\right )^3}d\frac {1}{x}}{5 a}+\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \left (a-\frac {1}{x}\right )^4}\right )}{7 a}+\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{7 \left (a-\frac {1}{x}\right )^5}\right )-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 \left (a-\frac {1}{x}\right )^6}\right )+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^4}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^5}\right )}{c^5}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {a \left (-a^2 \left (\frac {29}{3} \left (\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{7 \left (a-\frac {1}{x}\right )^5}+\frac {2 \left (\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{15 \left (a-\frac {1}{x}\right )^3}+\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \left (a-\frac {1}{x}\right )^4}\right )}{7 a}\right )-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 \left (a-\frac {1}{x}\right )^6}\right )+\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^4}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^5}\right )}{c^5}\)

Input:

Int[E^ArcCoth[a*x]/(c - a*c*x)^5,x]
 

Output:

(a*(-(a^2*((29*((2*((a*(1 - 1/(a^2*x^2))^(3/2))/(5*(a - x^(-1))^4) + (1 - 
1/(a^2*x^2))^(3/2)/(15*(a - x^(-1))^3)))/(7*a) + (a*(1 - 1/(a^2*x^2))^(3/2 
))/(7*(a - x^(-1))^5)))/3 - (a^2*(1 - 1/(a^2*x^2))^(3/2))/(9*(a - x^(-1))^ 
6))) + (a^3*(1 - 1/(a^2*x^2))^(3/2))/(a - x^(-1))^5 + (a^2*(1 - 1/(a^2*x^2 
))^(3/2))/(a - x^(-1))^4))/c^5
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 460

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

rule 461

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl 
ify[n + 2*p + 2]/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp 
lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
 

rule 581

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 
2*p + 1))), x] + Simp[1/(d^m*(m + n + 2*p + 1))   Int[(c + d*x)^n*(a + b*x^ 
2)^p*ExpandToSum[d^m*(m + n + 2*p + 1)*x^m - (m + n + 2*p + 1)*(c + d*x)^m 
+ c*(c + d*x)^(m - 2)*(c*(m + n - 1) + c*(m + n + 2*p + 1) + 2*d*(m + n + p 
)*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] & 
& IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[2*p] || ILtQ[m + n, 0] 
)
 

rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m 
 + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m 
+ p + 1))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, 
e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p 
 + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p 
 + 1, 0]
 

rule 2170

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + 
 p + q)*(d + e*x)^(q - 2)*(a*e - b*d*x), x], x], x] /; NeQ[m + q + 2*p + 1, 
 0]] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 
0] &&  !IGtQ[m, 0]
 

rule 6724

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.44

method	result	size
gosper	\(-\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x +1\right )}{315 \left (a x -1\right )^{4} c^{5} \sqrt {\frac {a x -1}{a x +1}}\, a}\)	\(58\)
default	\(-\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x +1\right )}{315 \left (a x -1\right )^{4} c^{5} \sqrt {\frac {a x -1}{a x +1}}\, a}\)	\(58\)
orering	\(\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x -1\right ) \left (a x +1\right )}{315 a \sqrt {\frac {a x -1}{a x +1}}\, \left (-a c x +c \right )^{5}}\)	\(62\)
trager	\(-\frac {\left (a x +1\right ) \left (2 a^{4} x^{4}-10 a^{3} x^{3}+21 a^{2} x^{2}-25 a x -58\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{315 a \,c^{5} \left (a x -1\right )^{5}}\)	\(68\)

Input:

int(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^5,x,method=_RETURNVERBOSE)
 

Output:

-1/315*(2*a^3*x^3-12*a^2*x^2+33*a*x-58)*(a*x+1)/(a*x-1)^4/c^5/((a*x-1)/(a* 
x+1))^(1/2)/a
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {{\left (2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} + 11 \, a^{3} x^{3} - 4 \, a^{2} x^{2} - 83 \, a x - 58\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\right )}} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^5,x, algorithm="fricas")
 

Output:

-1/315*(2*a^5*x^5 - 8*a^4*x^4 + 11*a^3*x^3 - 4*a^2*x^2 - 83*a*x - 58)*sqrt 
((a*x - 1)/(a*x + 1))/(a^6*c^5*x^5 - 5*a^5*c^5*x^4 + 10*a^4*c^5*x^3 - 10*a 
^3*c^5*x^2 + 5*a^2*c^5*x - a*c^5)
 

Sympy [F]

\[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=- \frac {\int \frac {1}{a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 5 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 10 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 10 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 5 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{5}} \] Input:

integrate(1/((a*x-1)/(a*x+1))**(1/2)/(-a*c*x+c)**5,x)
 

Output:

-Integral(1/(a**5*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 5*a**4*x**4*sqr 
t(a*x/(a*x + 1) - 1/(a*x + 1)) + 10*a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x 
+ 1)) - 10*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + 5*a*x*sqrt(a*x/(a 
*x + 1) - 1/(a*x + 1)) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))), x)/c**5
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {\frac {135 \, {\left (a x - 1\right )}}{a x + 1} - \frac {189 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {105 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 35}{2520 \, a c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^5,x, algorithm="maxima")
 

Output:

-1/2520*(135*(a*x - 1)/(a*x + 1) - 189*(a*x - 1)^2/(a*x + 1)^2 + 105*(a*x 
- 1)^3/(a*x + 1)^3 - 35)/(a*c^5*((a*x - 1)/(a*x + 1))^(9/2))
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {4 \, {\left (315 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{5} x^{5} + 189 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{4} x^{4} + 84 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{3} x^{3} - 36 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 9 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}}{315 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{9} a c^{5}} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^5,x, algorithm="giac")
 

Output:

4/315*(315*(a + sqrt(a^2 - 1/x^2))^5*x^5 + 189*(a + sqrt(a^2 - 1/x^2))^4*x 
^4 + 84*(a + sqrt(a^2 - 1/x^2))^3*x^3 - 36*(a + sqrt(a^2 - 1/x^2))^2*x^2 + 
 9*(a + sqrt(a^2 - 1/x^2))*x - 1)/(((a + sqrt(a^2 - 1/x^2))*x - 1)^9*a*c^5 
)
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {\frac {3\,{\left (a\,x-1\right )}^2}{5\,{\left (a\,x+1\right )}^2}-\frac {{\left (a\,x-1\right )}^3}{3\,{\left (a\,x+1\right )}^3}-\frac {3\,\left (a\,x-1\right )}{7\,\left (a\,x+1\right )}+\frac {1}{9}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \] Input:

int(1/((c - a*c*x)^5*((a*x - 1)/(a*x + 1))^(1/2)),x)
 

Output:

((3*(a*x - 1)^2)/(5*(a*x + 1)^2) - (a*x - 1)^3/(3*(a*x + 1)^3) - (3*(a*x - 
 1))/(7*(a*x + 1)) + 1/9)/(8*a*c^5*((a*x - 1)/(a*x + 1))^(9/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {2 \sqrt {a x -1}\, a^{4} x^{4}-8 \sqrt {a x -1}\, a^{3} x^{3}+12 \sqrt {a x -1}\, a^{2} x^{2}-8 \sqrt {a x -1}\, a x +2 \sqrt {a x -1}-2 \sqrt {a x +1}\, a^{4} x^{4}+10 \sqrt {a x +1}\, a^{3} x^{3}-21 \sqrt {a x +1}\, a^{2} x^{2}+25 \sqrt {a x +1}\, a x +58 \sqrt {a x +1}}{315 \sqrt {a x -1}\, a \,c^{5} \left (a^{4} x^{4}-4 a^{3} x^{3}+6 a^{2} x^{2}-4 a x +1\right )} \] Input:

int(1/((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^5,x)
 

Output:

(2*sqrt(a*x - 1)*a**4*x**4 - 8*sqrt(a*x - 1)*a**3*x**3 + 12*sqrt(a*x - 1)* 
a**2*x**2 - 8*sqrt(a*x - 1)*a*x + 2*sqrt(a*x - 1) - 2*sqrt(a*x + 1)*a**4*x 
**4 + 10*sqrt(a*x + 1)*a**3*x**3 - 21*sqrt(a*x + 1)*a**2*x**2 + 25*sqrt(a* 
x + 1)*a*x + 58*sqrt(a*x + 1))/(315*sqrt(a*x - 1)*a*c**5*(a**4*x**4 - 4*a* 
*3*x**3 + 6*a**2*x**2 - 4*a*x + 1))