Integrand size = 24, antiderivative size = 116 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=-\frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}+\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}} \] Output:
-2/a/c^2/(c-c/a/x)^(1/2)+x/c^2/(c-c/a/x)^(1/2)+arctanh((c-c/a/x)^(1/2)/c^( 1/2))/a/c^(5/2)+1/2*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)/a /c^(5/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.60 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {a x-\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a-\frac {1}{x}}{2 a}\right )-\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1-\frac {1}{a x}\right )}{a c^2 \sqrt {c-\frac {c}{a x}}} \] Input:
Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(5/2)),x]
Output:
(a*x - Hypergeometric2F1[-1/2, 1, 1/2, (a - x^(-1))/(2*a)] - Hypergeometri c2F1[-1/2, 1, 1/2, 1 - 1/(a*x)])/(a*c^2*Sqrt[c - c/(a*x)])
Time = 0.86 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6717, 6683, 1035, 281, 899, 114, 27, 169, 25, 27, 174, 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.
| \(\displaystyle \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle -\int \frac {1-a x}{\left (c-\frac {c}{a x}\right )^{5/2} (a x+1)}dx\) |
| \(\Big \downarrow \) 1035 |
| \(\displaystyle -\int \frac {\frac {1}{x}-a}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {a \int \frac {1}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}dx}{c}\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\frac {a \int \frac {x^2}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a \left (-\frac {\int -\frac {c \left (a+\frac {3}{x}\right ) x}{2 a \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{a c}-\frac {x}{a c \sqrt {c-\frac {c}{a x}}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {\left (a+\frac {3}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{2 a^2}-\frac {x}{a c \sqrt {c-\frac {c}{a x}}}\right )}{c}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a \left (\frac {\frac {4}{c \sqrt {c-\frac {c}{a x}}}-\frac {\int -\frac {c \left (a+\frac {2}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c^2}}{2 a^2}-\frac {x}{a c \sqrt {c-\frac {c}{a x}}}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {c \left (a+\frac {2}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c^2}+\frac {4}{c \sqrt {c-\frac {c}{a x}}}}{2 a^2}-\frac {x}{a c \sqrt {c-\frac {c}{a x}}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {\left (a+\frac {2}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c}+\frac {4}{c \sqrt {c-\frac {c}{a x}}}}{2 a^2}-\frac {x}{a c \sqrt {c-\frac {c}{a x}}}\right )}{c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}+\int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c}+\frac {4}{c \sqrt {c-\frac {c}{a x}}}}{2 a^2}-\frac {x}{a c \sqrt {c-\frac {c}{a x}}}\right )}{c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a \left (\frac {\frac {-\frac {2 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {2 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}}{c}+\frac {4}{c \sqrt {c-\frac {c}{a x}}}}{2 a^2}-\frac {x}{a c \sqrt {c-\frac {c}{a x}}}\right )}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \left (\frac {\frac {-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}}{c}+\frac {4}{c \sqrt {c-\frac {c}{a x}}}}{2 a^2}-\frac {x}{a c \sqrt {c-\frac {c}{a x}}}\right )}{c}\) |
Input:
Int[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(5/2)),x]
Output:
-((a*(-(x/(a*c*Sqrt[c - c/(a*x)])) + (4/(c*Sqrt[c - c/(a*x)]) + ((-2*ArcTa nh[Sqrt[c - c/(a*x)]/Sqrt[c]])/Sqrt[c] - (Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x) ]/(Sqrt[2]*Sqrt[c])])/Sqrt[c])/c)/(2*a^2)))/c)
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] :> 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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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]
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ mn, -n] && IntegerQ[p] && IntegerQ[r]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[u*(c + d/x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] && !G tQ[c, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(226\) vs. \(2(98)=196\).
Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.96
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {\ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{a^{5} c \left (x -\frac {1}{a}\right )}-\frac {\sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{4 a^{4} \sqrt {c}}\right ) a^{2} \sqrt {c \left (a x -1\right ) a x}}{c^{2} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(227\) |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-8 \sqrt {x \left (a x -1\right )}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}+4 \left (x \left (a x -1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} \sqrt {\frac {1}{a}}-2 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{2}+a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{2}+16 \sqrt {x \left (a x -1\right )}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x +4 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x -2 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x -8 \sqrt {x \left (a x -1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-2 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+\sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\right )}{4 a^{\frac {3}{2}} \sqrt {x \left (a x -1\right )}\, c^{3} \left (a x -1\right )^{2} \sqrt {\frac {1}{a}}}\) | \(368\) |
Input:
int((a*x-1)/(a*x+1)/(c-c/a/x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/a*(a*x-1)/c^2/(c*(a*x-1)/a/x)^(1/2)+(1/2/a^3*ln((-1/2*a*c+a^2*c*x)/(a^2* c)^(1/2)+(a^2*c*x^2-a*c*x)^(1/2))/(a^2*c)^(1/2)-1/a^5/c/(x-1/a)*((x-1/a)^2 *a^2*c+(x-1/a)*a*c)^(1/2)-1/4/a^4*2^(1/2)/c^(1/2)*ln((4*c-3*(x+1/a)*a*c+2* 2^(1/2)*c^(1/2)*((x+1/a)^2*a^2*c-3*(x+1/a)*a*c+2*c)^(1/2))/(x+1/a)))*a^2/c ^2/x/(c*(a*x-1)/a/x)^(1/2)*(c*(a*x-1)*a*x)^(1/2)
Time = 0.11 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.62 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 4 \, {\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}, -\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) - 2 \, {\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}\right ] \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a/x)^(5/2),x, algorithm="fricas")
Output:
[1/4*(sqrt(2)*(a*x - 1)*sqrt(c)*log(-(2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x + 1)) + 2*(a*x - 1)*sqrt(c)*log(-2*a*c*x - 2 *a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) + 4*(a^2*x^2 - 2*a*x)*sqrt((a*c* x - c)/(a*x)))/(a^2*c^3*x - a*c^3), -1/2*(sqrt(2)*(a*x - 1)*sqrt(-c)*arcta n(sqrt(2)*a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) + 2*(a*x - 1)* sqrt(-c)*arctan(a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) - 2*(a^2 *x^2 - 2*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^2*c^3*x - a*c^3)]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {a x - 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}\, dx \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a/x)**(5/2),x)
Output:
Integral((a*x - 1)/((-c*(-1 + 1/(a*x)))**(5/2)*(a*x + 1)), x)
\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {a x - 1}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a/x)^(5/2),x, algorithm="maxima")
Output:
integrate((a*x - 1)/((a*x + 1)*(c - c/(a*x))^(5/2)), x)
Exception generated. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a/x)^(5/2),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
Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {a\,x-1}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\left (a\,x+1\right )} \,d x \] Input:
int((a*x - 1)/((c - c/(a*x))^(5/2)*(a*x + 1)),x)
Output:
int((a*x - 1)/((c - c/(a*x))^(5/2)*(a*x + 1)), x)
Time = 0.17 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.31 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (\sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right )+\sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right )-\sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right )+4 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )-5 \sqrt {a x -1}+4 \sqrt {x}\, \sqrt {a}\, a x -8 \sqrt {x}\, \sqrt {a}\right )}{4 \sqrt {a x -1}\, a \,c^{3}} \] Input:
int((a*x-1)/(a*x+1)/(c-c/a/x)^(5/2),x)
Output:
(sqrt(c)*(sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) - sqrt (2)*i + i) + sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) + s qrt(2)*i - i) - sqrt(a*x - 1)*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a*x + 2) + 4*sqrt(a*x - 1)*log(sqrt(a*x - 1) + sqrt(x)*sqr t(a)) - 5*sqrt(a*x - 1) + 4*sqrt(x)*sqrt(a)*a*x - 8*sqrt(x)*sqrt(a)))/(4*s qrt(a*x - 1)*a*c**3)