\(\int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\) [515]

Optimal result

Integrand size = 27, antiderivative size = 303 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {4 a^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} a^4 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{\sqrt {1-\frac {1}{a x}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6317, 6315, 90, 52, 65, 212} \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {4 \sqrt {2} a^4 \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {4 a^4 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}} \]

[In]

Int[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^5,x]

[Out]

(4*a^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] + (2*a^4*(1 + 1/(a*x))^(3/2)*Sqrt[c - c/(a*x)])/
(3*Sqrt[1 - 1/(a*x)]) + (2*a^4*(1 + 1/(a*x))^(5/2)*Sqrt[c - c/(a*x)])/(5*Sqrt[1 - 1/(a*x)]) - (2*a^4*(1 + 1/(a
*x))^(7/2)*Sqrt[c - c/(a*x)])/(7*Sqrt[1 - 1/(a*x)]) + (2*a^4*(1 + 1/(a*x))^(9/2)*Sqrt[c - c/(a*x)])/(9*Sqrt[1
- 1/(a*x)]) - (4*Sqrt[2]*a^4*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/Sqrt[1 - 1/(a*x)]

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 90

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6315

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

Rule 6317

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a x}} \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{x^5} \, dx}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {x^3 \left (1+\frac {x}{a}\right )^{3/2}}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \left (-a^3 \left (1+\frac {x}{a}\right )^{3/2}+\frac {a^3 \left (1+\frac {x}{a}\right )^{3/2}}{1-\frac {x}{a}}+a^3 \left (1+\frac {x}{a}\right )^{5/2}-a^3 \left (1+\frac {x}{a}\right )^{7/2}\right ) \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 a^4 \left (1+\frac {1}{a x}\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {\left (a^3 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 a^4 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {\left (2 a^3 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {4 a^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {\left (4 a^3 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {4 a^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {\left (8 a^4 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {4 a^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} a^4 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{\sqrt {1-\frac {1}{a x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.59 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (35+95 a x+138 a^2 x^2+236 a^3 x^3+788 a^4 x^4\right )}{315 x^3 (-1+a x)}+2 \sqrt {2} a^4 \sqrt {c} \log \left ((-1+a x)^2\right )-2 \sqrt {2} a^4 \sqrt {c} \log \left (2 \sqrt {2} a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-2 a x+3 a^2 x^2\right )\right ) \]

[In]

Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^5,x]

[Out]

(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(35 + 95*a*x + 138*a^2*x^2 + 236*a^3*x^3 + 788*a^4*x^4))/(315*x^3
*(-1 + a*x)) + 2*Sqrt[2]*a^4*Sqrt[c]*Log[(-1 + a*x)^2] - 2*Sqrt[2]*a^4*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[
1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)]

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.68

method	result	size
risch	\(\frac {2 \left (788 a^{5} x^{5}+1024 a^{4} x^{4}+374 a^{3} x^{3}+233 a^{2} x^{2}+130 a x +35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {2 a^{4} \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{\sqrt {c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(206\)
default	\(\frac {2 \left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-315 a^{4} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x^{5}+788 x^{4} \sqrt {\left (a x +1\right ) x}\, a^{4} \sqrt {\frac {1}{a}}+236 a^{3} \sqrt {\frac {1}{a}}\, x^{3} \sqrt {\left (a x +1\right ) x}+138 a^{2} \sqrt {\frac {1}{a}}\, x^{2} \sqrt {\left (a x +1\right ) x}+95 x \sqrt {\left (a x +1\right ) x}\, a \sqrt {\frac {1}{a}}+35 \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}\right )}{315 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x^{4} \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}}\)	\(209\)

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

2/315*(788*a^5*x^5+1024*a^4*x^4+374*a^3*x^3+233*a^2*x^2+130*a*x+35)/x^4/((a*x-1)/(a*x+1))^(1/2)/(a*x+1)*(c*(a*
x-1)/a/x)^(1/2)-2*a^4*2^(1/2)/c^(1/2)*ln((4*c+3*(x-1/a)*a*c+2*2^(1/2)*c^(1/2)*(a^2*c*(x-1/a)^2+3*(x-1/a)*a*c+2
*c)^(1/2))/(x-1/a))/((a*x-1)/(a*x+1))^(1/2)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*((a*x+1)*a*c*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.36 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\left [\frac {315 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (788 \, a^{5} x^{5} + 1024 \, a^{4} x^{4} + 374 \, a^{3} x^{3} + 233 \, a^{2} x^{2} + 130 \, a x + 35\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{315 \, {\left (a x^{5} - x^{4}\right )}}, \frac {2 \, {\left (315 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + {\left (788 \, a^{5} x^{5} + 1024 \, a^{4} x^{4} + 374 \, a^{3} x^{3} + 233 \, a^{2} x^{2} + 130 \, a x + 35\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, {\left (a x^{5} - x^{4}\right )}}\right ] \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x^5,x, algorithm="fricas")

[Out]

[1/315*(315*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3
*x^3 + 4*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 +
3*a*x - 1)) + 2*(788*a^5*x^5 + 1024*a^4*x^4 + 374*a^3*x^3 + 233*a^2*x^2 + 130*a*x + 35)*sqrt((a*x - 1)/(a*x +
1))*sqrt((a*c*x - c)/(a*x)))/(a*x^5 - x^4), 2/315*(315*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(-c)*arctan(2*sqrt(2)*(
a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) + (788*
a^5*x^5 + 1024*a^4*x^4 + 374*a^3*x^3 + 233*a^2*x^2 + 130*a*x + 35)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/
(a*x)))/(a*x^5 - x^4)]

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\text {Timed out} \]

[In]

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**(1/2)/x**5,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{x^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(c - c/(a*x))/(x^5*((a*x - 1)/(a*x + 1))^(3/2)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x^5,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}}{x^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]

[In]

int((c - c/(a*x))^(1/2)/(x^5*((a*x - 1)/(a*x + 1))^(3/2)),x)

[Out]

int((c - c/(a*x))^(1/2)/(x^5*((a*x - 1)/(a*x + 1))^(3/2)), x)