\(\int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\) [541]

Optimal result

Integrand size = 27, antiderivative size = 289 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {8 a^4 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {32 a^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {50 a^4 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {38 a^4 \left (1+\frac {1}{a x}\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {2 a^4 \left (1+\frac {1}{a x}\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}} \]

[Out]

50/3*a^4*(1+1/a/x)^(3/2)*(c-c/a/x)^(1/2)/(1-1/a/x)^(1/2)-38/5*a^4*(1+1/a/x)^(5/2)*(c-c/a/x)^(1/2)/(1-1/a/x)^(1
/2)+2*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)-8*a^4*(c-c/a/x)^(1/2)/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-32*a^4*(1+1/a/x)^(1/2)*(c-c/a/x)^(1/2)/(1-1/a/x)^(1/
2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6317, 6315, 90} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{9/2} \sqrt {c-\frac {c}{a x}}}{9 \sqrt {1-\frac {1}{a x}}}+\frac {2 a^4 \left (\frac {1}{a x}+1\right )^{7/2} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {38 a^4 \left (\frac {1}{a x}+1\right )^{5/2} \sqrt {c-\frac {c}{a x}}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {50 a^4 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {32 a^4 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {8 a^4 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]

[In]

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

[Out]

(-8*a^4*Sqrt[c - c/(a*x)])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (32*a^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]
)/Sqrt[1 - 1/(a*x)] + (50*a^4*(1 + 1/(a*x))^(3/2)*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - 1/(a*x)]) - (38*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)])/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)])

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 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 )^2}{\left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \left (-\frac {4 a^3}{\left (1+\frac {x}{a}\right )^{3/2}}+\frac {16 a^3}{\sqrt {1+\frac {x}{a}}}-25 a^3 \sqrt {1+\frac {x}{a}}+19 a^3 \left (1+\frac {x}{a}\right )^{3/2}-7 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 {8 a^4 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {32 a^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {50 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 {38 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}}}{\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}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.30 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (5-20 a x+41 a^2 x^2-82 a^3 x^3+328 a^4 x^4+656 a^5 x^5\right )}{45 x^3 \left (-1+a^2 x^2\right )} \]

[In]

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

[Out]

(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(5 - 20*a*x + 41*a^2*x^2 - 82*a^3*x^3 + 328*a^4*x^4 + 656*a^5*x^
5))/(45*x^3*(-1 + a^2*x^2))

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.30

method	result	size
gosper	\(-\frac {2 \left (a x +1\right ) \left (656 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 x^{4} \left (a x -1\right )^{2}}\)	\(86\)
default	\(-\frac {2 \left (a x +1\right ) \left (656 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 x^{4} \left (a x -1\right )^{2}}\)	\(86\)
risch	\(-\frac {2 \left (476 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{45 x^{4} \left (a x -1\right )}-\frac {8 a^{5} x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}\)	\(125\)

[In]

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

[Out]

-2/45*(a*x+1)*(656*a^5*x^5+328*a^4*x^4-82*a^3*x^3+41*a^2*x^2-20*a*x+5)*(c*(a*x-1)/a/x)^(1/2)*((a*x-1)/(a*x+1))
^(3/2)/x^4/(a*x-1)^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.29 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {2 \, {\left (656 \, a^{5} x^{5} + 328 \, a^{4} x^{4} - 82 \, a^{3} x^{3} + 41 \, a^{2} x^{2} - 20 \, a x + 5\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{45 \, {\left (a x^{5} - x^{4}\right )}} \]

[In]

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

[Out]

-2/45*(656*a^5*x^5 + 328*a^4*x^4 - 82*a^3*x^3 + 41*a^2*x^2 - 20*a*x + 5)*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((c-c/a/x)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**5,x)

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(c - c/(a*x))*((a*x - 1)/(a*x + 1))^(3/2)/x^5, 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((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/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 [B] (verification not implemented)

Time = 4.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.37 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}\,\left (656\,a^4\,x^4+984\,a^3\,x^3+902\,a^2\,x^2+943\,a\,x+923\right )}{45\,x^4}-\frac {1856\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{45\,x^4\,\left (a\,x-1\right )} \]

[In]

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

[Out]

- (2*((a*x - 1)/(a*x + 1))^(1/2)*((c*(a*x - 1))/(a*x))^(1/2)*(943*a*x + 902*a^2*x^2 + 984*a^3*x^3 + 656*a^4*x^
4 + 923))/(45*x^4) - (1856*((a*x - 1)/(a*x + 1))^(1/2)*((c*(a*x - 1))/(a*x))^(1/2))/(45*x^4*(a*x - 1))