\(\int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\) [528]

Optimal result

Integrand size = 27, antiderivative size = 246 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {226 a^4 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{315 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {2 a^4 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {38 a^4 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{63 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a^4 c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}+\frac {2 a^4 c \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 \sqrt {c-\frac {c}{a x}}}-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right ) \] Output:

226/315*a^4*c^3*(1-1/a^2/x^2)^(5/2)/(c-c/a/x)^(5/2)+2/3*a^4*c^2*(1-1/a^2/x 
^2)^(3/2)/(c-c/a/x)^(3/2)-38/63*a^4*c^2*(1-1/a^2/x^2)^(5/2)/(c-c/a/x)^(3/2 
)+4*a^4*c*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2)+2/9*a^4*c*(1-1/a^2/x^2)^(5/2 
)/(c-c/a/x)^(1/2)-4*2^(1/2)*a^4*c^(1/2)*arctanh(1/2*c^(1/2)*(1-1/a^2/x^2)^ 
(1/2)*2^(1/2)/(c-c/a/x)^(1/2))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.72 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (35+95 a x+138 a^2 x^2+236 a^3 x^3+788 a^4 x^4\right )}{315 x^3 (-1+a x)}+2 \sqrt {2} a^4 \sqrt {c} \log \left ((-1+a x)^2\right )-2 \sqrt {2} a^4 \sqrt {c} \log \left (2 \sqrt {2} a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-2 a x+3 a^2 x^2\right )\right ) \] Input:

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

Output:

(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]*L 
og[(-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)]
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6733, 581, 27, 2170, 27, 672, 466, 466, 471, 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.

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

\(\Big \downarrow \) 6733

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

\(\Big \downarrow \) 581

\(\displaystyle -c^3 \left (-\frac {2 a^3 \int -\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (-\frac {11 c^3}{a x}+\frac {19 c^3}{a^2 x^2}+c^3\right )}{2 \left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{9 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 c^2 \sqrt {c-\frac {c}{a x}}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \left (\frac {a^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (-\frac {11 c^3}{a x}+\frac {19 c^3}{a^2 x^2}+c^3\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{9 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 c^2 \sqrt {c-\frac {c}{a x}}}\right )\)

\(\Big \downarrow \) 2170

\(\displaystyle -c^3 \left (\frac {a^3 \left (\frac {38 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {2 a^4 \int \frac {c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (50 a-\frac {113}{x}\right )}{2 a^5 \left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{7 c^2}\right )}{9 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 c^2 \sqrt {c-\frac {c}{a x}}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \left (\frac {a^3 \left (\frac {38 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (50 a-\frac {113}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{7 a}\right )}{9 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 c^2 \sqrt {c-\frac {c}{a x}}}\right )\)

\(\Big \downarrow \) 672

\(\displaystyle -c^3 \left (\frac {a^3 \left (\frac {38 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \left (\frac {226 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{5 \left (c-\frac {c}{a x}\right )^{5/2}}-63 a \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}\right )}{7 a}\right )}{9 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 c^2 \sqrt {c-\frac {c}{a x}}}\right )\)

\(\Big \downarrow \) 466

\(\displaystyle -c^3 \left (\frac {a^3 \left (\frac {38 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \left (\frac {226 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{5 \left (c-\frac {c}{a x}\right )^{5/2}}-63 a \left (\frac {2 \int \frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}}{c}-\frac {2 a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}\right )\right )}{7 a}\right )}{9 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 c^2 \sqrt {c-\frac {c}{a x}}}\right )\)

\(\Big \downarrow \) 466

\(\displaystyle -c^3 \left (\frac {a^3 \left (\frac {38 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \left (\frac {226 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{5 \left (c-\frac {c}{a x}\right )^{5/2}}-63 a \left (\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{c}-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}}}{c \sqrt {c-\frac {c}{a x}}}\right )}{c}-\frac {2 a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}\right )\right )}{7 a}\right )}{9 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 c^2 \sqrt {c-\frac {c}{a x}}}\right )\)

\(\Big \downarrow \) 471

\(\displaystyle -c^3 \left (\frac {a^3 \left (\frac {38 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \left (\frac {226 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{5 \left (c-\frac {c}{a x}\right )^{5/2}}-63 a \left (\frac {2 \left (-\frac {4 \int \frac {1}{\frac {c^2}{a^2 x^2}-\frac {2 c}{a^2}}d\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}}{a}-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}}}{c \sqrt {c-\frac {c}{a x}}}\right )}{c}-\frac {2 a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}\right )\right )}{7 a}\right )}{9 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 c^2 \sqrt {c-\frac {c}{a x}}}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -c^3 \left (\frac {a^3 \left (\frac {38 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \left (\frac {226 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{5 \left (c-\frac {c}{a x}\right )^{5/2}}-63 a \left (\frac {2 \left (\frac {2 \sqrt {2} a \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )}{c^{3/2}}-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}}}{c \sqrt {c-\frac {c}{a x}}}\right )}{c}-\frac {2 a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c \left (c-\frac {c}{a x}\right )^{3/2}}\right )\right )}{7 a}\right )}{9 c^3}-\frac {2 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{9 c^2 \sqrt {c-\frac {c}{a x}}}\right )\)

Input:

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

Output:

-(c^3*((-2*a^4*(1 - 1/(a^2*x^2))^(5/2))/(9*c^2*Sqrt[c - c/(a*x)]) + (a^3*( 
(38*a*c^2*(1 - 1/(a^2*x^2))^(5/2))/(7*(c - c/(a*x))^(3/2)) - (c^3*((226*a^ 
2*(1 - 1/(a^2*x^2))^(5/2))/(5*(c - c/(a*x))^(5/2)) - 63*a*((-2*a*(1 - 1/(a 
^2*x^2))^(3/2))/(3*c*(c - c/(a*x))^(3/2)) + (2*((-2*a*Sqrt[1 - 1/(a^2*x^2) 
])/(c*Sqrt[c - c/(a*x)]) + (2*Sqrt[2]*a*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x 
^2)])/(Sqrt[2]*Sqrt[c - c/(a*x)])])/c^(3/2)))/c)))/(7*a)))/(9*c^3)))
 

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 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 466

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

rule 471

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2*d   Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] 
], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 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 672

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), 
 x] + Simp[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2))   Int[(d + e*x 
)^m*(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] && NeQ[m + 2*p + 2, 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 6733

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.84

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

Input:

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

Output:

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

Time = 0.13 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.68 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\left [\frac {315 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (788 \, a^{5} x^{5} + 1024 \, a^{4} x^{4} + 374 \, a^{3} x^{3} + 233 \, a^{2} x^{2} + 130 \, a x + 35\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{315 \, {\left (a x^{5} - x^{4}\right )}}, \frac {2 \, {\left (315 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + {\left (788 \, a^{5} x^{5} + 1024 \, a^{4} x^{4} + 374 \, a^{3} x^{3} + 233 \, a^{2} x^{2} + 130 \, a x + 35\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, {\left (a x^{5} - x^{4}\right )}}\right ] \] Input:

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

Output:

[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 + 13 
0*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. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.84 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c}\, \left (788 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{4} x^{4}+236 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} x^{3}+138 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}+95 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +35 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+315 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a^{5} x^{5}-315 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a^{5} x^{5}-315 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a^{5} x^{5}+315 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a^{5} x^{5}-508 a^{5} x^{5}\right )}{315 a \,x^{5}} \] Input:

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

Output:

(2*sqrt(c)*(788*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**4*x**4 + 236*sqrt(x)*sqrt 
(a)*sqrt(a*x + 1)*a**3*x**3 + 138*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 
+ 95*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x + 35*sqrt(x)*sqrt(a)*sqrt(a*x + 1) 
+ 315*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1)*a**5*x**5 
 - 315*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a**5*x** 
5 - 315*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) - 1)*a**5*x* 
*5 + 315*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1)*a**5*x 
**5 - 508*a**5*x**5))/(315*a*x**5)