\(\int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx\) [527]

Optimal result

Integrand size = 27, antiderivative size = 209 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {4 a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {2 a^3 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {2 a^3 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a^3 c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right ) \] Output:

4/7*a^3*c^3*(1-1/a^2/x^2)^(5/2)/(c-c/a/x)^(5/2)+2/3*a^3*c^2*(1-1/a^2/x^2)^ 
(3/2)/(c-c/a/x)^(3/2)-2/7*a^3*c^2*(1-1/a^2/x^2)^(5/2)/(c-c/a/x)^(3/2)+4*a^ 
3*c*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2)-4*2^(1/2)*a^3*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 = 0.54 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.81 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (3+9 a x+16 a^2 x^2+52 a^3 x^3\right )}{21 x^2 (-1+a x)}+2 \sqrt {2} a^3 \sqrt {c} \log \left ((-1+a x)^2\right )-2 \sqrt {2} a^3 \sqrt {c} \log \left (2 \sqrt {2} a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-2 a x+3 a^2 x^2\right )\right ) \] Input:

Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4,x]
 

Output:

(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(3 + 9*a*x + 16*a^2*x^2 + 52* 
a^3*x^3))/(21*x^2*(-1 + a*x)) + 2*Sqrt[2]*a^3*Sqrt[c]*Log[(-1 + a*x)^2] - 
2*Sqrt[2]*a^3*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 = 0.93 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6733, 581, 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.

\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a x}} e^{3 \coth ^{-1}(a x)}}{x^4} \, dx\)

\(\Big \downarrow \) 6733

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

\(\Big \downarrow \) 581

\(\displaystyle -c^3 \left (\frac {2 a^2 \int -\frac {c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a-\frac {10}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{7 c^2}+\frac {2 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 c \left (c-\frac {c}{a x}\right )^{3/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -c^3 \left (\frac {2 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{7 c \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {1}{7} a \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a-\frac {10}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}\right )\)

\(\Big \downarrow \) 672

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

\(\Big \downarrow \) 466

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

\(\Big \downarrow \) 466

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

\(\Big \downarrow \) 471

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

\(\Big \downarrow \) 221

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

Input:

Int[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4,x]
 

Output:

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

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 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 = 187, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {2 \left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (21 a^{3} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x^{4}-52 a^{3} \sqrt {\frac {1}{a}}\, x^{3} \sqrt {x \left (a x +1\right )}-16 x^{2} \sqrt {x \left (a x +1\right )}\, a^{2} \sqrt {\frac {1}{a}}-9 x \sqrt {x \left (a x +1\right )}\, a \sqrt {\frac {1}{a}}-3 \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}\right )}{21 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x^{3} \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\)	\(187\)
risch	\(\frac {2 \left (52 a^{4} x^{4}+68 a^{3} x^{3}+25 a^{2} x^{2}+12 a x +3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{21 x^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {2 a^{3} \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{\sqrt {c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(198\)

Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-2/21/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)/x^3*(2 
1*a^3*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*(x*(a*x+1))^(1/2)*a+3*a*x+1)/(a*x- 
1))*x^4-52*a^3*(1/a)^(1/2)*x^3*(x*(a*x+1))^(1/2)-16*x^2*(x*(a*x+1))^(1/2)* 
a^2*(1/a)^(1/2)-9*x*(x*(a*x+1))^(1/2)*a*(1/a)^(1/2)-3*(x*(a*x+1))^(1/2)*(1 
/a)^(1/2))/(x*(a*x+1))^(1/2)/(1/a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.90 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\left [\frac {21 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (52 \, a^{4} x^{4} + 68 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 12 \, a x + 3\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{21 \, {\left (a x^{4} - x^{3}\right )}}, \frac {2 \, {\left (21 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + {\left (52 \, a^{4} x^{4} + 68 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 12 \, a x + 3\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}\right )}}{21 \, {\left (a x^{4} - x^{3}\right )}}\right ] \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x^4,x, algorithm="fric 
as")
 

Output:

[1/21*(21*sqrt(2)*(a^4*x^4 - a^3*x^3)*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*(52*a^4*x^4 + 68*a^3*x^3 + 25*a^2*x^2 + 12*a*x + 3)*sqrt((a*x 
- 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a*x^4 - x^3), 2/21*(21*sqrt(2)*( 
a^4*x^4 - a^3*x^3)*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)) 
 + (52*a^4*x^4 + 68*a^3*x^3 + 25*a^2*x^2 + 12*a*x + 3)*sqrt((a*x - 1)/(a*x 
 + 1))*sqrt((a*c*x - c)/(a*x)))/(a*x^4 - x^3)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\text {Timed out} \] Input:

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**(1/2)/x**4,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x^4,x, algorithm="maxi 
ma")
 

Output:

integrate(sqrt(c - c/(a*x))/(x^4*((a*x - 1)/(a*x + 1))^(3/2)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x^4,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. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}}{x^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:

int((c - c/(a*x))^(1/2)/(x^4*((a*x - 1)/(a*x + 1))^(3/2)),x)
 

Output:

int((c - c/(a*x))^(1/2)/(x^4*((a*x - 1)/(a*x + 1))^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 \sqrt {c}\, \left (52 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} x^{3}+16 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}+9 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +3 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+21 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a^{4} x^{4}-21 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a^{4} x^{4}-21 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a^{4} x^{4}+21 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a^{4} x^{4}-28 a^{4} x^{4}\right )}{21 a \,x^{4}} \] Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(1/2)/x^4,x)
 

Output:

(2*sqrt(c)*(52*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**3*x**3 + 16*sqrt(x)*sqrt(a 
)*sqrt(a*x + 1)*a**2*x**2 + 9*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x + 3*sqrt(x 
)*sqrt(a)*sqrt(a*x + 1) + 21*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - 
 sqrt(2) - 1)*a**4*x**4 - 21*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - 
 sqrt(2) + 1)*a**4*x**4 - 21*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + 
 sqrt(2) - 1)*a**4*x**4 + 21*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + 
 sqrt(2) + 1)*a**4*x**4 - 28*a**4*x**4))/(21*a*x**4)