\(\int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx\) [86]

Optimal result

Integrand size = 14, antiderivative size = 194 \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=\frac {2467 a^4 \sqrt [4]{1+a x}}{192 \sqrt [4]{1-a x}}-\frac {\sqrt [4]{1+a x}}{4 x^4 \sqrt [4]{1-a x}}-\frac {17 a \sqrt [4]{1+a x}}{24 x^3 \sqrt [4]{1-a x}}-\frac {113 a^2 \sqrt [4]{1+a x}}{96 x^2 \sqrt [4]{1-a x}}-\frac {521 a^3 \sqrt [4]{1+a x}}{192 x \sqrt [4]{1-a x}}-\frac {475}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac {475}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right ) \] Output:

2467/192*a^4*(a*x+1)^(1/4)/(-a*x+1)^(1/4)-1/4*(a*x+1)^(1/4)/x^4/(-a*x+1)^( 
1/4)-17/24*a*(a*x+1)^(1/4)/x^3/(-a*x+1)^(1/4)-113/96*a^2*(a*x+1)^(1/4)/x^2 
/(-a*x+1)^(1/4)-521/192*a^3*(a*x+1)^(1/4)/x/(-a*x+1)^(1/4)-475/64*a^4*arct 
an((a*x+1)^(1/4)/(-a*x+1)^(1/4))-475/64*a^4*arctanh((a*x+1)^(1/4)/(-a*x+1) 
^(1/4))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=\frac {-48-184 a x-362 a^2 x^2-747 a^3 x^3+1946 a^4 x^4+2467 a^5 x^5+950 a^4 x^4 (-1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {1-a x}{1+a x}\right )}{192 x^4 \sqrt [4]{1-a x} (1+a x)^{3/4}} \] Input:

Integrate[E^((5*ArcTanh[a*x])/2)/x^5,x]
 

Output:

(-48 - 184*a*x - 362*a^2*x^2 - 747*a^3*x^3 + 1946*a^4*x^4 + 2467*a^5*x^5 + 
 950*a^4*x^4*(-1 + a*x)*Hypergeometric2F1[3/4, 1, 7/4, (1 - a*x)/(1 + a*x) 
])/(192*x^4*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6676, 109, 27, 168, 27, 168, 27, 168, 27, 172, 27, 104, 756, 216, 219}

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.

\(\displaystyle \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx\)

\(\Big \downarrow \) 6676

\(\displaystyle \int \frac {(a x+1)^{5/4}}{x^5 (1-a x)^{5/4}}dx\)

\(\Big \downarrow \) 109

\(\displaystyle -\frac {1}{4} \int -\frac {a (16 a x+17)}{2 x^4 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} a \int \frac {16 a x+17}{x^4 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} a \left (-\frac {1}{3} \int -\frac {a (102 a x+113)}{2 x^3 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \int \frac {102 a x+113}{x^3 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{2} \int -\frac {a (452 a x+521)}{2 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \int \frac {452 a x+521}{x^2 (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (-\int -\frac {a (1042 a x+1425)}{2 x (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \int \frac {1042 a x+1425}{x (1-a x)^{5/4} (a x+1)^{3/4}}dx-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {2 \int -\frac {1425 a}{2 x \sqrt [4]{1-a x} (a x+1)^{3/4}}dx}{a}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (1425 \int \frac {1}{x \sqrt [4]{1-a x} (a x+1)^{3/4}}dx+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (5700 \int \frac {1}{\frac {a x+1}{1-a x}-1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (5700 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+1}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (5700 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a x+1}}{\sqrt {1-a x}}}d\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{8} a \left (\frac {1}{6} a \left (\frac {1}{4} a \left (\frac {1}{2} a \left (5700 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )\right )+\frac {4934 \sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac {521 \sqrt [4]{a x+1}}{x \sqrt [4]{1-a x}}\right )-\frac {113 \sqrt [4]{a x+1}}{2 x^2 \sqrt [4]{1-a x}}\right )-\frac {17 \sqrt [4]{a x+1}}{3 x^3 \sqrt [4]{1-a x}}\right )-\frac {\sqrt [4]{a x+1}}{4 x^4 \sqrt [4]{1-a x}}\)

Input:

Int[E^((5*ArcTanh[a*x])/2)/x^5,x]
 

Output:

-1/4*(1 + a*x)^(1/4)/(x^4*(1 - a*x)^(1/4)) + (a*((-17*(1 + a*x)^(1/4))/(3* 
x^3*(1 - a*x)^(1/4)) + (a*((-113*(1 + a*x)^(1/4))/(2*x^2*(1 - a*x)^(1/4)) 
+ (a*((-521*(1 + a*x)^(1/4))/(x*(1 - a*x)^(1/4)) + (a*((4934*(1 + a*x)^(1/ 
4))/(1 - a*x)^(1/4) + 5700*(-1/2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - 
 ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/2)))/2))/4))/6))/8
 

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 168

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] && ILtQ[m, -1]
 

rule 172

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, 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] + 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*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)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1 
])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
 

rule 216

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 6676

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) 
^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] &&  !Int 
egerQ[(n - 1)/2]
 

Maple [F]

\[\int \frac {{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {5}{2}}}{x^{5}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {5}{2} \text {arctanh}(a x)}}{x^5} \, dx=-\frac {2850 \, a^{4} x^{4} \arctan \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 1425 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) - 1425 \, a^{4} x^{4} \log \left (\sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) - 2 \, {\left (2467 \, a^{4} x^{4} - 521 \, a^{3} x^{3} - 226 \, a^{2} x^{2} - 136 \, a x - 48\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, x^{4}} \] Input:

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

Output:

-1/384*(2850*a^4*x^4*arctan(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 1425*a^ 
4*x^4*log(sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 1) - 1425*a^4*x^4*log(sqrt 
(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 1) - 2*(2467*a^4*x^4 - 521*a^3*x^3 - 226 
*a^2*x^2 - 136*a*x - 48)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/x^4
 

Sympy [F]

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

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

Output:

Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(5/2)/x**5, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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