\(\int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx\) [76]

Optimal result

Integrand size = 14, antiderivative size = 253 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {557 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x}{640 a^4}+\frac {157 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^2}{320 a^3}+\frac {5 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^3}{16 a^2}+\frac {11 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^4}{40 a}+\frac {1}{5} \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4} x^5-\frac {237 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {237 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5} \] Output:

557/640*(1-1/a/x)^(1/4)*(1+1/a/x)^(3/4)*x/a^4+157/320*(1-1/a/x)^(1/4)*(1+1 
/a/x)^(3/4)*x^2/a^3+5/16*(1-1/a/x)^(1/4)*(1+1/a/x)^(3/4)*x^3/a^2+11/40*(1- 
1/a/x)^(1/4)*(1+1/a/x)^(3/4)*x^4/a+1/5*(1-1/a/x)^(1/4)*(1+1/a/x)^(3/4)*x^5 
-237/128*arctan((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a^5+237/128*arctanh((1+1/ 
a/x)^(1/4)/(1-1/a/x)^(1/4))/a^5
 

Mathematica [A] (verified)

Time = 5.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.68 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {\frac {8192 e^{\frac {3}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^5}+\frac {22016 e^{\frac {3}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^4}+\frac {23936 e^{\frac {3}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^3}+\frac {14032 e^{\frac {3}{2} \coth ^{-1}(a x)}}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}+\frac {5500 e^{\frac {3}{2} \coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}-2370 \arctan \left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-1185 \log \left (1-e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+1185 \log \left (1+e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{1280 a^5} \] Input:

Integrate[E^((3*ArcCoth[a*x])/2)*x^4,x]
 

Output:

((8192*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^5 + (22016*E^((3* 
ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (23936*E^((3*ArcCoth[a*x]) 
/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (14032*E^((3*ArcCoth[a*x])/2))/(-1 + E^ 
(2*ArcCoth[a*x]))^2 + (5500*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x 
])) - 2370*ArcTan[E^(ArcCoth[a*x]/2)] - 1185*Log[1 - E^(ArcCoth[a*x]/2)] + 
 1185*Log[1 + E^(ArcCoth[a*x]/2)])/(1280*a^5)
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.08, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {6721, 110, 27, 168, 27, 168, 27, 168, 27, 168, 27, 104, 25, 827, 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.

\(\displaystyle \int x^4 e^{\frac {3}{2} \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6721

\(\displaystyle -\int \frac {\left (1+\frac {1}{a x}\right )^{3/4} x^6}{\left (1-\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}\)

\(\Big \downarrow \) 110

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {1}{5} \int \frac {\left (11 a+\frac {8}{x}\right ) x^5}{2 a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\int \frac {\left (11 a+\frac {8}{x}\right ) x^5}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}}{10 a^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {-\frac {1}{4} \int -\frac {3 \left (25 a+\frac {22}{x}\right ) x^4}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \int \frac {\left (25 a+\frac {22}{x}\right ) x^4}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (-\frac {1}{3} \int -\frac {\left (157 a+\frac {100}{x}\right ) x^3}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {\int \frac {\left (157 a+\frac {100}{x}\right ) x^3}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {-\frac {1}{2} \int -\frac {\left (557 a+\frac {314}{x}\right ) x^2}{2 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-\frac {157}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {\frac {\int \frac {\left (557 a+\frac {314}{x}\right ) x^2}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}}{4 a}-\frac {157}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {\frac {-\int -\frac {1185 x}{2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-557 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {157}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {\frac {\frac {1185}{2} \int \frac {x}{\left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}}d\frac {1}{x}-557 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {157}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {\frac {2370 \int -\frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-557 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {157}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {\frac {-2370 \int \frac {1}{\left (1-\frac {1}{x^4}\right ) x^2}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-557 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {157}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {\frac {2370 \left (\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-557 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {157}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {\frac {2370 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-557 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {157}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{5} x^5 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {\frac {3 \left (\frac {\frac {2370 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )\right )-557 a x \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{4 a}-\frac {157}{2} a x^2 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{6 a}-\frac {25}{3} a x^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}\right )}{8 a}-\frac {11}{4} a x^4 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}}{10 a^2}\)

Input:

Int[E^((3*ArcCoth[a*x])/2)*x^4,x]
 

Output:

((1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^5)/5 - ((-11*a*(1 - 1/(a*x))^(1 
/4)*(1 + 1/(a*x))^(3/4)*x^4)/4 + (3*((-25*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a* 
x))^(3/4)*x^3)/3 + ((-157*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/2 
 + (-557*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x + 2370*(ArcTan[(1 + 1 
/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/2 - ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/ 
(a*x))^(1/4)]/2))/(4*a))/(6*a)))/(8*a))/(10*a^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (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 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 827

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

rule 6721

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

Maple [F]

\[\int \frac {x^{4}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{4}}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.47 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {2 \, {\left (128 \, a^{5} x^{5} + 304 \, a^{4} x^{4} + 376 \, a^{3} x^{3} + 514 \, a^{2} x^{2} + 871 \, a x + 557\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 2370 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + 1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - 1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{1280 \, a^{5}} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/4)*x^4,x, algorithm="fricas")
 

Output:

1/1280*(2*(128*a^5*x^5 + 304*a^4*x^4 + 376*a^3*x^3 + 514*a^2*x^2 + 871*a*x 
 + 557)*((a*x - 1)/(a*x + 1))^(1/4) + 2370*arctan(((a*x - 1)/(a*x + 1))^(1 
/4)) + 1185*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) - 1185*log(((a*x - 1)/(a* 
x + 1))^(1/4) - 1))/a^5
 

Sympy [F]

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

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

Output:

Integral(x**4/((a*x - 1)/(a*x + 1))**(3/4), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.02 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=-\frac {1}{1280} \, a {\left (\frac {4 \, {\left (395 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} - 1440 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} + 3710 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} - 1992 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 1375 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{6}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{6}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{6}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{6}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{6}}{{\left (a x + 1\right )}^{5}} - a^{6}} - \frac {2370 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} - \frac {1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} + \frac {1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{6}}\right )} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/4)*x^4,x, algorithm="maxima")
 

Output:

-1/1280*a*(4*(395*((a*x - 1)/(a*x + 1))^(17/4) - 1440*((a*x - 1)/(a*x + 1) 
)^(13/4) + 3710*((a*x - 1)/(a*x + 1))^(9/4) - 1992*((a*x - 1)/(a*x + 1))^( 
5/4) + 1375*((a*x - 1)/(a*x + 1))^(1/4))/(5*(a*x - 1)*a^6/(a*x + 1) - 10*( 
a*x - 1)^2*a^6/(a*x + 1)^2 + 10*(a*x - 1)^3*a^6/(a*x + 1)^3 - 5*(a*x - 1)^ 
4*a^6/(a*x + 1)^4 + (a*x - 1)^5*a^6/(a*x + 1)^5 - a^6) - 2370*arctan(((a*x 
 - 1)/(a*x + 1))^(1/4))/a^6 - 1185*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^ 
6 + 1185*log(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^6)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.92 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {1}{1280} \, a {\left (\frac {2370 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} + \frac {1185 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} - \frac {1185 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{6}} + \frac {4 \, {\left (\frac {1992 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} - \frac {3710 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {1440 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{3}} - \frac {395 \, {\left (a x - 1\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{4}} - 1375 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{a^{6} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/4)*x^4,x, algorithm="giac")
 

Output:

1/1280*a*(2370*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 + 1185*log(((a*x - 
1)/(a*x + 1))^(1/4) + 1)/a^6 - 1185*log(abs(((a*x - 1)/(a*x + 1))^(1/4) - 
1))/a^6 + 4*(1992*(a*x - 1)*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1) - 3710*( 
a*x - 1)^2*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^2 + 1440*(a*x - 1)^3*((a* 
x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^3 - 395*(a*x - 1)^4*((a*x - 1)/(a*x + 1) 
)^(1/4)/(a*x + 1)^4 - 1375*((a*x - 1)/(a*x + 1))^(1/4))/(a^6*((a*x - 1)/(a 
*x + 1) - 1)^5))
 

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.91 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} x^4 \, dx=\frac {\frac {275\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{64}-\frac {249\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{40}+\frac {371\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{32}-\frac {9\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}}{2}+\frac {79\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}}{64}}{a^5+\frac {10\,a^5\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a^5\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a^5\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a^5\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {5\,a^5\,\left (a\,x-1\right )}{a\,x+1}}+\frac {237\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}+\frac {237\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5} \] Input:

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

Output:

((275*((a*x - 1)/(a*x + 1))^(1/4))/64 - (249*((a*x - 1)/(a*x + 1))^(5/4))/ 
40 + (371*((a*x - 1)/(a*x + 1))^(9/4))/32 - (9*((a*x - 1)/(a*x + 1))^(13/4 
))/2 + (79*((a*x - 1)/(a*x + 1))^(17/4))/64)/(a^5 + (10*a^5*(a*x - 1)^2)/( 
a*x + 1)^2 - (10*a^5*(a*x - 1)^3)/(a*x + 1)^3 + (5*a^5*(a*x - 1)^4)/(a*x + 
 1)^4 - (a^5*(a*x - 1)^5)/(a*x + 1)^5 - (5*a^5*(a*x - 1))/(a*x + 1)) + (23 
7*atan(((a*x - 1)/(a*x + 1))^(1/4)))/(128*a^5) + (237*atanh(((a*x - 1)/(a* 
x + 1))^(1/4)))/(128*a^5)
 

Reduce [F]

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

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

Output:

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