3.77 \(\int e^{\frac{5}{2} \coth ^{-1}(a x)} x^4 \, dx\)

Optimal. Leaf size=287 \[ \frac{181 x^3 \sqrt [4]{\frac{1}{a x}+1}}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1189 x^2 \sqrt [4]{\frac{1}{a x}+1}}{960 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{5533 x \sqrt [4]{\frac{1}{a x}+1}}{1920 a^4 \sqrt [4]{1-\frac{1}{a x}}}-\frac{26111 \sqrt [4]{\frac{1}{a x}+1}}{1920 a^5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1003 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1003 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{x^5 \sqrt [4]{\frac{1}{a x}+1}}{5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 x^4 \sqrt [4]{\frac{1}{a x}+1}}{40 a \sqrt [4]{1-\frac{1}{a x}}} \]

[Out]

(-26111*(1 + 1/(a*x))^(1/4))/(1920*a^5*(1 - 1/(a*x))^(1/4)) + (5533*(1 + 1/(a*x))^(1/4)*x)/(1920*a^4*(1 - 1/(a
*x))^(1/4)) + (1189*(1 + 1/(a*x))^(1/4)*x^2)/(960*a^3*(1 - 1/(a*x))^(1/4)) + (181*(1 + 1/(a*x))^(1/4)*x^3)/(24
0*a^2*(1 - 1/(a*x))^(1/4)) + (21*(1 + 1/(a*x))^(1/4)*x^4)/(40*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^
5)/(5*(1 - 1/(a*x))^(1/4)) + (1003*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) + (1003*ArcTanh[
(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)

________________________________________________________________________________________

Rubi [A]  time = 0.16532, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6171, 98, 151, 155, 12, 93, 212, 206, 203} \[ \frac{181 x^3 \sqrt [4]{\frac{1}{a x}+1}}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1189 x^2 \sqrt [4]{\frac{1}{a x}+1}}{960 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{5533 x \sqrt [4]{\frac{1}{a x}+1}}{1920 a^4 \sqrt [4]{1-\frac{1}{a x}}}-\frac{26111 \sqrt [4]{\frac{1}{a x}+1}}{1920 a^5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1003 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1003 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{x^5 \sqrt [4]{\frac{1}{a x}+1}}{5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 x^4 \sqrt [4]{\frac{1}{a x}+1}}{40 a \sqrt [4]{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Int[E^((5*ArcCoth[a*x])/2)*x^4,x]

[Out]

(-26111*(1 + 1/(a*x))^(1/4))/(1920*a^5*(1 - 1/(a*x))^(1/4)) + (5533*(1 + 1/(a*x))^(1/4)*x)/(1920*a^4*(1 - 1/(a
*x))^(1/4)) + (1189*(1 + 1/(a*x))^(1/4)*x^2)/(960*a^3*(1 - 1/(a*x))^(1/4)) + (181*(1 + 1/(a*x))^(1/4)*x^3)/(24
0*a^2*(1 - 1/(a*x))^(1/4)) + (21*(1 + 1/(a*x))^(1/4)*x^4)/(40*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^
5)/(5*(1 - 1/(a*x))^(1/4)) + (1003*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) + (1003*ArcTanh[
(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)

Rule 6171

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]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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] && LtQ[m, -1] && IntegerQ[m]

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 212

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

Rule 206

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

Rule 203

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

Rubi steps

\begin{align*} \int e^{\frac{5}{2} \coth ^{-1}(a x)} x^4 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/4}}{x^6 \left (1-\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1}{5} \operatorname{Subst}\left (\int \frac{-\frac{21}{2 a}-\frac{10 x}{a^2}}{x^5 \left (1-\frac{x}{a}\right )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{21 \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{20} \operatorname{Subst}\left (\int \frac{\frac{181}{4 a^2}+\frac{42 x}{a^3}}{x^4 \left (1-\frac{x}{a}\right )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{181 \sqrt [4]{1+\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1}{60} \operatorname{Subst}\left (\int \frac{-\frac{1189}{8 a^3}-\frac{543 x}{4 a^4}}{x^3 \left (1-\frac{x}{a}\right )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1189 \sqrt [4]{1+\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{181 \sqrt [4]{1+\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{120} \operatorname{Subst}\left (\int \frac{\frac{5533}{16 a^4}+\frac{1189 x}{4 a^5}}{x^2 \left (1-\frac{x}{a}\right )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5533 \sqrt [4]{1+\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1189 \sqrt [4]{1+\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{181 \sqrt [4]{1+\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1}{120} \operatorname{Subst}\left (\int \frac{-\frac{15045}{32 a^5}-\frac{5533 x}{16 a^6}}{x \left (1-\frac{x}{a}\right )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{26111 \sqrt [4]{1+\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1+\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1189 \sqrt [4]{1+\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{181 \sqrt [4]{1+\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{60} a \operatorname{Subst}\left (\int \frac{15045}{64 a^6 x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{26111 \sqrt [4]{1+\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1+\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1189 \sqrt [4]{1+\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{181 \sqrt [4]{1+\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}-\frac{1003 \operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{256 a^5}\\ &=-\frac{26111 \sqrt [4]{1+\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1+\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1189 \sqrt [4]{1+\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{181 \sqrt [4]{1+\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}-\frac{1003 \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^5}\\ &=-\frac{26111 \sqrt [4]{1+\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1+\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1189 \sqrt [4]{1+\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{181 \sqrt [4]{1+\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1003 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1003 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}\\ &=-\frac{26111 \sqrt [4]{1+\frac{1}{a x}}}{1920 a^5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{5533 \sqrt [4]{1+\frac{1}{a x}} x}{1920 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1189 \sqrt [4]{1+\frac{1}{a x}} x^2}{960 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{181 \sqrt [4]{1+\frac{1}{a x}} x^3}{240 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{21 \sqrt [4]{1+\frac{1}{a x}} x^4}{40 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^5}{5 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1003 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}+\frac{1003 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{128 a^5}\\ \end{align*}

Mathematica [A]  time = 5.2503, size = 198, normalized size = 0.69 \[ \frac{-8 e^{\frac{1}{2} \coth ^{-1}(a x)}+\frac{4117 e^{\frac{1}{2} \coth ^{-1}(a x)}}{192 \left (e^{2 \coth ^{-1}(a x)}-1\right )}+\frac{1661 e^{\frac{1}{2} \coth ^{-1}(a x)}}{48 \left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+\frac{233 e^{\frac{1}{2} \coth ^{-1}(a x)}}{6 \left (e^{2 \coth ^{-1}(a x)}-1\right )^3}+\frac{122 e^{\frac{1}{2} \coth ^{-1}(a x)}}{5 \left (e^{2 \coth ^{-1}(a x)}-1\right )^4}+\frac{32 e^{\frac{1}{2} \coth ^{-1}(a x)}}{5 \left (e^{2 \coth ^{-1}(a x)}-1\right )^5}-\frac{1003}{256} \log \left (1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right )+\frac{1003}{256} \log \left (e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )+\frac{1003}{128} \tan ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )}{a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^((5*ArcCoth[a*x])/2)*x^4,x]

[Out]

(-8*E^(ArcCoth[a*x]/2) + (32*E^(ArcCoth[a*x]/2))/(5*(-1 + E^(2*ArcCoth[a*x]))^5) + (122*E^(ArcCoth[a*x]/2))/(5
*(-1 + E^(2*ArcCoth[a*x]))^4) + (233*E^(ArcCoth[a*x]/2))/(6*(-1 + E^(2*ArcCoth[a*x]))^3) + (1661*E^(ArcCoth[a*
x]/2))/(48*(-1 + E^(2*ArcCoth[a*x]))^2) + (4117*E^(ArcCoth[a*x]/2))/(192*(-1 + E^(2*ArcCoth[a*x]))) + (1003*Ar
cTan[E^(ArcCoth[a*x]/2)])/128 - (1003*Log[1 - E^(ArcCoth[a*x]/2)])/256 + (1003*Log[1 + E^(ArcCoth[a*x]/2)])/25
6)/a^5

________________________________________________________________________________________

Maple [F]  time = 0.319, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x)

[Out]

int(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x)

________________________________________________________________________________________

Maxima [A]  time = 1.54584, size = 371, normalized size = 1.29 \begin{align*} -\frac{1}{3840} \, a{\left (\frac{4 \,{\left (\frac{58985 \,{\left (a x - 1\right )}}{a x + 1} - \frac{125920 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{137930 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{72216 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + \frac{15045 \,{\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} - 7680\right )}}{a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{21}{4}} - 5 \, a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{17}{4}} + 10 \, a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{4}} - 10 \, a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} + 5 \, a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} - a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} + \frac{30090 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{6}} - \frac{15045 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{6}} + \frac{15045 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x, algorithm="maxima")

[Out]

-1/3840*a*(4*(58985*(a*x - 1)/(a*x + 1) - 125920*(a*x - 1)^2/(a*x + 1)^2 + 137930*(a*x - 1)^3/(a*x + 1)^3 - 72
216*(a*x - 1)^4/(a*x + 1)^4 + 15045*(a*x - 1)^5/(a*x + 1)^5 - 7680)/(a^6*((a*x - 1)/(a*x + 1))^(21/4) - 5*a^6*
((a*x - 1)/(a*x + 1))^(17/4) + 10*a^6*((a*x - 1)/(a*x + 1))^(13/4) - 10*a^6*((a*x - 1)/(a*x + 1))^(9/4) + 5*a^
6*((a*x - 1)/(a*x + 1))^(5/4) - a^6*((a*x - 1)/(a*x + 1))^(1/4)) + 30090*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a
^6 - 15045*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^6 + 15045*log(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^6)

________________________________________________________________________________________

Fricas [A]  time = 1.70466, size = 423, normalized size = 1.47 \begin{align*} -\frac{30090 \,{\left (a x - 1\right )} \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 15045 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + 15045 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right ) - 2 \,{\left (384 \, a^{6} x^{6} + 1392 \, a^{5} x^{5} + 2456 \, a^{4} x^{4} + 3826 \, a^{3} x^{3} + 7911 \, a^{2} x^{2} - 20578 \, a x - 26111\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{3840 \,{\left (a^{6} x - a^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x, algorithm="fricas")

[Out]

-1/3840*(30090*(a*x - 1)*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 15045*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4)
+ 1) + 15045*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4) - 1) - 2*(384*a^6*x^6 + 1392*a^5*x^5 + 2456*a^4*x^4 +
3826*a^3*x^3 + 7911*a^2*x^2 - 20578*a*x - 26111)*((a*x - 1)/(a*x + 1))^(3/4))/(a^6*x - a^5)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(5/4)*x**4,x)

[Out]

Integral(x**4/((a*x - 1)/(a*x + 1))**(5/4), x)

________________________________________________________________________________________

Giac [A]  time = 1.20408, size = 343, normalized size = 1.2 \begin{align*} -\frac{1}{3840} \, a{\left (\frac{30090 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{6}} - \frac{15045 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{6}} + \frac{15045 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{6}} + \frac{30720}{a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} - \frac{4 \,{\left (\frac{49120 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{a x + 1} - \frac{61130 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{2}} + \frac{33816 \,{\left (a x - 1\right )}^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{3}} - \frac{7365 \,{\left (a x - 1\right )}^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{4}} - 20585 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}\right )}}{a^{6}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(5/4)*x^4,x, algorithm="giac")

[Out]

-1/3840*a*(30090*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 - 15045*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^6 + 15
045*log(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^6 + 30720/(a^6*((a*x - 1)/(a*x + 1))^(1/4)) - 4*(49120*(a*x -
1)*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1) - 61130*(a*x - 1)^2*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^2 + 33816*(
a*x - 1)^3*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^3 - 7365*(a*x - 1)^4*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^4
- 20585*((a*x - 1)/(a*x + 1))^(3/4))/(a^6*((a*x - 1)/(a*x + 1) - 1)^5))