\(\int \frac {e^{3 \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^{5/2}} \, dx\) [462]

Optimal result

Integrand size = 24, antiderivative size = 335 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=-\frac {5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{3 \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {29 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{12 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {73 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {249 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{16 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}} \]

[Out]

11*(1-1/a/x)^(5/2)*arctanh((1+1/a/x)^(1/2))/a/(c-c/a/x)^(5/2)-249/32*(1-1/a/x)^(5/2)*arctanh(1/2*(1+1/a/x)^(1/
2)*2^(1/2))/a/(c-c/a/x)^(5/2)*2^(1/2)-5/3*a^2*(1-1/a/x)^(5/2)*(1+1/a/x)^(1/2)/(a-1/x)^3/(c-c/a/x)^(5/2)-29/12*
a*(1-1/a/x)^(5/2)*(1+1/a/x)^(1/2)/(a-1/x)^2/(c-c/a/x)^(5/2)-73/16*(1-1/a/x)^(5/2)*(1+1/a/x)^(1/2)/(a-1/x)/(c-c
/a/x)^(5/2)+a^3*(1-1/a/x)^(5/2)*x*(1+1/a/x)^(1/2)/(a-1/x)^3/(c-c/a/x)^(5/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6317, 6314, 100, 156, 162, 65, 214, 212} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {a^3 x \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}{\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}{3 \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {249 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{16 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {29 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}{12 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {73 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}} \]

[In]

Int[E^(3*ArcCoth[a*x])/(c - c/(a*x))^(5/2),x]

[Out]

(-5*a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(3*(a - x^(-1))^3*(c - c/(a*x))^(5/2)) - (29*a*(1 - 1/(a*x))^(5
/2)*Sqrt[1 + 1/(a*x)])/(12*(a - x^(-1))^2*(c - c/(a*x))^(5/2)) - (73*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(1
6*(a - x^(-1))*(c - c/(a*x))^(5/2)) + (a^3*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x)/((a - x^(-1))^3*(c - c/(a*
x))^(5/2)) + (11*(1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(5/2)) - (249*(1 - 1/(a*x))^
(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(16*Sqrt[2]*a*(c - c/(a*x))^(5/2))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 100

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 156

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, -1]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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 6314

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[-c^p, Subst[Int[(1 + d*(x/c))^p
*((1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2))), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0
] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0])

Rule 6317

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Dist[(c + d/x)^p/(1 + d/(c*x))^
p, Int[u*(1 + d/(c*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&
!IntegerQ[n/2] &&  !(IntegerQ[p] || GtQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1-\frac {1}{a x}\right )^{5/2} \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{5/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^2 \left (1-\frac {x}{a}\right )^4} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}} \\ & = \frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} \text {Subst}\left (\int \frac {-\frac {11}{2 a}-\frac {9 x}{2 a^2}}{x \left (1-\frac {x}{a}\right )^4 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{3 \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {33}{a^2}+\frac {25 x}{a^3}}{x \left (1-\frac {x}{a}\right )^3 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{3 \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {29 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{12 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (a^2 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {-\frac {132}{a^3}-\frac {87 x}{a^4}}{x \left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{24 \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{3 \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {29 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{12 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {73 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (a^3 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {264}{a^4}+\frac {219 x}{2 a^5}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{48 \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{3 \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {29 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{12 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {73 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (249 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{32 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (11 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{3 \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {29 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{12 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {73 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (11 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (249 \left (1-\frac {1}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{16 a \left (c-\frac {c}{a x}\right )^{5/2}} \\ & = -\frac {5 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{3 \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {29 a \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{12 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {73 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}{16 \left (a-\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {249 \left (1-\frac {1}{a x}\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{16 \sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.43 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a \sqrt {1+\frac {1}{a x}} x \left (-219+554 a x-415 a^2 x^2+48 a^3 x^3\right )+1056 (-1+a x)^3 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-747 \sqrt {2} (-1+a x)^3 \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{96 a c^2 \sqrt {c-\frac {c}{a x}} (-1+a x)^3} \]

[In]

Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^(5/2),x]

[Out]

(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(-219 + 554*a*x - 415*a^2*x^2 + 48*a^3*x^3) + 1056*(-1 + a*x)^3*Ar
cTanh[Sqrt[1 + 1/(a*x)]] - 747*Sqrt[2]*(-1 + a*x)^3*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(96*a*c^2*Sqrt[c - c/
(a*x)]*(-1 + a*x)^3)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.12

method	result	size
risch	\(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {11 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {2 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{3 a^{7} c \left (x -\frac {1}{a}\right )^{3}}-\frac {11 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{4 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {271 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{48 a^{5} c \left (x -\frac {1}{a}\right )}-\frac {249 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{64 a^{4} \sqrt {c}}\right ) a^{2} \sqrt {\left (a x +1\right ) a c x}\, \left (a x -1\right )}{c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\)	\(376\)
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (192 \sqrt {\left (a x +1\right ) x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{3}-747 a^{\frac {7}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x^{3}-1660 \sqrt {\left (a x +1\right ) x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}+1056 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{4} \sqrt {\frac {1}{a}}\, x^{3}+2241 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x^{2}+2216 \sqrt {\left (a x +1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x -3168 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {1}{a}}\, x^{2}-2241 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x -876 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+3168 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x -1056 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+747 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{192 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x -1\right )^{2} \left (a x +1\right ) a^{\frac {3}{2}} c^{3} \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}}\)	\(480\)

[In]

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

[Out]

1/a/c^2/((a*x-1)/(a*x+1))^(1/2)/(c*(a*x-1)/a/x)^(1/2)*(a*x-1)+(11/2/a^3*ln((1/2*a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^
2*c*x^2+a*c*x)^(1/2))/(a^2*c)^(1/2)-2/3/a^7/c/(x-1/a)^3*(a^2*c*(x-1/a)^2+3*(x-1/a)*a*c+2*c)^(1/2)-11/4/a^6/c/(
x-1/a)^2*(a^2*c*(x-1/a)^2+3*(x-1/a)*a*c+2*c)^(1/2)-271/48/a^5/c/(x-1/a)*(a^2*c*(x-1/a)^2+3*(x-1/a)*a*c+2*c)^(1
/2)-249/64/a^4/c^(1/2)*2^(1/2)*ln((4*c+3*(x-1/a)*a*c+2*2^(1/2)*c^(1/2)*(a^2*c*(x-1/a)^2+3*(x-1/a)*a*c+2*c)^(1/
2))/(x-1/a)))*a^2/c^2/((a*x-1)/(a*x+1))^(1/2)/(a*x+1)/x/(c*(a*x-1)/a/x)^(1/2)*((a*x+1)*a*c*x)^(1/2)*(a*x-1)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 736, normalized size of antiderivative = 2.20 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {747 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\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 ) + 1056 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, 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 x - 1}\right ) + 8 \, {\left (48 \, a^{5} x^{5} - 367 \, a^{4} x^{4} + 139 \, a^{3} x^{3} + 335 \, a^{2} x^{2} - 219 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{384 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}, \frac {747 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\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 ) - 1056 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {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}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (48 \, a^{5} x^{5} - 367 \, a^{4} x^{4} + 139 \, a^{3} x^{3} + 335 \, a^{2} x^{2} - 219 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{192 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \]

[In]

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

[Out]

[1/384*(747*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 1
3*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)) + 1056*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*log(-(8*a^3
*c*x^3 - 7*a*c*x + 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) -
c)/(a*x - 1)) + 8*(48*a^5*x^5 - 367*a^4*x^4 + 139*a^3*x^3 + 335*a^2*x^2 - 219*a*x)*sqrt((a*x - 1)/(a*x + 1))*
sqrt((a*c*x - c)/(a*x)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3), 1/192*(747*sqrt(
2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*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)) - 1056*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 -
4*a*x + 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^
2*c*x^2 - a*c*x - c)) + 4*(48*a^5*x^5 - 367*a^4*x^4 + 139*a^3*x^3 + 335*a^2*x^2 - 219*a*x)*sqrt((a*x - 1)/(a*x
+ 1))*sqrt((a*c*x - c)/(a*x)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)]

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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