\(\int e^{n \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx\) [372]

Optimal result

Integrand size = 20, antiderivative size = 98 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2}{7} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (-5+n)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},\frac {1}{2} (-5+n),-\frac {5}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \]

[Out]

2/7*((a-1/x)/(a+1/x))^(-5/2+1/2*n)*(1+1/a/x)^(1+1/2*n)*x*(-a*c*x+c)^(5/2)*hypergeom([-7/2, -5/2+1/2*n],[-5/2],
2/(a+1/x)/x)/((1-1/a/x)^(1/2*n))

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6311, 6316, 134} \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2}{7} x (c-a c x)^{5/2} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n-5}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},\frac {n-5}{2},-\frac {5}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \]

[In]

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

[Out]

(2*((a - x^(-1))/(a + x^(-1)))^((-5 + n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^(5/2)*Hypergeometric2F1[-7
/2, (-5 + n)/2, -5/2, 2/((a + x^(-1))*x)])/(7*(1 - 1/(a*x))^(n/2))

Rule 134

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x
)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c
*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f*x))))^n, x] /; FreeQ[{a
, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] &&  !IntegerQ[n]

Rule 6311

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

Rule 6316

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

Rubi steps \begin{align*} \text {integral}& = \frac {(c-a c x)^{5/2} \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{5/2} x^{5/2} \, dx}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {5}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^{9/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{5/2}} \\ & = \frac {2}{7} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (-5+n)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},\frac {1}{2} (-5+n),-\frac {5}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 c^2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {-1+a x}{1+a x}\right )^{\frac {1}{2} (-1+n)} (1+a x)^3 \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},\frac {1}{2} (-5+n),-\frac {5}{2},\frac {2}{1+a x}\right )}{7 a} \]

[In]

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

[Out]

(2*c^2*(1 + 1/(a*x))^(n/2)*((-1 + a*x)/(1 + a*x))^((-1 + n)/2)*(1 + a*x)^3*Sqrt[c - a*c*x]*Hypergeometric2F1[-
7/2, (-5 + n)/2, -5/2, 2/(1 + a*x)])/(7*a*(1 - 1/(a*x))^(n/2))

Maple [F]

\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{\frac {5}{2}}d x\]

[In]

int(exp(n*arccoth(a*x))*(-a*c*x+c)^(5/2),x)

[Out]

int(exp(n*arccoth(a*x))*(-a*c*x+c)^(5/2),x)

Fricas [F]

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

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(5/2),x, algorithm="fricas")

[Out]

integral((a^2*c^2*x^2 - 2*a*c^2*x + c^2)*sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

Sympy [F(-2)]

Exception generated. \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(exp(n*acoth(a*x))*(-a*c*x+c)**(5/2),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{-1,[0,6,1,0,0]%%%}+%%%{3,[0,4,1,1,0]%%%}+%%%{-3,[0,2,1,2
,0]%%%}+%%%

Mupad [F(-1)]

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

[In]

int(exp(n*acoth(a*x))*(c - a*c*x)^(5/2),x)

[Out]

int(exp(n*acoth(a*x))*(c - a*c*x)^(5/2), x)