\(\int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} (e x)^m \, dx\) [555]

Optimal result

Integrand size = 29, antiderivative size = 119 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} (e x)^m \, dx=-\frac {2 c \sqrt {1-\frac {1}{a^2 x^2}} x (e x)^m}{(1+2 m) \sqrt {c-\frac {c}{a x}}}+\frac {(3+4 m) \sqrt {c-\frac {c}{a x}} x (e x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-1-m,-m,-\frac {1}{a x}\right )}{(1+m) (1+2 m) \sqrt {1-\frac {1}{a x}}} \] Output:

-2*c*(1-1/a^2/x^2)^(1/2)*x*(e*x)^m/(1+2*m)/(c-c/a/x)^(1/2)+(3+4*m)*(c-c/a/ 
x)^(1/2)*x*(e*x)^m*hypergeom([1/2, -1-m],[-m],-1/a/x)/(1+m)/(1+2*m)/(1-1/a 
/x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.80 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} (e x)^m \, dx=\frac {\sqrt {c-\frac {c}{a x}} (e x)^m \left (2 a m \sqrt {1+\frac {1}{a x}} x-(3+4 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,-\frac {1}{a x}\right )\right )}{2 a m (1+m) \sqrt {1-\frac {1}{a x}}} \] Input:

Integrate[(Sqrt[c - c/(a*x)]*(e*x)^m)/E^ArcCoth[a*x],x]
 

Output:

(Sqrt[c - c/(a*x)]*(e*x)^m*(2*a*m*Sqrt[1 + 1/(a*x)]*x - (3 + 4*m)*Hypergeo 
metric2F1[1/2, -m, 1 - m, -(1/(a*x))]))/(2*a*m*(1 + m)*Sqrt[1 - 1/(a*x)])
 

Rubi [F]

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.

\(\displaystyle \int \sqrt {c-\frac {c}{a x}} e^{-\coth ^{-1}(a x)} (e x)^m \, dx\)

\(\Big \downarrow \) 6736

\(\displaystyle \frac {\sqrt {c-\frac {c}{a x}} \int e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} (e x)^mdx}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 7268

\(\displaystyle \frac {2 \sqrt {c-\frac {c}{a x}} \int a^2 e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) x^2 (e x)^md\sqrt {1-\frac {1}{a x}}}{a \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 2044

\(\displaystyle \frac {2 \sqrt {c-\frac {c}{a x}} \left (\frac {1}{a x}\right )^m (e x)^m \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) \left (\frac {1}{a x}\right )^{-m-2}d\sqrt {1-\frac {1}{a x}}}{a \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {2 \sqrt {c-\frac {c}{a x}} \left (\frac {1}{a x}\right )^m (e x)^m \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) \left (\frac {1}{a x}\right )^{-m-2}d\sqrt {1-\frac {1}{a x}}}{a \sqrt {1-\frac {1}{a x}}}\)

Input:

Int[(Sqrt[c - c/(a*x)]*(e*x)^m)/E^ArcCoth[a*x],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2044

Int[(u_.)*((c_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[S 
imp[(c*(a + b*x^n)^q)^p/(a + b*x^n)^(p*q)]   Int[u*(a + b*x^n)^(p*q), x], x 
] /; FreeQ[{a, b, c, n, p, q}, x] && GeQ[a, 0]
 

rule 6736

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] 
 :> Simp[(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] &&  !Int 
egerQ[n/2] &&  !(IntegerQ[p] || GtQ[c, 0])
 

rule 7268

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears 
[u, x]}, Simp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls 
t[[2]])], x] /;  !FalseQ[lst]]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \sqrt {c -\frac {c}{a x}}\, \left (e x \right )^{m} \sqrt {\frac {a x -1}{a x +1}}d x\]

Input:

int((c-c/a/x)^(1/2)*(e*x)^m*((a*x-1)/(a*x+1))^(1/2),x)
 

Output:

int((c-c/a/x)^(1/2)*(e*x)^m*((a*x-1)/(a*x+1))^(1/2),x)
 

Fricas [F]

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

integrate((c-c/a/x)^(1/2)*(e*x)^m*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fr 
icas")
 

Output:

integral((e*x)^m*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)), x)
 

Sympy [F(-1)]

Timed out. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} (e x)^m \, dx=\text {Timed out} \] Input:

integrate((c-c/a/x)**(1/2)*(e*x)**m*((a*x-1)/(a*x+1))**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((c-c/a/x)^(1/2)*(e*x)^m*((a*x-1)/(a*x+1))^(1/2),x, algorithm="ma 
xima")
 

Output:

integrate((e*x)^m*sqrt(c - c/(a*x))*sqrt((a*x - 1)/(a*x + 1)), x)
 

Giac [F]

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

integrate((c-c/a/x)^(1/2)*(e*x)^m*((a*x-1)/(a*x+1))^(1/2),x, algorithm="gi 
ac")
 

Output:

integrate((e*x)^m*sqrt(c - c/(a*x))*sqrt((a*x - 1)/(a*x + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} (e x)^m \, dx=\int \sqrt {c-\frac {c}{a\,x}}\,{\left (e\,x\right )}^m\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \] Input:

int((c - c/(a*x))^(1/2)*(e*x)^m*((a*x - 1)/(a*x + 1))^(1/2),x)
 

Output:

int((c - c/(a*x))^(1/2)*(e*x)^m*((a*x - 1)/(a*x + 1))^(1/2), x)
 

Reduce [F]

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

int((c-c/a/x)^(1/2)*(e*x)^m*((a*x-1)/(a*x+1))^(1/2),x)
 

Output:

(e**m*sqrt(c)*( - int(x**m/(sqrt(x)*sqrt(a*x + 1)),x) + int((x**m*x)/(sqrt 
(x)*sqrt(a*x + 1)),x)*a))/sqrt(a)