\(\int e^{\coth ^{-1}(a x)} (e x)^m \sqrt {c-a c x} \, dx\) [390]

Optimal result

Integrand size = 23, antiderivative size = 66 \[ \int e^{\coth ^{-1}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {2 x (e x)^m \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2}-m,-\frac {1}{2}-m,-\frac {1}{a x}\right )}{(3+2 m) \sqrt {1-\frac {1}{a x}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int e^{\coth ^{-1}(a x)} (e x)^m \sqrt {c-a c x} \, dx=-\frac {x (e x)^m \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2}-m,-\frac {1}{2}-m,-\frac {1}{a x}\right )}{\left (-\frac {3}{2}-m\right ) \sqrt {1-\frac {1}{a x}}} \] Input:

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

Output:

-((x*(e*x)^m*Sqrt[c - a*c*x]*Hypergeometric2F1[-1/2, -3/2 - m, -1/2 - m, - 
(1/(a*x))])/((-3/2 - m)*Sqrt[1 - 1/(a*x)]))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6730, 74}

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.

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

\(\Big \downarrow \) 6730

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{m+\frac {1}{2}} \sqrt {c-a c x} (e x)^m \int \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{-m-\frac {5}{2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {2 x \sqrt {c-a c x} (e x)^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-m-\frac {3}{2},-m-\frac {1}{2},-\frac {1}{a x}\right )}{(2 m+3) \sqrt {1-\frac {1}{a x}}}\)

Input:

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

Output:

(2*x*(e*x)^m*Sqrt[c - a*c*x]*Hypergeometric2F1[-1/2, -3/2 - m, -1/2 - m, - 
(1/(a*x))])/((3 + 2*m)*Sqrt[1 - 1/(a*x)])
 

Defintions of rubi rules used

rule 74

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

rule 6730

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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