\(\int e^{-\coth ^{-1}(a x)} (e x)^m \sqrt {c-a c x} \, dx\) [393]

Optimal result

Integrand size = 25, antiderivative size = 132 \[ \int e^{-\coth ^{-1}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} x (e x)^m \sqrt {c-a c x}}{(3+2 m) \sqrt {1-\frac {1}{a x}}}-\frac {2 (5+4 m) (e x)^m \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-m,\frac {1}{2}-m,-\frac {1}{a x}\right )}{a \left (3+8 m+4 m^2\right ) \sqrt {1-\frac {1}{a x}}} \] Output:

2*(1+1/a/x)^(1/2)*x*(e*x)^m*(-a*c*x+c)^(1/2)/(3+2*m)/(1-1/a/x)^(1/2)-2*(5+ 
4*m)*(e*x)^m*(-a*c*x+c)^(1/2)*hypergeom([1/2, -1/2-m],[1/2-m],-1/a/x)/a/(4 
*m^2+8*m+3)/(1-1/a/x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.79 \[ \int e^{-\coth ^{-1}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {2 (e x)^m \sqrt {c-a c x} \left (a (1+2 m) \sqrt {1+\frac {1}{a x}} x-(5+4 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-m,\frac {1}{2}-m,-\frac {1}{a x}\right )\right )}{a (1+2 m) (3+2 m) \sqrt {1-\frac {1}{a x}}} \] Input:

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

Output:

(2*(e*x)^m*Sqrt[c - a*c*x]*(a*(1 + 2*m)*Sqrt[1 + 1/(a*x)]*x - (5 + 4*m)*Hy 
pergeometric2F1[1/2, -1/2 - m, 1/2 - m, -(1/(a*x))]))/(a*(1 + 2*m)*(3 + 2* 
m)*Sqrt[1 - 1/(a*x)])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6730, 27, 88, 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

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 88

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{m+\frac {1}{2}} \sqrt {c-a c x} (e x)^m \left (-\frac {(4 m+5) \int \frac {\left (\frac {1}{x}\right )^{-m-\frac {3}{2}}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 m+3}-\frac {2 a \sqrt {\frac {1}{a x}+1} \left (\frac {1}{x}\right )^{-m-\frac {3}{2}}}{2 m+3}\right )}{a \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 74

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{m+\frac {1}{2}} \sqrt {c-a c x} (e x)^m \left (\frac {2 (4 m+5) \left (\frac {1}{x}\right )^{-m-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m-\frac {1}{2},\frac {1}{2}-m,-\frac {1}{a x}\right )}{(2 m+1) (2 m+3)}-\frac {2 a \sqrt {\frac {1}{a x}+1} \left (\frac {1}{x}\right )^{-m-\frac {3}{2}}}{2 m+3}\right )}{a \sqrt {1-\frac {1}{a x}}}\)

Input:

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

Output:

-(((x^(-1))^(1/2 + m)*(e*x)^m*Sqrt[c - a*c*x]*((-2*a*Sqrt[1 + 1/(a*x)]*(x^ 
(-1))^(-3/2 - m))/(3 + 2*m) + (2*(5 + 4*m)*(x^(-1))^(-1/2 - m)*Hypergeomet 
ric2F1[1/2, -1/2 - m, 1/2 - m, -(1/(a*x))])/((1 + 2*m)*(3 + 2*m))))/(a*Sqr 
t[1 - 1/(a*x)]))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

Output:

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