\(\int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx\) [667]

Optimal result

Integrand size = 25, antiderivative size = 57 \[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}+m,\frac {3}{2}+m,-a x\right )}{(1+2 m) \sqrt {1-a x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}+m,\frac {3}{2}+m,-a x\right )}{\left (\frac {1}{2}+m\right ) \sqrt {1-a x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6684, 6678, 516, 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.

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

\(\Big \downarrow \) 6684

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int e^{\text {arctanh}(a x)} x^{m-\frac {1}{2}} \sqrt {1-a x}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 6678

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {x^{m-\frac {1}{2}} \sqrt {1-a^2 x^2}}{\sqrt {1-a x}}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 516

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int x^{m-\frac {1}{2}} \sqrt {a x+1}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 74

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

Input:

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

Output:

(2*Sqrt[c - c/(a*x)]*x^(1 + m)*Hypergeometric2F1[-1/2, 1/2 + m, 3/2 + m, - 
(a*x)])/((1 + 2*m)*Sqrt[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 516

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free 
Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || 
(GtQ[a, 0] && GtQ[c, 0] &&  !IntegerQ[n]))
 

rule 6678

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

rule 6684

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

Maple [F]

\[\int \frac {\left (a x +1\right ) \sqrt {c -\frac {c}{a x}}\, x^{m}}{\sqrt {-a^{2} x^{2}+1}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x^{m}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x^{m}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x^{m}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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