\(\int e^{-2 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx\) [485]

Optimal result

Integrand size = 25, antiderivative size = 55 \[ \int e^{-2 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {(e x)^{1+m} \sqrt {c-a c x} \operatorname {AppellF1}\left (1+m,-\frac {3}{2},1,2+m,a x,-a x\right )}{e (1+m) \sqrt {1-a x}} \] Output:

(e*x)^(1+m)*(-a*c*x+c)^(1/2)*AppellF1(1+m,-3/2,1,2+m,a*x,-a*x)/e/(1+m)/(-a 
*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.25 \[ \int e^{-2 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {x (e x)^m \sqrt {c-a c x} \left (2 \operatorname {AppellF1}\left (1+m,-\frac {1}{2},1,2+m,a x,-a x\right )-\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1+m,2+m,a x\right )\right )}{(1+m) \sqrt {1-a x}} \] Input:

Integrate[((e*x)^m*Sqrt[c - a*c*x])/E^(2*ArcTanh[a*x]),x]
 

Output:

(x*(e*x)^m*Sqrt[c - a*c*x]*(2*AppellF1[1 + m, -1/2, 1, 2 + m, a*x, -(a*x)] 
 - Hypergeometric2F1[-1/2, 1 + m, 2 + m, a*x]))/((1 + m)*Sqrt[1 - a*x])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 55, 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 = {6680, 35, 152, 150}

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.

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

\(\Big \downarrow \) 6680

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

\(\Big \downarrow \) 35

\(\displaystyle \frac {\int \frac {(e x)^m (c-a c x)^{3/2}}{a x+1}dx}{c}\)

\(\Big \downarrow \) 152

\(\displaystyle \frac {\sqrt {c-a c x} \int \frac {(e x)^m (1-a x)^{3/2}}{a x+1}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {\sqrt {c-a c x} (e x)^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {3}{2},1,m+2,a x,-a x\right )}{e (m+1) \sqrt {1-a x}}\)

Input:

Int[((e*x)^m*Sqrt[c - a*c*x])/E^(2*ArcTanh[a*x]),x]
 

Output:

((e*x)^(1 + m)*Sqrt[c - a*c*x]*AppellF1[1 + m, -3/2, 1, 2 + m, a*x, -(a*x) 
])/(e*(1 + m)*Sqrt[1 - a*x])
 

Defintions of rubi rules used

rule 35

Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} 
, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] &&  !(IntegerQ[n] && SimplerQ[a + 
b*x, c + d*x])
 

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

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

rule 6680

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{-2 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {2 e^{m} \sqrt {c}\, \left (-x^{m} \sqrt {-a x +1}\, a x -4 x^{m} \sqrt {-a x +1}\, m -5 x^{m} \sqrt {-a x +1}+8 \left (\int \frac {x^{m} \sqrt {-a x +1}\, x}{2 a^{2} m \,x^{2}+3 a^{2} x^{2}-2 m -3}d x \right ) a^{2} m^{3}+30 \left (\int \frac {x^{m} \sqrt {-a x +1}\, x}{2 a^{2} m \,x^{2}+3 a^{2} x^{2}-2 m -3}d x \right ) a^{2} m^{2}+39 \left (\int \frac {x^{m} \sqrt {-a x +1}\, x}{2 a^{2} m \,x^{2}+3 a^{2} x^{2}-2 m -3}d x \right ) a^{2} m +18 \left (\int \frac {x^{m} \sqrt {-a x +1}\, x}{2 a^{2} m \,x^{2}+3 a^{2} x^{2}-2 m -3}d x \right ) a^{2}-8 \left (\int \frac {x^{m} \sqrt {-a x +1}}{2 a^{2} m \,x^{3}+3 a^{2} x^{3}-2 m x -3 x}d x \right ) m^{3}-22 \left (\int \frac {x^{m} \sqrt {-a x +1}}{2 a^{2} m \,x^{3}+3 a^{2} x^{3}-2 m x -3 x}d x \right ) m^{2}-15 \left (\int \frac {x^{m} \sqrt {-a x +1}}{2 a^{2} m \,x^{3}+3 a^{2} x^{3}-2 m x -3 x}d x \right ) m \right )}{a \left (2 m +3\right )} \] Input:

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

Output:

(2*e**m*sqrt(c)*( - x**m*sqrt( - a*x + 1)*a*x - 4*x**m*sqrt( - a*x + 1)*m 
- 5*x**m*sqrt( - a*x + 1) + 8*int((x**m*sqrt( - a*x + 1)*x)/(2*a**2*m*x**2 
 + 3*a**2*x**2 - 2*m - 3),x)*a**2*m**3 + 30*int((x**m*sqrt( - a*x + 1)*x)/ 
(2*a**2*m*x**2 + 3*a**2*x**2 - 2*m - 3),x)*a**2*m**2 + 39*int((x**m*sqrt( 
- a*x + 1)*x)/(2*a**2*m*x**2 + 3*a**2*x**2 - 2*m - 3),x)*a**2*m + 18*int(( 
x**m*sqrt( - a*x + 1)*x)/(2*a**2*m*x**2 + 3*a**2*x**2 - 2*m - 3),x)*a**2 - 
 8*int((x**m*sqrt( - a*x + 1))/(2*a**2*m*x**3 + 3*a**2*x**3 - 2*m*x - 3*x) 
,x)*m**3 - 22*int((x**m*sqrt( - a*x + 1))/(2*a**2*m*x**3 + 3*a**2*x**3 - 2 
*m*x - 3*x),x)*m**2 - 15*int((x**m*sqrt( - a*x + 1))/(2*a**2*m*x**3 + 3*a* 
*2*x**3 - 2*m*x - 3*x),x)*m))/(a*(2*m + 3))