\(\int e^{-\text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx\) [458]

Optimal result

Integrand size = 24, antiderivative size = 65 \[ \int e^{-\text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx=-\frac {2 c (a x)^{-m} (e x)^m (1-a x)^{3/2} \sqrt {1+a x} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-m,\frac {5}{2},1-a x\right )}{3 a \sqrt {c+a c x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.68 \[ \int e^{-\text {arctanh}(a x)} (e x)^m \sqrt {c+a c x} \, dx=\frac {x (e x)^m \sqrt {c+a c x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1+m,2+m,a x\right )}{(1+m) \sqrt {1+a x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6680, 37, 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.

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

\(\Big \downarrow \) 6680

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

\(\Big \downarrow \) 37

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

\(\Big \downarrow \) 74

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 37

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*e**m*sqrt(c)*(2*x**m*sqrt( - a*x + 1)*a*m*x + x**m*sqrt( - a*x + 1)*a*x 
 - x**m*sqrt( - a*x + 1) - 4*int((x**m*sqrt( - a*x + 1))/(4*a*m**2*x**2 + 
8*a*m*x**2 + 3*a*x**2 - 4*m**2*x - 8*m*x - 3*x),x)*m**3 - 8*int((x**m*sqrt 
( - a*x + 1))/(4*a*m**2*x**2 + 8*a*m*x**2 + 3*a*x**2 - 4*m**2*x - 8*m*x - 
3*x),x)*m**2 - 3*int((x**m*sqrt( - a*x + 1))/(4*a*m**2*x**2 + 8*a*m*x**2 + 
 3*a*x**2 - 4*m**2*x - 8*m*x - 3*x),x)*m))/(a*(4*m**2 + 8*m + 3))