\(\int e^{\text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx\) [480]

Optimal result

Integrand size = 23, antiderivative size = 52 \[ \int e^{\text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {c (e x)^{1+m} \sqrt {1-a x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1+m,2+m,-a x\right )}{e (1+m) \sqrt {c-a c x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90 \[ \int e^{\text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {x (e x)^m \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1+m,2+m,-a x\right )}{(1+m) \sqrt {1-a x}} \] Input:

Integrate[E^ArcTanh[a*x]*(e*x)^m*Sqrt[c - a*c*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 = 0.53 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6678, 585, 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.

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 585

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

\(\Big \downarrow \) 74

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

Input:

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

Output:

(c*(e*x)^(1 + m)*Sqrt[1 - a*x]*Hypergeometric2F1[-1/2, 1 + m, 2 + m, -(a*x 
)])/(e*(1 + m)*Sqrt[c - a*c*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 585

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) 
, x_Symbol] :> Simp[a^p*c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^F 
racPart[n])   Int[(e*x)^m*(1 - d*(x/c))^p*(1 + d*(x/c))^(n + p), x], x] /; 
FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[a, 0]
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{\text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {2 e^{m} \sqrt {c}\, \left (2 x^{m} \sqrt {a x +1}\, a m x +x^{m} \sqrt {a x +1}\, a x +x^{m} \sqrt {a x +1}+8 \left (\int \frac {x^{m} \sqrt {a x +1}}{4 a \,m^{2} x^{2}+8 a m \,x^{2}+3 a \,x^{2}+4 m^{2} x +8 m x +3 x}d x \right ) m^{4}+24 \left (\int \frac {x^{m} \sqrt {a x +1}}{4 a \,m^{2} x^{2}+8 a m \,x^{2}+3 a \,x^{2}+4 m^{2} x +8 m x +3 x}d x \right ) m^{3}+22 \left (\int \frac {x^{m} \sqrt {a x +1}}{4 a \,m^{2} x^{2}+8 a m \,x^{2}+3 a \,x^{2}+4 m^{2} x +8 m x +3 x}d x \right ) m^{2}+6 \left (\int \frac {x^{m} \sqrt {a x +1}}{4 a \,m^{2} x^{2}+8 a m \,x^{2}+3 a \,x^{2}+4 m^{2} x +8 m x +3 x}d x \right ) m -4 \left (\int \frac {x^{m} \sqrt {a x +1}}{2 a m \,x^{2}+a \,x^{2}+2 m x +x}d x \right ) m^{3}-8 \left (\int \frac {x^{m} \sqrt {a x +1}}{2 a m \,x^{2}+a \,x^{2}+2 m x +x}d x \right ) m^{2}-3 \left (\int \frac {x^{m} \sqrt {a x +1}}{2 a m \,x^{2}+a \,x^{2}+2 m x +x}d x \right ) m \right )}{a \left (4 m^{2}+8 m +3\right )} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(e*x)^m*(-a*c*x+c)^(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) + 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**4 + 24*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 + 
 22*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 + 6*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 - 4*int((x**m*sqrt(a* 
x + 1))/(2*a*m*x**2 + a*x**2 + 2*m*x + x),x)*m**3 - 8*int((x**m*sqrt(a*x + 
 1))/(2*a*m*x**2 + a*x**2 + 2*m*x + x),x)*m**2 - 3*int((x**m*sqrt(a*x + 1) 
)/(2*a*m*x**2 + a*x**2 + 2*m*x + x),x)*m))/(a*(4*m**2 + 8*m + 3))