\(\int e^{-3 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx\) [482]

Optimal result

Integrand size = 25, antiderivative size = 161 \[ \int e^{-3 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {8 (e x)^{1+m} \sqrt {c-a c x}}{e \sqrt {1-a x} \sqrt {1+a x}}+\frac {2 (e x)^{1+m} \sqrt {1+a x} \sqrt {c-a c x}}{e (3+2 m) \sqrt {1-a x}}-\frac {\left (23+40 m+16 m^2\right ) (e x)^{1+m} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,2+m,-a x\right )}{e (1+m) (3+2 m) \sqrt {1-a x}} \] Output:

8*(e*x)^(1+m)*(-a*c*x+c)^(1/2)/e/(-a*x+1)^(1/2)/(a*x+1)^(1/2)+2*(e*x)^(1+m 
)*(a*x+1)^(1/2)*(-a*c*x+c)^(1/2)/e/(3+2*m)/(-a*x+1)^(1/2)-(16*m^2+40*m+23) 
*(e*x)^(1+m)*(-a*c*x+c)^(1/2)*hypergeom([1/2, 1+m],[2+m],-a*x)/e/(1+m)/(3+ 
2*m)/(-a*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.61 \[ \int e^{-3 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\frac {c x (e x)^m \sqrt {1-a x} \left (2 (1+m) (13+8 m+a x)-\left (23+40 m+16 m^2\right ) \sqrt {1+a x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,2+m,-a x\right )\right )}{(1+m) (3+2 m) \sqrt {1+a x} \sqrt {c-a c x}} \] Input:

Integrate[((e*x)^m*Sqrt[c - a*c*x])/E^(3*ArcTanh[a*x]),x]
 

Output:

(c*x*(e*x)^m*Sqrt[1 - a*x]*(2*(1 + m)*(13 + 8*m + a*x) - (23 + 40*m + 16*m 
^2)*Sqrt[1 + a*x]*Hypergeometric2F1[1/2, 1 + m, 2 + m, -(a*x)]))/((1 + m)* 
(3 + 2*m)*Sqrt[1 + a*x]*Sqrt[c - a*c*x])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.75, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6680, 37, 100, 27, 90, 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.

\(\displaystyle \int e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x} (e x)^m \, dx\)

\(\Big \downarrow \) 6680

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

\(\Big \downarrow \) 37

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {8 (e x)^{m+1}}{e \sqrt {a x+1}}-\frac {2 \int \frac {a^2 e (e x)^m (8 m-a x+7)}{2 \sqrt {a x+1}}dx}{a^2 e}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {8 (e x)^{m+1}}{e \sqrt {a x+1}}-\int \frac {(e x)^m (8 m-a x+7)}{\sqrt {a x+1}}dx\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {\left (16 m^2+40 m+23\right ) \int \frac {(e x)^m}{\sqrt {a x+1}}dx}{2 m+3}+\frac {2 \sqrt {a x+1} (e x)^{m+1}}{e (2 m+3)}+\frac {8 (e x)^{m+1}}{e \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {\left (16 m^2+40 m+23\right ) (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},m+1,m+2,-a x\right )}{e (m+1) (2 m+3)}+\frac {2 \sqrt {a x+1} (e x)^{m+1}}{e (2 m+3)}+\frac {8 (e x)^{m+1}}{e \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\)

Input:

Int[((e*x)^m*Sqrt[c - a*c*x])/E^(3*ArcTanh[a*x]),x]
 

Output:

(Sqrt[c - a*c*x]*((8*(e*x)^(1 + m))/(e*Sqrt[1 + a*x]) + (2*(e*x)^(1 + m)*S 
qrt[1 + a*x])/(e*(3 + 2*m)) - ((23 + 40*m + 16*m^2)*(e*x)^(1 + m)*Hypergeo 
metric2F1[1/2, 1 + m, 2 + m, -(a*x)])/(e*(1 + m)*(3 + 2*m))))/Sqrt[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 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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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]

\[\int \frac {\left (e x \right )^{m} \sqrt {-a c x +c}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (a x +1\right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^m*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorit 
hm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int e^{-3 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

integrate((e*x)^m*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorit 
hm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{-3 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x)^m*(-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorit 
hm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{-3 \text {arctanh}(a x)} (e x)^m \sqrt {c-a c x} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*e**m*sqrt(c)*(4*x**m*sqrt(a*x + 1)*a**2*m**2*x**2 - x**m*sqrt(a*x + 1)* 
a**2*x**2 - 12*x**m*sqrt(a*x + 1)*a*m**2*x - 14*x**m*sqrt(a*x + 1)*a*m*x + 
 10*x**m*sqrt(a*x + 1)*a*x + 16*x**m*sqrt(a*x + 1)*m**2 + 40*x**m*sqrt(a*x 
 + 1)*m + 23*x**m*sqrt(a*x + 1) - 128*int((x**m*sqrt(a*x + 1))/(8*a**2*m** 
3*x**3 + 12*a**2*m**2*x**3 - 2*a**2*m*x**3 - 3*a**2*x**3 + 16*a*m**3*x**2 
+ 24*a*m**2*x**2 - 4*a*m*x**2 - 6*a*x**2 + 8*m**3*x + 12*m**2*x - 2*m*x - 
3*x),x)*a*m**6*x - 512*int((x**m*sqrt(a*x + 1))/(8*a**2*m**3*x**3 + 12*a** 
2*m**2*x**3 - 2*a**2*m*x**3 - 3*a**2*x**3 + 16*a*m**3*x**2 + 24*a*m**2*x** 
2 - 4*a*m*x**2 - 6*a*x**2 + 8*m**3*x + 12*m**2*x - 2*m*x - 3*x),x)*a*m**5* 
x - 632*int((x**m*sqrt(a*x + 1))/(8*a**2*m**3*x**3 + 12*a**2*m**2*x**3 - 2 
*a**2*m*x**3 - 3*a**2*x**3 + 16*a*m**3*x**2 + 24*a*m**2*x**2 - 4*a*m*x**2 
- 6*a*x**2 + 8*m**3*x + 12*m**2*x - 2*m*x - 3*x),x)*a*m**4*x - 148*int((x* 
*m*sqrt(a*x + 1))/(8*a**2*m**3*x**3 + 12*a**2*m**2*x**3 - 2*a**2*m*x**3 - 
3*a**2*x**3 + 16*a*m**3*x**2 + 24*a*m**2*x**2 - 4*a*m*x**2 - 6*a*x**2 + 8* 
m**3*x + 12*m**2*x - 2*m*x - 3*x),x)*a*m**3*x + 166*int((x**m*sqrt(a*x + 1 
))/(8*a**2*m**3*x**3 + 12*a**2*m**2*x**3 - 2*a**2*m*x**3 - 3*a**2*x**3 + 1 
6*a*m**3*x**2 + 24*a*m**2*x**2 - 4*a*m*x**2 - 6*a*x**2 + 8*m**3*x + 12*m** 
2*x - 2*m*x - 3*x),x)*a*m**2*x + 69*int((x**m*sqrt(a*x + 1))/(8*a**2*m**3* 
x**3 + 12*a**2*m**2*x**3 - 2*a**2*m*x**3 - 3*a**2*x**3 + 16*a*m**3*x**2 + 
24*a*m**2*x**2 - 4*a*m*x**2 - 6*a*x**2 + 8*m**3*x + 12*m**2*x - 2*m*x -...