\(\int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^2 \, dx\) [383]

Optimal result

Integrand size = 21, antiderivative size = 94 \[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^2 \, dx=\frac {a^2 c^2 x^3 (e x)^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-3-m),\frac {1}{2} (-1-m),\frac {1}{a^2 x^2}\right )}{3+m}-\frac {a c^2 x^2 (e x)^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-2-m),-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{2+m} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.93 \[ \int e^{\coth ^{-1}(a x)} (e x)^m (c-a c x)^2 \, dx=\frac {a c^2 x^2 (e x)^m \left (a (2+m) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2}-\frac {m}{2},-\frac {1}{2}-\frac {m}{2},\frac {1}{a^2 x^2}\right )-(3+m) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-1-\frac {m}{2},-\frac {m}{2},\frac {1}{a^2 x^2}\right )\right )}{(2+m) (3+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6729, 92, 135, 278}

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.

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

\(\Big \downarrow \) 6729

\(\displaystyle -a^2 c^2 \left (\frac {1}{x}\right )^m (e x)^m \int \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{-m-4}d\frac {1}{x}\)

\(\Big \downarrow \) 92

\(\displaystyle -a^2 c^2 \left (\frac {1}{x}\right )^m (e x)^m \left (\int \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{-m-4}d\frac {1}{x}-\frac {\int \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{-m-3}d\frac {1}{x}}{a}\right )\)

\(\Big \downarrow \) 135

\(\displaystyle -a^2 c^2 \left (\frac {1}{x}\right )^m (e x)^m \left (\int \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {1}{x}\right )^{-m-4}d\frac {1}{x}-\frac {\int \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {1}{x}\right )^{-m-3}d\frac {1}{x}}{a}\right )\)

\(\Big \downarrow \) 278

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

Input:

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

Output:

-(a^2*c^2*(x^(-1))^m*(e*x)^m*(-(((x^(-1))^(-3 - m)*Hypergeometric2F1[-1/2, 
 (-3 - m)/2, (-1 - m)/2, 1/(a^2*x^2)])/(3 + m)) + ((x^(-1))^(-2 - m)*Hyper 
geometric2F1[-1/2, (-2 - m)/2, -1/2*m, 1/(a^2*x^2)])/(a*(2 + m))))
 

Defintions of rubi rules used

rule 92

Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), 
x_] :> Simp[a   Int[(a + b*x)^n*(c + d*x)^n*(f*x)^p, x], x] + Simp[b/f   In 
t[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, f, m, 
 n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] &&  !RationalQ[p] && 
!IGtQ[m, 0] && NeQ[m + n + p + 2, 0]
 

rule 135

Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), 
x_] :> Int[(a*c + b*d*x^2)^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, 
 x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && GtQ[a, 0] && GtQ[c, 0]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 6729

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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