\(\int e^{\frac {3}{2} \text {arctanh}(a x)} (c x)^m \, dx\) [141]

Optimal result

Integrand size = 16, antiderivative size = 36 \[ \int e^{\frac {3}{2} \text {arctanh}(a x)} (c x)^m \, dx=\frac {(c x)^{1+m} \operatorname {AppellF1}\left (1+m,\frac {3}{4},-\frac {3}{4},2+m,a x,-a x\right )}{c (1+m)} \] Output:

(c*x)^(1+m)*AppellF1(1+m,3/4,-3/4,2+m,a*x,-a*x)/c/(1+m)
 

Mathematica [F]

\[ \int e^{\frac {3}{2} \text {arctanh}(a x)} (c x)^m \, dx=\int e^{\frac {3}{2} \text {arctanh}(a x)} (c x)^m \, dx \] Input:

Integrate[E^((3*ArcTanh[a*x])/2)*(c*x)^m,x]
 

Output:

Integrate[E^((3*ArcTanh[a*x])/2)*(c*x)^m, x]
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6676, 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.

\(\displaystyle \int e^{\frac {3}{2} \text {arctanh}(a x)} (c x)^m \, dx\)

\(\Big \downarrow \) 6676

\(\displaystyle \int \frac {(a x+1)^{3/4} (c x)^m}{(1-a x)^{3/4}}dx\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {(c x)^{m+1} \operatorname {AppellF1}\left (m+1,\frac {3}{4},-\frac {3}{4},m+2,a x,-a x\right )}{c (m+1)}\)

Input:

Int[E^((3*ArcTanh[a*x])/2)*(c*x)^m,x]
 

Output:

((c*x)^(1 + m)*AppellF1[1 + m, 3/4, -3/4, 2 + m, a*x, -(a*x)])/(c*(1 + m))
 

Defintions of rubi rules used

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 6676

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) 
^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] &&  !Int 
egerQ[(n - 1)/2]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int e^{\frac {3}{2} \text {arctanh}(a x)} (c x)^m \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(c**m*( - 2*x**m*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(1/4) - 2*int((x**m*(a* 
x + 1)**(3/4)*( - a*x + 1)**(1/4))/(a**2*m*x**3 + a**2*x**3 - m*x - x),x)* 
m**2 - 2*int((x**m*(a*x + 1)**(3/4)*( - a*x + 1)**(1/4))/(a**2*m*x**3 + a* 
*2*x**3 - m*x - x),x)*m - 2*int((x**m*(a*x + 1)**(3/4)*( - a*x + 1)**(1/4) 
)/(a**2*m*x**2 + a**2*x**2 - m - 1),x)*a*m**2 - 5*int((x**m*(a*x + 1)**(3/ 
4)*( - a*x + 1)**(1/4))/(a**2*m*x**2 + a**2*x**2 - m - 1),x)*a*m - 3*int(( 
x**m*(a*x + 1)**(3/4)*( - a*x + 1)**(1/4))/(a**2*m*x**2 + a**2*x**2 - m - 
1),x)*a))/(2*a*(m + 1))