Integrand size = 16, antiderivative size = 42 \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx=\frac {x (c x)^m \operatorname {AppellF1}\left (-1-m,\frac {3}{4},-\frac {3}{4},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{1+m} \] Output:
x*(c*x)^m*AppellF1(-1-m,3/4,-3/4,-m,1/a/x,-1/a/x)/(1+m)
\[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx=\int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx \] Input:
Integrate[E^((3*ArcCoth[a*x])/2)*(c*x)^m,x]
Output:
Integrate[E^((3*ArcCoth[a*x])/2)*(c*x)^m, x]
Time = 0.39 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6723, 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.
\(\displaystyle \int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx\) |
\(\Big \downarrow \) 6723 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (c x)^m \int \frac {\left (1+\frac {1}{a x}\right )^{3/4} \left (\frac {1}{x}\right )^{-m-2}}{\left (1-\frac {1}{a x}\right )^{3/4}}d\frac {1}{x}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {x (c x)^m \operatorname {AppellF1}\left (-m-1,\frac {3}{4},-\frac {3}{4},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{m+1}\) |
Input:
Int[E^((3*ArcCoth[a*x])/2)*(c*x)^m,x]
Output:
(x*(c*x)^m*AppellF1[-1 - m, 3/4, -3/4, -m, 1/(a*x), -(1/(a*x))])/(1 + m)
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])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_), x_Symbol] :> Simp[(-(c* x)^m)*(1/x)^m Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, m, n}, x] && !IntegerQ[n] && !IntegerQ[m]
\[\int \frac {\left (x c \right )^{m}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{4}}}d x\]
Input:
int(1/((a*x-1)/(a*x+1))^(3/4)*(x*c)^m,x)
Output:
int(1/((a*x-1)/(a*x+1))^(3/4)*(x*c)^m,x)
\[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx=\int { \frac {\left (c x\right )^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/4)*(c*x)^m,x, algorithm="fricas")
Output:
integral((a*x + 1)*(c*x)^m*((a*x - 1)/(a*x + 1))^(1/4)/(a*x - 1), x)
\[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx=\int \frac {\left (c x\right )^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/4)*(c*x)**m,x)
Output:
Integral((c*x)**m/((a*x - 1)/(a*x + 1))**(3/4), x)
\[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx=\int { \frac {\left (c x\right )^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/4)*(c*x)^m,x, algorithm="maxima")
Output:
integrate((c*x)^m/((a*x - 1)/(a*x + 1))^(3/4), x)
\[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx=\int { \frac {\left (c x\right )^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/4)*(c*x)^m,x, algorithm="giac")
Output:
integrate((c*x)^m/((a*x - 1)/(a*x + 1))^(3/4), x)
Timed out. \[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx=\int \frac {{\left (c\,x\right )}^m}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4}} \,d x \] Input:
int((c*x)^m/((a*x - 1)/(a*x + 1))^(3/4),x)
Output:
int((c*x)^m/((a*x - 1)/(a*x + 1))^(3/4), x)
\[ \int e^{\frac {3}{2} \coth ^{-1}(a x)} (c x)^m \, dx=c^{m} \left (\int \frac {x^{m} \left (a x +1\right )^{\frac {3}{4}}}{\left (a x -1\right )^{\frac {3}{4}}}d x \right ) \] Input:
int(1/((a*x-1)/(a*x+1))^(3/4)*(c*x)^m,x)
Output:
c**m*int((x**m*(a*x + 1)**(3/4))/(a*x - 1)**(3/4),x)