Integrand size = 17, antiderivative size = 42 \[ \int e^{4 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {(c+a c x)^{3+p} \operatorname {Hypergeometric2F1}\left (2,3+p,4+p,\frac {1}{2} (1+a x)\right )}{4 a c^3 (3+p)} \] Output:
1/4*(a*c*x+c)^(3+p)*hypergeom([2, 3+p],[4+p],1/2*a*x+1/2)/a/c^3/(3+p)
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.55 \[ \int e^{4 \text {arctanh}(a x)} (c+a c x)^p \, dx=-\frac {(1+a x) (c+a c x)^p \left (3+2 p-a x+(2+p) (-1+a x) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {1}{2} (1+a x)\right )\right )}{a (1+p) (-1+a x)} \] Input:
Integrate[E^(4*ArcTanh[a*x])*(c + a*c*x)^p,x]
Output:
-(((1 + a*x)*(c + a*c*x)^p*(3 + 2*p - a*x + (2 + p)*(-1 + a*x)*Hypergeomet ric2F1[1, 1 + p, 2 + p, (1 + a*x)/2]))/(a*(1 + p)*(-1 + a*x)))
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6680, 35, 78}
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.
\(\displaystyle \int e^{4 \text {arctanh}(a x)} (a c x+c)^p \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(a x+1)^2 (a c x+c)^p}{(1-a x)^2}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {\int \frac {(a x c+c)^{p+2}}{(1-a x)^2}dx}{c^2}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {(a c x+c)^{p+3} \operatorname {Hypergeometric2F1}\left (2,p+3,p+4,\frac {1}{2} (a x+1)\right )}{4 a c^3 (p+3)}\) |
Input:
Int[E^(4*ArcTanh[a*x])*(c + a*c*x)^p,x]
Output:
((c + a*c*x)^(3 + p)*Hypergeometric2F1[2, 3 + p, 4 + p, (1 + a*x)/2])/(4*a *c^3*(3 + p))
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
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])
\[\int \frac {\left (a x +1\right )^{4} \left (a c x +c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{2}}d x\]
Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(a*c*x+c)^p,x)
Output:
int((a*x+1)^4/(-a^2*x^2+1)^2*(a*c*x+c)^p,x)
\[ \int e^{4 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (a c x + c\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(a*c*x+c)^p,x, algorithm="fricas")
Output:
integral((a^2*x^2 + 2*a*x + 1)*(a*c*x + c)^p/(a^2*x^2 - 2*a*x + 1), x)
\[ \int e^{4 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int \frac {\left (c \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \] Input:
integrate((a*x+1)**4/(-a**2*x**2+1)**2*(a*c*x+c)**p,x)
Output:
Integral((c*(a*x + 1))**p*(a*x + 1)**2/(a*x - 1)**2, x)
\[ \int e^{4 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (a c x + c\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(a*c*x+c)^p,x, algorithm="maxima")
Output:
integrate((a*x + 1)^4*(a*c*x + c)^p/(a^2*x^2 - 1)^2, x)
\[ \int e^{4 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (a c x + c\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(a*c*x+c)^p,x, algorithm="giac")
Output:
integrate((a*x + 1)^4*(a*c*x + c)^p/(a^2*x^2 - 1)^2, x)
Timed out. \[ \int e^{4 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int \frac {{\left (c+a\,c\,x\right )}^p\,{\left (a\,x+1\right )}^4}{{\left (a^2\,x^2-1\right )}^2} \,d x \] Input:
int(((c + a*c*x)^p*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)
Output:
int(((c + a*c*x)^p*(a*x + 1)^4)/(a^2*x^2 - 1)^2, x)
\[ \int e^{4 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {\left (a c x +c \right )^{p} a^{2} p \,x^{2}+4 \left (a c x +c \right )^{p} a p x +4 \left (a c x +c \right )^{p} a x -\left (a c x +c \right )^{p} p -4 \left (a c x +c \right )^{p}+8 \left (\int \frac {\left (a c x +c \right )^{p} x}{a^{3} p \,x^{3}+a^{3} x^{3}-a^{2} p \,x^{2}-a^{2} x^{2}-a p x -a x +p +1}d x \right ) a^{3} p^{3} x +24 \left (\int \frac {\left (a c x +c \right )^{p} x}{a^{3} p \,x^{3}+a^{3} x^{3}-a^{2} p \,x^{2}-a^{2} x^{2}-a p x -a x +p +1}d x \right ) a^{3} p^{2} x +16 \left (\int \frac {\left (a c x +c \right )^{p} x}{a^{3} p \,x^{3}+a^{3} x^{3}-a^{2} p \,x^{2}-a^{2} x^{2}-a p x -a x +p +1}d x \right ) a^{3} p x -8 \left (\int \frac {\left (a c x +c \right )^{p} x}{a^{3} p \,x^{3}+a^{3} x^{3}-a^{2} p \,x^{2}-a^{2} x^{2}-a p x -a x +p +1}d x \right ) a^{2} p^{3}-24 \left (\int \frac {\left (a c x +c \right )^{p} x}{a^{3} p \,x^{3}+a^{3} x^{3}-a^{2} p \,x^{2}-a^{2} x^{2}-a p x -a x +p +1}d x \right ) a^{2} p^{2}-16 \left (\int \frac {\left (a c x +c \right )^{p} x}{a^{3} p \,x^{3}+a^{3} x^{3}-a^{2} p \,x^{2}-a^{2} x^{2}-a p x -a x +p +1}d x \right ) a^{2} p}{a p \left (a p x +a x -p -1\right )} \] Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(a*c*x+c)^p,x)
Output:
((a*c*x + c)**p*a**2*p*x**2 + 4*(a*c*x + c)**p*a*p*x + 4*(a*c*x + c)**p*a* x - (a*c*x + c)**p*p - 4*(a*c*x + c)**p + 8*int(((a*c*x + c)**p*x)/(a**3*p *x**3 + a**3*x**3 - a**2*p*x**2 - a**2*x**2 - a*p*x - a*x + p + 1),x)*a**3 *p**3*x + 24*int(((a*c*x + c)**p*x)/(a**3*p*x**3 + a**3*x**3 - a**2*p*x**2 - a**2*x**2 - a*p*x - a*x + p + 1),x)*a**3*p**2*x + 16*int(((a*c*x + c)** p*x)/(a**3*p*x**3 + a**3*x**3 - a**2*p*x**2 - a**2*x**2 - a*p*x - a*x + p + 1),x)*a**3*p*x - 8*int(((a*c*x + c)**p*x)/(a**3*p*x**3 + a**3*x**3 - a** 2*p*x**2 - a**2*x**2 - a*p*x - a*x + p + 1),x)*a**2*p**3 - 24*int(((a*c*x + c)**p*x)/(a**3*p*x**3 + a**3*x**3 - a**2*p*x**2 - a**2*x**2 - a*p*x - a* x + p + 1),x)*a**2*p**2 - 16*int(((a*c*x + c)**p*x)/(a**3*p*x**3 + a**3*x* *3 - a**2*p*x**2 - a**2*x**2 - a*p*x - a*x + p + 1),x)*a**2*p)/(a*p*(a*p*x + a*x - p - 1))