Integrand size = 17, antiderivative size = 42 \[ \int e^{2 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {(c+a c x)^{2+p} \operatorname {Hypergeometric2F1}\left (1,2+p,3+p,\frac {1}{2} (1+a x)\right )}{2 a c^2 (2+p)} \] Output:
1/2*(a*c*x+c)^(2+p)*hypergeom([1, 2+p],[3+p],1/2*a*x+1/2)/a/c^2/(2+p)
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98 \[ \int e^{2 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {(1+a x) (c+a c x)^p \left (-1+\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {1}{2} (1+a x)\right )\right )}{a (1+p)} \] Input:
Integrate[E^(2*ArcTanh[a*x])*(c + a*c*x)^p,x]
Output:
((1 + a*x)*(c + a*c*x)^p*(-1 + Hypergeometric2F1[1, 1 + p, 2 + p, (1 + a*x )/2]))/(a*(1 + p))
Time = 0.37 (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.
\(\displaystyle \int e^{2 \text {arctanh}(a x)} (a c x+c)^p \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(a x+1) (a c x+c)^p}{1-a x}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {\int \frac {(a x c+c)^{p+1}}{1-a x}dx}{c}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {(a c x+c)^{p+2} \operatorname {Hypergeometric2F1}\left (1,p+2,p+3,\frac {1}{2} (a x+1)\right )}{2 a c^2 (p+2)}\) |
Input:
Int[E^(2*ArcTanh[a*x])*(c + a*c*x)^p,x]
Output:
((c + a*c*x)^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, (1 + a*x)/2])/(2*a *c^2*(2 + 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])
\[\int \frac {\left (a x +1\right )^{2} \left (a c x +c \right )^{p}}{-a^{2} x^{2}+1}d x\]
Input:
int((a*x+1)^2/(-a^2*x^2+1)*(a*c*x+c)^p,x)
Output:
int((a*x+1)^2/(-a^2*x^2+1)*(a*c*x+c)^p,x)
\[ \int e^{2 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int { -\frac {{\left (a x + 1\right )}^{2} {\left (a c x + c\right )}^{p}}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(a*c*x+c)^p,x, algorithm="fricas")
Output:
integral(-(a*x + 1)*(a*c*x + c)^p/(a*x - 1), x)
\[ \int e^{2 \text {arctanh}(a x)} (c+a c x)^p \, dx=- \int \frac {\left (a c x + c\right )^{p}}{a x - 1}\, dx - \int \frac {a x \left (a c x + c\right )^{p}}{a x - 1}\, dx \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(a*c*x+c)**p,x)
Output:
-Integral((a*c*x + c)**p/(a*x - 1), x) - Integral(a*x*(a*c*x + c)**p/(a*x - 1), x)
\[ \int e^{2 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int { -\frac {{\left (a x + 1\right )}^{2} {\left (a c x + c\right )}^{p}}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(a*c*x+c)^p,x, algorithm="maxima")
Output:
-integrate((a*x + 1)^2*(a*c*x + c)^p/(a^2*x^2 - 1), x)
\[ \int e^{2 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int { -\frac {{\left (a x + 1\right )}^{2} {\left (a c x + c\right )}^{p}}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(a*c*x+c)^p,x, algorithm="giac")
Output:
integrate(-(a*x + 1)^2*(a*c*x + c)^p/(a^2*x^2 - 1), x)
Timed out. \[ \int e^{2 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int -\frac {{\left (c+a\,c\,x\right )}^p\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \] Input:
int(-((c + a*c*x)^p*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
int(-((c + a*c*x)^p*(a*x + 1)^2)/(a^2*x^2 - 1), x)
\[ \int e^{2 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {-\left (a c x +c \right )^{p} a p x +\left (a c x +c \right )^{p} p +2 \left (a c x +c \right )^{p}-4 \left (\int \frac {\left (a c x +c \right )^{p} x}{a^{2} x^{2}-1}d x \right ) a^{2} p^{2}-4 \left (\int \frac {\left (a c x +c \right )^{p} x}{a^{2} x^{2}-1}d x \right ) a^{2} p}{a p \left (p +1\right )} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*(a*c*x+c)^p,x)
Output:
( - (a*c*x + c)**p*a*p*x + (a*c*x + c)**p*p + 2*(a*c*x + c)**p - 4*int(((a *c*x + c)**p*x)/(a**2*x**2 - 1),x)*a**2*p**2 - 4*int(((a*c*x + c)**p*x)/(a **2*x**2 - 1),x)*a**2*p)/(a*p*(p + 1))