\(\int e^{\text {arctanh}(a x)} (c+a c x)^p \, dx\) [284]

Optimal result

Integrand size = 15, antiderivative size = 63 \[ \int e^{\text {arctanh}(a x)} (c+a c x)^p \, dx=-\frac {2^{\frac {3}{2}+p} \sqrt {1-a x} (1+a x)^{-p} (c+a c x)^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-p,\frac {3}{2},\frac {1}{2} (1-a x)\right )}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int e^{\text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {\sqrt {2} (1+a x)^{3/2} (c+a c x)^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}+p,\frac {5}{2}+p,\frac {1}{2} (1+a x)\right )}{a (3+2 p)} \] Input:

Integrate[E^ArcTanh[a*x]*(c + a*c*x)^p,x]
 

Output:

(Sqrt[2]*(1 + a*x)^(3/2)*(c + a*c*x)^p*Hypergeometric2F1[1/2, 3/2 + p, 5/2 
 + p, (1 + a*x)/2])/(a*(3 + 2*p))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6680, 37, 79}

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.

\(\displaystyle \int e^{\text {arctanh}(a x)} (a c x+c)^p \, dx\)

\(\Big \downarrow \) 6680

\(\displaystyle \int \frac {\sqrt {a x+1} (a c x+c)^p}{\sqrt {1-a x}}dx\)

\(\Big \downarrow \) 37

\(\displaystyle (a x+1)^{-p} (a c x+c)^p \int \frac {(a x+1)^{p+\frac {1}{2}}}{\sqrt {1-a x}}dx\)

\(\Big \downarrow \) 79

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

Input:

Int[E^ArcTanh[a*x]*(c + a*c*x)^p,x]
 

Output:

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

Defintions of rubi rules used

rule 37

Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S 
imp[(a + b*x)^m/(c + d*x)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, 
b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !SimplerQ[a + b*x, c + d*x]
 

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 6680

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((c*(a*x + 1))**p*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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