\(\int e^{5 \text {arctanh}(a x)} (c+a c x)^p \, dx\) [282]

Optimal result

Integrand size = 17, antiderivative size = 65 \[ \int e^{5 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {2^{\frac {7}{2}+p} (1+a x)^{-p} (c+a c x)^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {5}{2}-p,-\frac {1}{2},\frac {1}{2} (1-a x)\right )}{3 a (1-a x)^{3/2}} \] Output:

1/3*2^(7/2+p)*(a*c*x+c)^p*hypergeom([-3/2, -5/2-p],[-1/2],-1/2*a*x+1/2)/a/ 
(-a*x+1)^(3/2)/((a*x+1)^p)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91 \[ \int e^{5 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {(1+a x)^{7/2} (c+a c x)^p \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {7}{2}+p,\frac {9}{2}+p,\frac {1}{2} (1+a x)\right )}{2 \sqrt {2} a (7+2 p)} \] Input:

Integrate[E^(5*ArcTanh[a*x])*(c + a*c*x)^p,x]
 

Output:

((1 + a*x)^(7/2)*(c + a*c*x)^p*Hypergeometric2F1[5/2, 7/2 + p, 9/2 + p, (1 
 + a*x)/2])/(2*Sqrt[2]*a*(7 + 2*p))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 65, 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, 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.

\(\displaystyle \int e^{5 \text {arctanh}(a x)} (a c x+c)^p \, dx\)

\(\Big \downarrow \) 6680

\(\displaystyle \int \frac {(a x+1)^{5/2} (a c x+c)^p}{(1-a x)^{5/2}}dx\)

\(\Big \downarrow \) 37

\(\displaystyle (a x+1)^{-p} (a c x+c)^p \int \frac {(a x+1)^{p+\frac {5}{2}}}{(1-a x)^{5/2}}dx\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {2^{p+\frac {7}{2}} (a x+1)^{-p} (a c x+c)^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-p-\frac {5}{2},-\frac {1}{2},\frac {1}{2} (1-a x)\right )}{3 a (1-a x)^{3/2}}\)

Input:

Int[E^(5*ArcTanh[a*x])*(c + a*c*x)^p,x]
 

Output:

(2^(7/2 + p)*(c + a*c*x)^p*Hypergeometric2F1[-3/2, -5/2 - p, -1/2, (1 - a* 
x)/2])/(3*a*(1 - a*x)^(3/2)*(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]

\[\int \frac {\left (a x +1\right )^{5} \left (a c x +c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}d x\]

Input:

int((a*x+1)^5/(-a^2*x^2+1)^(5/2)*(a*c*x+c)^p,x)
 

Output:

int((a*x+1)^5/(-a^2*x^2+1)^(5/2)*(a*c*x+c)^p,x)
 

Fricas [F]

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

integrate((a*x+1)^5/(-a^2*x^2+1)^(5/2)*(a*c*x+c)^p,x, algorithm="fricas")
 

Output:

integral(-(a^2*x^2 + 2*a*x + 1)*sqrt(-a^2*x^2 + 1)*(a*c*x + c)^p/(a^3*x^3 
- 3*a^2*x^2 + 3*a*x - 1), x)
 

Sympy [F]

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

integrate((a*x+1)**5/(-a**2*x**2+1)**(5/2)*(a*c*x+c)**p,x)
 

Output:

Integral((c*(a*x + 1))**p*(a*x + 1)**5/(-(a*x - 1)*(a*x + 1))**(5/2), x)
 

Maxima [F]

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

integrate((a*x+1)^5/(-a^2*x^2+1)^(5/2)*(a*c*x+c)^p,x, algorithm="maxima")
 

Output:

integrate((a*x + 1)^5*(a*c*x + c)^p/(-a^2*x^2 + 1)^(5/2), x)
 

Giac [F]

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

integrate((a*x+1)^5/(-a^2*x^2+1)^(5/2)*(a*c*x+c)^p,x, algorithm="giac")
 

Output:

integrate((a*x + 1)^5*(a*c*x + c)^p/(-a^2*x^2 + 1)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{5 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int \frac {{\left (c+a\,c\,x\right )}^p\,{\left (a\,x+1\right )}^5}{{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \] Input:

int(((c + a*c*x)^p*(a*x + 1)^5)/(1 - a^2*x^2)^(5/2),x)
 

Output:

int(((c + a*c*x)^p*(a*x + 1)^5)/(1 - a^2*x^2)^(5/2), x)
 

Reduce [F]

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

int((a*x+1)^5/(-a^2*x^2+1)^(5/2)*(a*c*x+c)^p,x)
 

Output:

int((a*c*x + c)**p/(sqrt( - a**2*x**2 + 1)*a**2*x**2 - 2*sqrt( - a**2*x**2 
 + 1)*a*x + sqrt( - a**2*x**2 + 1)),x) + int(((a*c*x + c)**p*x**3)/(sqrt( 
- a**2*x**2 + 1)*a**2*x**2 - 2*sqrt( - a**2*x**2 + 1)*a*x + sqrt( - a**2*x 
**2 + 1)),x)*a**3 + 3*int(((a*c*x + c)**p*x**2)/(sqrt( - a**2*x**2 + 1)*a* 
*2*x**2 - 2*sqrt( - a**2*x**2 + 1)*a*x + sqrt( - a**2*x**2 + 1)),x)*a**2 + 
 3*int(((a*c*x + c)**p*x)/(sqrt( - a**2*x**2 + 1)*a**2*x**2 - 2*sqrt( - a* 
*2*x**2 + 1)*a*x + sqrt( - a**2*x**2 + 1)),x)*a