\(\int e^{3 \text {arctanh}(a x)} x (c-a^2 c x^2)^p \, dx\) [1214]

Optimal result

Integrand size = 23, antiderivative size = 138 \[ \int e^{3 \text {arctanh}(a x)} x \left (c-a^2 c x^2\right )^p \, dx=-\frac {(1+a x)^3 \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p) \sqrt {1-a^2 x^2}}+\frac {3\ 2^{\frac {3}{2}+p} (1-a x)^{-\frac {1}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-\frac {1}{2}+p,\frac {1}{2}+p,\frac {1}{2} (1-a x)\right )}{a^2 \left (1-p-2 p^2\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.97 \[ \int e^{3 \text {arctanh}(a x)} x \left (c-a^2 c x^2\right )^p \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {\left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \left (\frac {4}{1-2 p}+\frac {3-3 a^2 x^2}{1+2 p}\right )}{a^2}+a x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {3}{2}-p,\frac {5}{2},a^2 x^2\right )+\frac {1}{5} a^3 x^5 \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {3}{2}-p,\frac {7}{2},a^2 x^2\right )\right ) \] Input:

Integrate[E^(3*ArcTanh[a*x])*x*(c - a^2*c*x^2)^p,x]
 

Output:

((c - a^2*c*x^2)^p*(((1 - a^2*x^2)^(-1/2 + p)*(4/(1 - 2*p) + (3 - 3*a^2*x^ 
2)/(1 + 2*p)))/a^2 + a*x^3*Hypergeometric2F1[3/2, 3/2 - p, 5/2, a^2*x^2] + 
 (a^3*x^5*Hypergeometric2F1[5/2, 3/2 - p, 7/2, a^2*x^2])/5))/(1 - a^2*x^2) 
^p
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6703, 6698, 572, 472, 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.

\(\displaystyle \int x e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx\)

\(\Big \downarrow \) 6703

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{3 \text {arctanh}(a x)} x \left (1-a^2 x^2\right )^pdx\)

\(\Big \downarrow \) 6698

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int x (a x+1)^3 \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx\)

\(\Big \downarrow \) 572

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {3 \int (a x+1)^3 \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx}{2 a (p+1)}-\frac {(a x+1)^3 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{2 a^2 (p+1)}\right )\)

\(\Big \downarrow \) 472

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {3 \int (1-a x)^{p-\frac {3}{2}} (a x+1)^{p+\frac {3}{2}}dx}{2 a (p+1)}-\frac {(a x+1)^3 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{2 a^2 (p+1)}\right )\)

\(\Big \downarrow \) 79

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

Input:

Int[E^(3*ArcTanh[a*x])*x*(c - a^2*c*x^2)^p,x]
 

Output:

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

Defintions of rubi rules used

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 472

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
a^(p + 1)*c^(n - 1)*(((c - d*x)/c)^(p + 1)/(a/c + b*(x/d))^(p + 1))   Int[( 
1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] 
 && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) && GtQ[a, 0] &&  !( 
IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
 

rule 572

Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 2))), x] + Simp[c*(n/(d 
*(n + 2*p + 2)))   Int[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && NeQ[n + 2*p + 2, 0]
 

rule 6698

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[c^p   Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] 
/; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 
 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]
 

rule 6703

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar 
t[p])   Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, 
d, m, n, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) &&  !I 
ntegerQ[n/2]
 

Maple [F]

\[\int \frac {\left (a x +1\right )^{3} x \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^p,x, algorithm="fr 
icas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^p,x, algorithm="ma 
xima")
 

Output:

-(-a^2*x^2 + 1)^p*c^p/(sqrt(-a^2*x^2 + 1)*a^2*(2*p - 1)) - integrate((a^3* 
c^p*x^4 + 3*a^2*c^p*x^3 + 3*a*c^p*x^2)*e^(p*log(a*x + 1) + p*log(-a*x + 1) 
)/((a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(-a*x + 1)), x)
 

Giac [F]

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

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^p,x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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