\(\int e^{\text {arctanh}(a x)} x^3 (c-a^2 c x^2)^p \, dx\) [1049]

Optimal result

Integrand size = 23, antiderivative size = 134 \[ \int e^{\text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^p \, dx=-\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^4 (1+2 p)}+\frac {\left (1-a^2 x^2\right )^{3/2} \left (c-a^2 c x^2\right )^p}{a^4 (3+2 p)}+\frac {1}{5} a x^5 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{2}-p,\frac {7}{2},a^2 x^2\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.78 \[ \int e^{\text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^p \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p} \left (2+a^2 (1+2 p) x^2\right )}{a^4 \left (3+8 p+4 p^2\right )}+\frac {1}{5} a x^5 \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{2}-p,\frac {7}{2},a^2 x^2\right )\right ) \] Input:

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

Output:

((c - a^2*c*x^2)^p*(-(((1 - a^2*x^2)^(1/2 + p)*(2 + a^2*(1 + 2*p)*x^2))/(a 
^4*(3 + 8*p + 4*p^2))) + (a*x^5*Hypergeometric2F1[5/2, 1/2 - p, 7/2, a^2*x 
^2])/5))/(1 - a^2*x^2)^p
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6703, 6698, 542, 243, 53, 278, 2009}

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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 542

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a \int x^4 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx+\int x^3 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx\right )\)

\(\Big \downarrow \) 243

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} \int x^2 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx^2+a \int x^4 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx\right )\)

\(\Big \downarrow \) 53

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} \int \left (\frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{a^2}-\frac {\left (1-a^2 x^2\right )^{p+\frac {1}{2}}}{a^2}\right )dx^2+a \int x^4 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx\right )\)

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 2009

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{5} a x^5 \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{2}-p,\frac {7}{2},a^2 x^2\right )+\frac {1}{2} \left (\frac {2 \left (1-a^2 x^2\right )^{p+\frac {3}{2}}}{a^4 (2 p+3)}-\frac {2 \left (1-a^2 x^2\right )^{p+\frac {1}{2}}}{a^4 (2 p+1)}\right )\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 542

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[c   Int[x^m*(a + b*x^2)^p, x], x] + Simp[d   Int[x^(m + 1)*(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && IntegerQ[m] &&  !IntegerQ[2*p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.20 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.03 \[ \int e^{\text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^p \, dx=- \frac {a a^{2 p} c^{p} x^{2 p + 5} e^{i \pi p} \Gamma \left (- p - \frac {5}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p - \frac {5}{2} \\ \frac {1}{2}, - p - \frac {3}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p + 5} c^{p} x^{2 p + 5} e^{i \pi p} \Gamma \left (- p - \frac {5}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p + \frac {5}{2} \\ p + 1, p + \frac {7}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } a^{4} \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + 1\right )} - \frac {c^{p} {G_{3, 3}^{2, 2}\left (\begin {matrix} - p - 1, 1 & -1 \\- p - \frac {3}{2}, - p - 1 & 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{a^{2} x^{2}}} \right )} \Gamma \left (p + \frac {1}{2}\right )}{2 \pi a^{4}} - \frac {c^{p} {G_{3, 3}^{1, 3}\left (\begin {matrix} -1, - p - 2, 1 & \\- p - 2 & - p - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{a^{2} x^{2}}} \right )} \Gamma \left (p + \frac {1}{2}\right )}{2 a^{4} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} \] Input:

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

Output:

-a*a**(2*p)*c**p*x**(2*p + 5)*exp(I*pi*p)*gamma(-p - 5/2)*gamma(p + 1/2)*h 
yper((1, -p, -p - 5/2), (1/2, -p - 3/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma( 
-p - 3/2)*gamma(p + 1)) - a**(2*p + 5)*c**p*x**(2*p + 5)*exp(I*pi*p)*gamma 
(-p - 5/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 5/2), (p + 1, p + 7/2), a**2* 
x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*a**4*gamma(-p - 3/2)*gamma(p + 1)) - c 
**p*meijerg(((-p - 1, 1), (-1,)), ((-p - 3/2, -p - 1), (0,)), exp_polar(-I 
*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*pi*a**4) - c**p*meijerg(((-1, -p - 2, 
1), ()), ((-p - 2,), (-p - 3/2, 0)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p 
 + 1/2)/(2*a**4*gamma(-p)*gamma(p + 1))
 

Maxima [F]

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

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

Output:

a*c^p*integrate(x^4*e^(p*log(a*x + 1) + p*log(-a*x + 1))/(sqrt(a*x + 1)*sq 
rt(-a*x + 1)), x) + (a^4*c^p*(2*p + 1)*x^4 - a^2*c^p*(2*p - 1)*x^2 - 2*c^p 
)*(-a^2*x^2 + 1)^p/(sqrt(-a^2*x^2 + 1)*(4*p^2 + 8*p + 3)*a^4)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{\text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^p \, dx=\frac {-2 c^{p} \left (-a^{2} x^{2}+1\right )^{p +\frac {1}{2}} a^{2} p \,x^{2}-c^{p} \left (-a^{2} x^{2}+1\right )^{p +\frac {1}{2}} a^{2} x^{2}-2 c^{p} \left (-a^{2} x^{2}+1\right )^{p +\frac {1}{2}}+4 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x^{4}}{\sqrt {-a^{2} x^{2}+1}}d x \right ) a^{5} p^{2}+8 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x^{4}}{\sqrt {-a^{2} x^{2}+1}}d x \right ) a^{5} p +3 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x^{4}}{\sqrt {-a^{2} x^{2}+1}}d x \right ) a^{5}}{a^{4} \left (4 p^{2}+8 p +3\right )} \] Input:

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

Output:

( - 2*c**p*( - a**2*x**2 + 1)**((2*p + 1)/2)*a**2*p*x**2 - c**p*( - a**2*x 
**2 + 1)**((2*p + 1)/2)*a**2*x**2 - 2*c**p*( - a**2*x**2 + 1)**((2*p + 1)/ 
2) + 4*int((( - a**2*c*x**2 + c)**p*x**4)/sqrt( - a**2*x**2 + 1),x)*a**5*p 
**2 + 8*int((( - a**2*c*x**2 + c)**p*x**4)/sqrt( - a**2*x**2 + 1),x)*a**5* 
p + 3*int((( - a**2*c*x**2 + c)**p*x**4)/sqrt( - a**2*x**2 + 1),x)*a**5)/( 
a**4*(4*p**2 + 8*p + 3))