\(\int e^{3 \text {arctanh}(a x)} x^m (c-a^2 c x^2)^p \, dx\) [1211]

Optimal result

Integrand size = 25, antiderivative size = 251 \[ \int e^{3 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=-\frac {3 x^{1+m} \left (c-a^2 c x^2\right )^p}{(m+2 p) \sqrt {1-a^2 x^2}}-\frac {a x^{2+m} \left (c-a^2 c x^2\right )^p}{(1+m+2 p) \sqrt {1-a^2 x^2}}+\frac {(3+4 m+2 p) x^{1+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {3}{2}-p,\frac {3+m}{2},a^2 x^2\right )}{(1+m) (m+2 p)}+\frac {a (5+4 m+6 p) x^{2+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},\frac {3}{2}-p,\frac {4+m}{2},a^2 x^2\right )}{(2+m) (1+m+2 p)} \] Output:

-3*x^(1+m)*(-a^2*c*x^2+c)^p/(m+2*p)/(-a^2*x^2+1)^(1/2)-a*x^(2+m)*(-a^2*c*x 
^2+c)^p/(1+m+2*p)/(-a^2*x^2+1)^(1/2)+(3+4*m+2*p)*x^(1+m)*(-a^2*c*x^2+c)^p* 
hypergeom([3/2-p, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/(1+m)/(m+2*p)/((-a^2*x^2 
+1)^p)+a*(5+4*m+6*p)*x^(2+m)*(-a^2*c*x^2+c)^p*hypergeom([1+1/2*m, 3/2-p],[ 
2+1/2*m],a^2*x^2)/(2+m)/(1+m+2*p)/((-a^2*x^2+1)^p)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.74 \[ \int e^{3 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=x^{1+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {\operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {3}{2}-p,\frac {3+m}{2},a^2 x^2\right )}{1+m}+a x \left (\frac {3 \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},\frac {3}{2}-p,\frac {4+m}{2},a^2 x^2\right )}{2+m}+a x \left (\frac {3 \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},\frac {3}{2}-p,\frac {5+m}{2},a^2 x^2\right )}{3+m}+\frac {a x \operatorname {Hypergeometric2F1}\left (\frac {4+m}{2},\frac {3}{2}-p,\frac {6+m}{2},a^2 x^2\right )}{4+m}\right )\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6703, 6698, 559, 25, 2340, 25, 27, 557, 278}

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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 559

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\frac {\int -x^m \left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 (m+2 p+1) x^2 a^4+(4 m+6 p+5) x a^3+(m+2 p+1) a^2\right )dx}{a^2 (m+2 p+1)}-\frac {a x^{m+2} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p+1}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {\int x^m \left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 (m+2 p+1) x^2 a^4+(4 m+6 p+5) x a^3+(m+2 p+1) a^2\right )dx}{a^2 (m+2 p+1)}-\frac {a x^{m+2} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p+1}\right )\)

\(\Big \downarrow \) 2340

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {-\frac {\int -a^4 x^m ((m+2 p+1) (4 m+2 p+3)+a (m+2 p) (4 m+6 p+5) x) \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx}{a^2 (m+2 p)}-\frac {3 a^2 (m+2 p+1) x^{m+1} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p}}{a^2 (m+2 p+1)}-\frac {a x^{m+2} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p+1}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {\frac {\int a^4 x^m ((m+2 p+1) (4 m+2 p+3)+a (m+2 p) (4 m+6 p+5) x) \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx}{a^2 (m+2 p)}-\frac {3 a^2 (m+2 p+1) x^{m+1} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p}}{a^2 (m+2 p+1)}-\frac {a x^{m+2} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p+1}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {\frac {a^2 \int x^m ((m+2 p+1) (4 m+2 p+3)+a (m+2 p) (4 m+6 p+5) x) \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx}{m+2 p}-\frac {3 a^2 (m+2 p+1) x^{m+1} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p}}{a^2 (m+2 p+1)}-\frac {a x^{m+2} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p+1}\right )\)

\(\Big \downarrow \) 557

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {\frac {a^2 \left (a (m+2 p) (4 m+6 p+5) \int x^{m+1} \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+(m+2 p+1) (4 m+2 p+3) \int x^m \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx\right )}{m+2 p}-\frac {3 a^2 (m+2 p+1) x^{m+1} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p}}{a^2 (m+2 p+1)}-\frac {a x^{m+2} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p+1}\right )\)

\(\Big \downarrow \) 278

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {\frac {a^2 \left (\frac {(m+2 p+1) (4 m+2 p+3) x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},\frac {3}{2}-p,\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a (m+2 p) (4 m+6 p+5) x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},\frac {3}{2}-p,\frac {m+4}{2},a^2 x^2\right )}{m+2}\right )}{m+2 p}-\frac {3 a^2 (m+2 p+1) x^{m+1} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p}}{a^2 (m+2 p+1)}-\frac {a x^{m+2} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{m+2 p+1}\right )\)

Input:

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

Output:

((c - a^2*c*x^2)^p*(-((a*x^(2 + m)*(1 - a^2*x^2)^(-1/2 + p))/(1 + m + 2*p) 
) + ((-3*a^2*(1 + m + 2*p)*x^(1 + m)*(1 - a^2*x^2)^(-1/2 + p))/(m + 2*p) + 
 (a^2*(((1 + m + 2*p)*(3 + 4*m + 2*p)*x^(1 + m)*Hypergeometric2F1[(1 + m)/ 
2, 3/2 - p, (3 + m)/2, a^2*x^2])/(1 + m) + (a*(m + 2*p)*(5 + 4*m + 6*p)*x^ 
(2 + m)*Hypergeometric2F1[(2 + m)/2, 3/2 - p, (4 + m)/2, a^2*x^2])/(2 + m) 
))/(m + 2*p))/(a^2*(1 + m + 2*p))))/(1 - a^2*x^2)^p
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 557

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

rule 559

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( 
m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1))   Int[(e*x)^m*(a + b* 
x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 
)*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && IGtQ[n, 1] &&  !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{3 \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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