Integrand size = 27, antiderivative size = 136 \[ \int e^{3 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=-\frac {3 x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}-\frac {a x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}}+\frac {4 x^{1+m} \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}(1,1+m,2+m,a x)}{(1+m) \sqrt {1-a^2 x^2}} \] Output:
-3*x^(1+m)*(-a^2*c*x^2+c)^(1/2)/(1+m)/(-a^2*x^2+1)^(1/2)-a*x^(2+m)*(-a^2*c *x^2+c)^(1/2)/(2+m)/(-a^2*x^2+1)^(1/2)+4*x^(1+m)*(-a^2*c*x^2+c)^(1/2)*hype rgeom([1, 1+m],[2+m],a*x)/(1+m)/(-a^2*x^2+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.54 \[ \int e^{3 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=-\frac {x^{1+m} \sqrt {c-a^2 c x^2} (6+a x+m (3+a x)-4 (2+m) \operatorname {Hypergeometric2F1}(1,1+m,2+m,a x))}{(1+m) (2+m) \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^(3*ArcTanh[a*x])*x^m*Sqrt[c - a^2*c*x^2],x]
Output:
-((x^(1 + m)*Sqrt[c - a^2*c*x^2]*(6 + a*x + m*(3 + a*x) - 4*(2 + m)*Hyperg eometric2F1[1, 1 + m, 2 + m, a*x]))/((1 + m)*(2 + m)*Sqrt[1 - a^2*x^2]))
Time = 0.48 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6703, 6700, 99, 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.
\(\displaystyle \int x^m e^{3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int e^{3 \text {arctanh}(a x)} x^m \sqrt {1-a^2 x^2}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \frac {x^m (a x+1)^2}{1-a x}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \left (\frac {4 x^m}{1-a x}-3 x^m-a x^{m+1}\right )dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \left (\frac {4 x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,a x)}{m+1}-\frac {a x^{m+2}}{m+2}-\frac {3 x^{m+1}}{m+1}\right )}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[E^(3*ArcTanh[a*x])*x^m*Sqrt[c - a^2*c*x^2],x]
Output:
(Sqrt[c - a^2*c*x^2]*((-3*x^(1 + m))/(1 + m) - (a*x^(2 + m))/(2 + m) + (4* x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(1 + m)))/Sqrt[1 - a^2* x^2]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
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]
\[\int \frac {\left (a x +1\right )^{3} x^{m} \sqrt {-a^{2} c \,x^{2}+c}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}d x\]
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^(1/2),x)
Output:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^(1/2),x)
\[ \int e^{3 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}^{3} x^{m}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^(1/2),x, algorit hm="fricas")
Output:
integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x + 1)*x^m/(a^2*x^2 - 2*a*x + 1), x)
\[ \int e^{3 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^{m} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**m*(-a**2*c*x**2+c)**(1/2),x)
Output:
Integral(x**m*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)
\[ \int e^{3 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}^{3} x^{m}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^(1/2),x, algorit hm="maxima")
Output:
integrate(sqrt(-a^2*c*x^2 + c)*(a*x + 1)^3*x^m/(-a^2*x^2 + 1)^(3/2), x)
Exception generated. \[ \int e^{3 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^(1/2),x, algorit hm="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
Timed out. \[ \int e^{3 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^m\,\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((x^m*(c - a^2*c*x^2)^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
int((x^m*(c - a^2*c*x^2)^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2), x)
\[ \int e^{3 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c}\, \left (-x^{m} a^{2} m^{2} x^{2}-x^{m} a^{2} m \,x^{2}-3 x^{m} a \,m^{2} x -6 x^{m} a m x -4 x^{m} m^{2}-12 x^{m} m -8 x^{m}-4 \left (\int \frac {x^{m}}{a \,x^{2}-x}d x \right ) m^{3}-12 \left (\int \frac {x^{m}}{a \,x^{2}-x}d x \right ) m^{2}-8 \left (\int \frac {x^{m}}{a \,x^{2}-x}d x \right ) m \right )}{a m \left (m^{2}+3 m +2\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^m*(-a^2*c*x^2+c)^(1/2),x)
Output:
(sqrt(c)*( - x**m*a**2*m**2*x**2 - x**m*a**2*m*x**2 - 3*x**m*a*m**2*x - 6* x**m*a*m*x - 4*x**m*m**2 - 12*x**m*m - 8*x**m - 4*int(x**m/(a*x**2 - x),x) *m**3 - 12*int(x**m/(a*x**2 - x),x)*m**2 - 8*int(x**m/(a*x**2 - x),x)*m))/ (a*m*(m**2 + 3*m + 2))