Integrand size = 25, antiderivative size = 133 \[ \int e^{-\text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^p \, dx=\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^3 (1+2 p)}-\frac {\left (1-a^2 x^2\right )^{3/2} \left (c-a^2 c x^2\right )^p}{a^3 (3+2 p)}+\frac {1}{3} x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right ) \]
-(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/a^3/(3+2*p)+1/3*x^3*(-a^2*c*x^2+c)^p* hypergeom([3/2, 1/2-p],[5/2],a^2*x^2)/((-a^2*x^2+1)^p)+(-a^2*c*x^2+c)^p*(- a^2*x^2+1)^(1/2)/a^3/(1+2*p)
Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.77 \[ \int e^{-\text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^p \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p} \left (2+a^2 (1+2 p) x^2\right )}{a^3 \left (3+8 p+4 p^2\right )}+\frac {1}{3} x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right )\right ) \]
((c - a^2*c*x^2)^p*(((1 - a^2*x^2)^(1/2 + p)*(2 + a^2*(1 + 2*p)*x^2))/(a^3 *(3 + 8*p + 4*p^2)) + (x^3*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2])/ 3))/(1 - a^2*x^2)^p
Time = 0.46 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6703, 6699, 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.
\(\displaystyle \int x^2 e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{-\text {arctanh}(a x)} x^2 \left (1-a^2 x^2\right )^pdx\) |
\(\Big \downarrow \) 6699 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int x^2 (1-a x) \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx\) |
\(\Big \downarrow \) 542 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int x^2 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx-a \int x^3 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int x^2 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx-\frac {1}{2} a \int x^2 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx^2\right )\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int x^2 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx-\frac {1}{2} a \int \left (\frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{a^2}-\frac {\left (1-a^2 x^2\right )^{p+\frac {1}{2}}}{a^2}\right )dx^2\right )\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{3} x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right )-\frac {1}{2} a \int \left (\frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{a^2}-\frac {\left (1-a^2 x^2\right )^{p+\frac {1}{2}}}{a^2}\right )dx^2\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{3} x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right )-\frac {1}{2} a \left (\frac {2 \left (1-a^2 x^2\right )^{p+\frac {3}{2}}}{a^4 (2 p+3)}-\frac {2 \left (1-a^2 x^2\right )^{p+\frac {1}{2}}}{a^4 (2 p+1)}\right )\right )\) |
((c - a^2*c*x^2)^p*(-1/2*(a*((-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)))) + (x^3*Hypergeometric2F1[3 /2, 1/2 - p, 5/2, a^2*x^2])/3))/(1 - a^2*x^2)^p
3.13.23.3.1 Defintions of rubi rules used
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])
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]
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])
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]
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]) && ILtQ[(n - 1)/2, 0] && !IntegerQ[p - n/2]
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]
\[\int \frac {x^{2} \left (-a^{2} c \,x^{2}+c \right )^{p} \sqrt {-a^{2} x^{2}+1}}{a x +1}d x\]
\[ \int e^{-\text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p} x^{2}}{a x + 1} \,d x } \]
\[ \int e^{-\text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^p \, dx=\int \frac {x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{a x + 1}\, dx \]
\[ \int e^{-\text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p} x^{2}}{a x + 1} \,d x } \]
\[ \int e^{-\text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p} x^{2}}{a x + 1} \,d x } \]
Timed out. \[ \int e^{-\text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^p \, dx=\int \frac {x^2\,{\left (c-a^2\,c\,x^2\right )}^p\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]