Integrand size = 23, antiderivative size = 114 \[ \int e^{-\text {arctanh}(a x)} x^m \sqrt {c-a c x} \, dx=-\frac {2 c x^{1+m} \sqrt {1-a^2 x^2}}{(3+2 m) \sqrt {c-a c x}}+\frac {2 (5+4 m) x^m (-a x)^{-m} (1+a x) \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1+a x\right )}{a (3+2 m) \sqrt {1-a^2 x^2}} \]
2*(5+4*m)*x^m*(a*x+1)*hypergeom([1/2, -m],[3/2],a*x+1)*(-a*c*x+c)^(1/2)/a/ (3+2*m)/((-a*x)^m)/(-a^2*x^2+1)^(1/2)-2*c*x^(1+m)*(-a^2*x^2+1)^(1/2)/(3+2* m)/(-a*c*x+c)^(1/2)
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.68 \[ \int e^{-\text {arctanh}(a x)} x^m \sqrt {c-a c x} \, dx=-\frac {c x^{1+m} \sqrt {1-a x} \left (2 (1+m) \sqrt {1+a x}-(5+4 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,2+m,-a x\right )\right )}{(1+m) (3+2 m) \sqrt {c-a c x}} \]
-((c*x^(1 + m)*Sqrt[1 - a*x]*(2*(1 + m)*Sqrt[1 + a*x] - (5 + 4*m)*Hypergeo metric2F1[1/2, 1 + m, 2 + m, -(a*x)]))/((1 + m)*(3 + 2*m)*Sqrt[c - a*c*x]) )
Time = 0.35 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6678, 574, 585, 74}
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.
\(\displaystyle \int x^m e^{-\text {arctanh}(a x)} \sqrt {c-a c x} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle \frac {\int \frac {x^m (c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}}dx}{c}\) |
\(\Big \downarrow \) 574 |
\(\displaystyle \frac {\frac {c (4 m+5) \int \frac {x^m \sqrt {c-a c x}}{\sqrt {1-a^2 x^2}}dx}{2 m+3}-\frac {2 c^2 \sqrt {1-a^2 x^2} x^{m+1}}{(2 m+3) \sqrt {c-a c x}}}{c}\) |
\(\Big \downarrow \) 585 |
\(\displaystyle \frac {\frac {c (4 m+5) \sqrt {c-a c x} \int \frac {x^m}{\sqrt {a x+1}}dx}{(2 m+3) \sqrt {1-a x}}-\frac {2 c^2 \sqrt {1-a^2 x^2} x^{m+1}}{(2 m+3) \sqrt {c-a c x}}}{c}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {\frac {c (4 m+5) x^{m+1} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},m+1,m+2,-a x\right )}{(m+1) (2 m+3) \sqrt {1-a x}}-\frac {2 c^2 \sqrt {1-a^2 x^2} x^{m+1}}{(2 m+3) \sqrt {c-a c x}}}{c}\) |
((-2*c^2*x^(1 + m)*Sqrt[1 - a^2*x^2])/((3 + 2*m)*Sqrt[c - a*c*x]) + (c*(5 + 4*m)*x^(1 + m)*Sqrt[c - a*c*x]*Hypergeometric2F1[1/2, 1 + m, 2 + m, -(a* x)])/((1 + m)*(3 + 2*m)*Sqrt[1 - a*x]))/c
3.5.11.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((e_.)*(x_))^(n_)*((c_) + (d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d^2*(e*x)^(n + 1)*(c + d*x)^(m - 2)*((a + b*x^2)^(p + 1)/ (b*e*(n + p + 2))), x] + Simp[c*((2*n + p + 3)/(n + p + 2)) Int[(e*x)^n*( c + d*x)^(m - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x ] && EqQ[b*c^2 + a*d^2, 0] && EqQ[m + p - 1, 0] && !LtQ[n, -1] && IntegerQ [2*p]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[a^p*c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^F racPart[n]) Int[(e*x)^m*(1 - d*(x/c))^p*(1 + d*(x/c))^(n + p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[a, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
\[\int \frac {x^{m} \sqrt {-a c x +c}\, \sqrt {-a^{2} x^{2}+1}}{a x +1}d x\]
\[ \int e^{-\text {arctanh}(a x)} x^m \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} x^{m}}{a x + 1} \,d x } \]
\[ \int e^{-\text {arctanh}(a x)} x^m \sqrt {c-a c x} \, dx=\int \frac {x^{m} \sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \]
\[ \int e^{-\text {arctanh}(a x)} x^m \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} x^{m}}{a x + 1} \,d x } \]
\[ \int e^{-\text {arctanh}(a x)} x^m \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} x^{m}}{a x + 1} \,d x } \]
Timed out. \[ \int e^{-\text {arctanh}(a x)} x^m \sqrt {c-a c x} \, dx=\int \frac {x^m\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{a\,x+1} \,d x \]