Integrand size = 27, antiderivative size = 126 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^{1+m}}{(1+m) \sqrt {1-\frac {1}{a x}}}-\frac {(3+4 m) \sqrt {c-\frac {c}{a x}} x^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,-\frac {1}{a x}\right )}{2 a m (1+m) \sqrt {1-\frac {1}{a x}}} \]
-1/2*(3+4*m)*x^m*hypergeom([1/2, -m],[1-m],-1/a/x)*(c-c/a/x)^(1/2)/a/m/(1+ m)/(1-1/a/x)^(1/2)+x^(1+m)*(1+1/a/x)^(1/2)*(c-c/a/x)^(1/2)/(1+m)/(1-1/a/x) ^(1/2)
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\frac {\sqrt {c-\frac {c}{a x}} x^m \left (2 a m \sqrt {1+\frac {1}{a x}} x-(3+4 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,-\frac {1}{a x}\right )\right )}{2 a m (1+m) \sqrt {1-\frac {1}{a x}}} \]
(Sqrt[c - c/(a*x)]*x^m*(2*a*m*Sqrt[1 + 1/(a*x)]*x - (3 + 4*m)*Hypergeometr ic2F1[1/2, -m, 1 - m, -(1/(a*x))]))/(2*a*m*(1 + m)*Sqrt[1 - 1/(a*x)])
Time = 0.52 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6736, 6735, 27, 88, 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.
\(\displaystyle \int x^m \sqrt {c-\frac {c}{a x}} e^{-\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6736 |
\(\displaystyle \frac {\sqrt {c-\frac {c}{a x}} \int e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^mdx}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 6735 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^m x^m \sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{-m-2}}{a \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^m x^m \sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{-m-2}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^m x^m \sqrt {c-\frac {c}{a x}} \left (-\frac {(4 m+3) \int \frac {\left (\frac {1}{x}\right )^{-m-1}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 (m+1)}-\frac {a \sqrt {\frac {1}{a x}+1} \left (\frac {1}{x}\right )^{-m-1}}{m+1}\right )}{a \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^m x^m \sqrt {c-\frac {c}{a x}} \left (\frac {(4 m+3) \left (\frac {1}{x}\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,-\frac {1}{a x}\right )}{2 m (m+1)}-\frac {a \sqrt {\frac {1}{a x}+1} \left (\frac {1}{x}\right )^{-m-1}}{m+1}\right )}{a \sqrt {1-\frac {1}{a x}}}\) |
-((Sqrt[c - c/(a*x)]*(x^(-1))^m*x^m*(-((a*Sqrt[1 + 1/(a*x)]*(x^(-1))^(-1 - m))/(1 + m)) + ((3 + 4*m)*Hypergeometric2F1[1/2, -m, 1 - m, -(1/(a*x))])/ (2*m*(1 + m)*(x^(-1))^m)))/(a*Sqrt[1 - 1/(a*x)]))
3.6.16.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Sy mbol] :> Simp[(-c^p)*x^m*(1/x)^m Subst[Int[(1 + d*(x/c))^p*((1 + x/a)^(n/ 2)/(x^(m + 2)*(1 - x/a)^(n/2))), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n, p }, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegerQ[m]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[(c + d/x)^p/(1 + d/(c*x))^p Int[u*(1 + d/(c*x))^p*E^(n*ArcCoth[a *x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !Int egerQ[n/2] && !(IntegerQ[p] || GtQ[c, 0])
\[\int x^{m} \sqrt {c -\frac {c}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}d x\]
\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\int { \sqrt {c - \frac {c}{a x}} x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
Timed out. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\text {Timed out} \]
\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\int { \sqrt {c - \frac {c}{a x}} x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\int { \sqrt {c - \frac {c}{a x}} x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
Timed out. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx=\int x^m\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]