Integrand size = 20, antiderivative size = 167 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=-\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(3+n) (c-a c x)^{5/2}}+\frac {a \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3+n}{2}} \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {3}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(3+n) (c-a c x)^{5/2}} \]
-a*(1-1/a/x)^(1-1/2*n)*(1+1/a/x)^(1+1/2*n)*x^2/(3+n)/(-a*c*x+c)^(5/2)+a*(( a-1/x)/(a+1/x))^(3/2+1/2*n)*(1-1/a/x)^(1-1/2*n)*(1+1/a/x)^(1+1/2*n)*x^2*hy pergeom([1/2, 3/2+1/2*n],[3/2],2/(a+1/x)/x)/(3+n)/(-a*c*x+c)^(5/2)
Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.70 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (-1-a x+(-1+a x) \left (\frac {-1+a x}{1+a x}\right )^{\frac {1+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {3}{2},\frac {2}{1+a x}\right )\right )}{a c^2 (3+n) (-1+a x) \sqrt {c-a c x}} \]
((1 + 1/(a*x))^(n/2)*(-1 - a*x + (-1 + a*x)*((-1 + a*x)/(1 + a*x))^((1 + n )/2)*Hypergeometric2F1[1/2, (3 + n)/2, 3/2, 2/(1 + a*x)]))/(a*c^2*(3 + n)* (1 - 1/(a*x))^(n/2)*(-1 + a*x)*Sqrt[c - a*c*x])
Time = 0.31 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6727, 105, 142}
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.
| \(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6727 |
| \(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{n/2} \sqrt {\frac {1}{x}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {a \sqrt {\frac {1}{x}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+3}-\frac {a \int \frac {\left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (1+\frac {1}{a x}\right )^{n/2}}{\sqrt {\frac {1}{x}}}d\frac {1}{x}}{2 (n+3)}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
| \(\Big \downarrow \) 142 |
| \(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {a \sqrt {\frac {1}{x}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+3}-\frac {a \sqrt {\frac {1}{x}} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n+3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+3}{2},\frac {3}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{n+3}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\) |
-(((1 - 1/(a*x))^(5/2)*((a*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*Sqrt[x^(-1)])/(3 + n) - (a*((a - x^(-1))/(a + x^(-1)))^((3 + n)/2)*( 1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*Sqrt[x^(-1)]*Hypergeom etric2F1[1/2, (3 + n)/2, 3/2, 2/((a + x^(-1))*x)])/(3 + n)))/((x^(-1))^(5/ 2)*(c - a*c*x)^(5/2)))
3.4.77.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f *x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] && !IntegerQ[n]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Si mp[(-(1/x)^p)*((c + d*x)^p/(1 + c/(d*x))^p) Subst[Int[((1 + c*(x/d))^p*(( 1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a)^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[p]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (-a c x +c \right )^{\frac {5}{2}}}d x\]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {5}{2}}} \,d x } \]
integral(-sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^3*c^3*x^3 - 3* a^2*c^3*x^2 + 3*a*c^3*x - c^3), x)
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a\,c\,x\right )}^{5/2}} \,d x \]