Integrand size = 27, antiderivative size = 463 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {(6+n) \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {\left (15+6 n+n^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {\left (18+7 n-2 n^2-n^3\right ) \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{(1-n) \left (c-a^2 c x^2\right )^{5/2}} \]
-(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*x^5/(3+ n)/(-a^2*c*x^2+c)^(5/2)-(6+n)*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-1/2-1/2*n)*( 1+1/a/x)^(-3/2+1/2*n)*x^5/(n^2+4*n+3)/(-a^2*c*x^2+c)^(5/2)+(n^2+6*n+15)*(1 -1/a^2/x^2)^(5/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*x^5/(-n^3-3 *n^2+n+3)/(-a^2*c*x^2+c)^(5/2)-(-n^3-2*n^2+7*n+18)*(1-1/a^2/x^2)^(5/2)*(1- 1/a/x)^(3/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*x^5/(n^4-10*n^2+9)/(-a^2*c*x^2+c )^(5/2)-2*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n) *x^5*hypergeom([1, -1/2+1/2*n],[1/2+1/2*n],(a+1/x)/(a-1/x))/(1-n)/(-a^2*c* x^2+c)^(5/2)
Time = 2.45 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.43 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (-1+a^2 x^2\right ) \left (\frac {8 e^{n \coth ^{-1}(a x)} (n-a x)}{-1+n^2}+\frac {e^{n \coth ^{-1}(a x)} \left (26 n-2 n^3-27 a x+3 a n^2 x+2 n \left (-1+n^2\right ) \cosh \left (2 \coth ^{-1}(a x)\right )-3 a \left (-1+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (3 \coth ^{-1}(a x)\right )\right )}{9-10 n^2+n^4}-\frac {8 a e^{(1+n) \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )}{1+n}\right )}{4 a^5 c \left (c-a^2 c x^2\right )^{3/2}} \]
((-1 + a^2*x^2)*((8*E^(n*ArcCoth[a*x])*(n - a*x))/(-1 + n^2) + (E^(n*ArcCo th[a*x])*(26*n - 2*n^3 - 27*a*x + 3*a*n^2*x + 2*n*(-1 + n^2)*Cosh[2*ArcCot h[a*x]] - 3*a*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(9 - 10*n^2 + n^4) - (8*a*E^((1 + n)*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*x*H ypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])])/(1 + n)))/( 4*a^5*c*(c - a^2*c*x^2)^(3/2))
Time = 0.78 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.78, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {6746, 6749, 144, 25, 27, 172, 25, 27, 172, 27, 172, 27, 141}
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.
| \(\displaystyle \int \frac {x^4 e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6746 |
| \(\displaystyle \frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}dx}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 6749 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}} xd\frac {1}{x}}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 144 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}}}{n+3}-\frac {a \int -\frac {\left (a (n+3)+\frac {3}{x}\right ) \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}} x}{a^2}d\frac {1}{x}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {a \int \frac {\left (a (n+3)+\frac {3}{x}\right ) \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}} x}{a^2}d\frac {1}{x}}{n+3}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\int \left (a (n+3)+\frac {3}{x}\right ) \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}} xd\frac {1}{x}}{a (n+3)}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\frac {a (n+6) \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}}}{n+1}-\frac {a \int -\frac {\left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}} \left (a (n+1) (n+3)+\frac {2 (n+6)}{x}\right ) x}{a}d\frac {1}{x}}{n+1}}{a (n+3)}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\frac {a \int \frac {\left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}} \left (a (n+1) (n+3)+\frac {2 (n+6)}{x}\right ) x}{a}d\frac {1}{x}}{n+1}+\frac {a (n+6) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}}{a (n+3)}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\frac {\int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}} \left (a (n+1) (n+3)+\frac {2 (n+6)}{x}\right ) xd\frac {1}{x}}{n+1}+\frac {a (n+6) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}}{a (n+3)}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\frac {\frac {a \int \frac {\left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}} \left (a (1-n) (n+1) (n+3)-\frac {n^2+6 n+15}{x}\right ) x}{a}d\frac {1}{x}}{1-n}-\frac {a \left (n^2+6 n+15\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {a (n+6) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}}{a (n+3)}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\frac {\frac {\int \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}} \left (a (1-n) (n+1) (n+3)-\frac {n^2+6 n+15}{x}\right ) xd\frac {1}{x}}{1-n}-\frac {a \left (n^2+6 n+15\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {a (n+6) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}}{a (n+3)}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\frac {\frac {\frac {a \int \left (n^4-10 n^2+9\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n-3}{2}} xd\frac {1}{x}}{3-n}+\frac {a \left (-n^3-2 n^2+7 n+18\right ) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}}}{3-n}}{1-n}-\frac {a \left (n^2+6 n+15\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {a (n+6) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}}{a (n+3)}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\frac {\frac {\frac {a \left (n^4-10 n^2+9\right ) \int \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n-3}{2}} xd\frac {1}{x}}{3-n}+\frac {a \left (-n^3-2 n^2+7 n+18\right ) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}}}{3-n}}{1-n}-\frac {a \left (n^2+6 n+15\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {a (n+6) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}}{a (n+3)}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 141 |
| \(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {\frac {\frac {\frac {2 a \left (n^4-10 n^2+9\right ) \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-1}{2},\frac {n+1}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{(1-n) (3-n)}+\frac {a \left (-n^3-2 n^2+7 n+18\right ) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}}}{3-n}}{1-n}-\frac {a \left (n^2+6 n+15\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {a (n+6) \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}}{a (n+3)}+\frac {\left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
-(((1 - 1/(a^2*x^2))^(5/2)*x^5*(((1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^ ((-3 + n)/2))/(3 + n) + ((a*(6 + n)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x ))^((-3 + n)/2))/(1 + n) + (-((a*(15 + 6*n + n^2)*(1 - 1/(a*x))^((1 - n)/2 )*(1 + 1/(a*x))^((-3 + n)/2))/(1 - n)) + ((a*(18 + 7*n - 2*n^2 - n^3)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2))/(3 - n) + (2*a*(9 - 10*n ^2 + n^4)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*Hypergeomet ric2F1[1, (-1 + n)/2, (1 + n)/2, (a + x^(-1))/(a - x^(-1))])/((1 - n)*(3 - n)))/(1 - n))/(1 + n))/(a*(3 + n))))/(c - a^2*c*x^2)^(5/2))
3.8.56.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f ))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] || !Su mSimplerQ[p, 1]) && !ILtQ[m, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[b*(a + b*x)^(m + 1)*( c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x )^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !SumSimplerQ[n, 1] && !SumSimplerQ[p, 1]))] /; FreeQ[{a, b, c, d, e, f , m, n, p}, x] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ (b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1) *(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f )*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1 ])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p) Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_.), x _Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/ x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2] && IntegerQ[m]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x^{4}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
\[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
integral(-sqrt(-a^2*c*x^2 + c)*x^4*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^6*c^3* x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)
\[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{4} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^4\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]