Integrand size = 20, antiderivative size = 368 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=-\frac {16 \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {94208 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^5}-\frac {40960 (c-a c x)^{9/2}}{231 a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^4}+\frac {4096 (c-a c x)^{9/2}}{231 a^4 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {1024 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {320 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}} \]
-16/33*(a-1/x)^5*(-a*c*x+c)^(9/2)/a^6/(1-1/a/x)^(9/2)/(1+1/a/x)^(1/2)-9420 8/231*(-a*c*x+c)^(9/2)/a^6/(1-1/a/x)^(9/2)/x^5/(1+1/a/x)^(1/2)-40960/231*( -a*c*x+c)^(9/2)/a^5/(1-1/a/x)^(9/2)/x^4/(1+1/a/x)^(1/2)+4096/231*(-a*c*x+c )^(9/2)/a^4/(1-1/a/x)^(9/2)/x^3/(1+1/a/x)^(1/2)-1024/231*(a-1/x)^3*(-a*c*x +c)^(9/2)/a^6/(1-1/a/x)^(9/2)/x^2/(1+1/a/x)^(1/2)+320/231*(a-1/x)^4*(-a*c* x+c)^(9/2)/a^6/(1-1/a/x)^(9/2)/x/(1+1/a/x)^(1/2)+2/11*(a-1/x)^6*x*(-a*c*x+ c)^(9/2)/a^6/(1-1/a/x)^(9/2)/(1+1/a/x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.23 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 c^4 \sqrt {c-a c x} \left (-46355-23062 a x+5419 a^2 x^2-2132 a^3 x^3+755 a^4 x^4-182 a^5 x^5+21 a^6 x^6\right )}{231 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
(2*c^4*Sqrt[c - a*c*x]*(-46355 - 23062*a*x + 5419*a^2*x^2 - 2132*a^3*x^3 + 755*a^4*x^4 - 182*a^5*x^5 + 21*a^6*x^6))/(231*a^2*Sqrt[1 - 1/(a^2*x^2)]*x )
Time = 0.35 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.73, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6727, 27, 105, 105, 105, 105, 100, 27, 87, 48}
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.
| \(\displaystyle \int (c-a c x)^{9/2} e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6727 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \int \frac {\left (a-\frac {1}{x}\right )^6}{a^6 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{13/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \int \frac {\left (a-\frac {1}{x}\right )^6}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{13/2}}d\frac {1}{x}}{a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {24}{11} \int \frac {\left (a-\frac {1}{x}\right )^5}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^6}{11 \left (\frac {1}{x}\right )^{11/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {24}{11} \left (-\frac {20}{9} \int \frac {\left (a-\frac {1}{x}\right )^4}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^6}{11 \left (\frac {1}{x}\right )^{11/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {24}{11} \left (-\frac {20}{9} \left (-\frac {16}{7} \int \frac {\left (a-\frac {1}{x}\right )^3}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^6}{11 \left (\frac {1}{x}\right )^{11/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {24}{11} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \int \frac {\left (a-\frac {1}{x}\right )^2}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^6}{11 \left (\frac {1}{x}\right )^{11/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {24}{11} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (\frac {2}{3} \int -\frac {10 a-\frac {3}{x}}{2 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^6}{11 \left (\frac {1}{x}\right )^{11/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {24}{11} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (-\frac {1}{3} \int \frac {10 a-\frac {3}{x}}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^6}{11 \left (\frac {1}{x}\right )^{11/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {24}{11} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (\frac {1}{3} \left (23 \int \frac {1}{\left (1+\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{x}}}d\frac {1}{x}+\frac {20 a}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^6}{11 \left (\frac {1}{x}\right )^{11/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} \left (-\frac {24}{11} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (\frac {1}{3} \left (\frac {20 a}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}+\frac {46 \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^6}{11 \left (\frac {1}{x}\right )^{11/2} \sqrt {\frac {1}{a x}+1}}\right ) (c-a c x)^{9/2}}{a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
-((((-24*((-20*((-16*((-12*(((20*a)/(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]) + (46 *Sqrt[x^(-1)])/Sqrt[1 + 1/(a*x)])/3 - (2*a^2)/(3*Sqrt[1 + 1/(a*x)]*(x^(-1) )^(3/2))))/5 - (2*(a - x^(-1))^3)/(5*Sqrt[1 + 1/(a*x)]*(x^(-1))^(5/2))))/7 - (2*(a - x^(-1))^4)/(7*Sqrt[1 + 1/(a*x)]*(x^(-1))^(7/2))))/9 - (2*(a - x ^(-1))^5)/(9*Sqrt[1 + 1/(a*x)]*(x^(-1))^(9/2))))/11 - (2*(a - x^(-1))^6)/( 11*Sqrt[1 + 1/(a*x)]*(x^(-1))^(11/2)))*(x^(-1))^(9/2)*(c - a*c*x)^(9/2))/( a^6*(1 - 1/(a*x))^(9/2)))
3.3.70.3.1 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_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 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)/((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[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]
Time = 0.46 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.24
| method | result | size |
| gosper | \(\frac {2 \left (a x +1\right ) \left (21 a^{6} x^{6}-182 a^{5} x^{5}+755 a^{4} x^{4}-2132 a^{3} x^{3}+5419 a^{2} x^{2}-23062 a x -46355\right ) \left (-a c x +c \right )^{\frac {9}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{231 a \left (a x -1\right )^{6}}\) | \(88\) |
| default | \(\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{4} \left (21 a^{6} x^{6}-182 a^{5} x^{5}+755 a^{4} x^{4}-2132 a^{3} x^{3}+5419 a^{2} x^{2}-23062 a x -46355\right )}{231 \left (a x -1\right )^{2} a}\) | \(92\) |
| risch | \(-\frac {2 \left (21 a^{5} x^{5}-203 a^{4} x^{4}+958 a^{3} x^{3}-3090 a^{2} x^{2}+8509 a x -31571\right ) \left (a x +1\right ) c^{5} \sqrt {\frac {a x -1}{a x +1}}}{231 a \sqrt {-c \left (a x -1\right )}}+\frac {128 c^{5} \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-c \left (a x -1\right )}}\) | \(111\) |
2/231*(a*x+1)*(21*a^6*x^6-182*a^5*x^5+755*a^4*x^4-2132*a^3*x^3+5419*a^2*x^ 2-23062*a*x-46355)*(-a*c*x+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2)/a/(a*x-1)^6
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.29 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (21 \, a^{6} c^{4} x^{6} - 182 \, a^{5} c^{4} x^{5} + 755 \, a^{4} c^{4} x^{4} - 2132 \, a^{3} c^{4} x^{3} + 5419 \, a^{2} c^{4} x^{2} - 23062 \, a c^{4} x - 46355 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{231 \, {\left (a^{2} x - a\right )}} \]
2/231*(21*a^6*c^4*x^6 - 182*a^5*c^4*x^5 + 755*a^4*c^4*x^4 - 2132*a^3*c^4*x ^3 + 5419*a^2*c^4*x^2 - 23062*a*c^4*x - 46355*c^4)*sqrt(-a*c*x + c)*sqrt(( a*x - 1)/(a*x + 1))/(a^2*x - a)
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.41 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (21 \, a^{7} \sqrt {-c} c^{4} x^{7} - 161 \, a^{6} \sqrt {-c} c^{4} x^{6} + 573 \, a^{5} \sqrt {-c} c^{4} x^{5} - 1377 \, a^{4} \sqrt {-c} c^{4} x^{4} + 3287 \, a^{3} \sqrt {-c} c^{4} x^{3} - 17643 \, a^{2} \sqrt {-c} c^{4} x^{2} - 69417 \, a \sqrt {-c} c^{4} x - 46355 \, \sqrt {-c} c^{4}\right )} {\left (a x - 1\right )}^{2}}{231 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
2/231*(21*a^7*sqrt(-c)*c^4*x^7 - 161*a^6*sqrt(-c)*c^4*x^6 + 573*a^5*sqrt(- c)*c^4*x^5 - 1377*a^4*sqrt(-c)*c^4*x^4 + 3287*a^3*sqrt(-c)*c^4*x^3 - 17643 *a^2*sqrt(-c)*c^4*x^2 - 69417*a*sqrt(-c)*c^4*x - 46355*sqrt(-c)*c^4)*(a*x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*(a*x + 1)^(3/2))
Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 4.47 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.30 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (21\,a^5\,x^5-161\,a^4\,x^4+594\,a^3\,x^3-1538\,a^2\,x^2+3881\,a\,x-19181\right )}{231\,a}-\frac {131072\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{231\,a\,\left (a\,x-1\right )} \]