Integrand size = 20, antiderivative size = 197 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=-\frac {8 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{33 a \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {856 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{1155 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {16 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{9/2}}{21 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{9/2}}{11 a^3 \left (1-\frac {1}{a x}\right )^{9/2}} \]
-8/33*(1+1/a/x)^(5/2)*(-a*c*x+c)^(9/2)/a/(1-1/a/x)^(9/2)-856/1155*(1+1/a/x )^(5/2)*(-a*c*x+c)^(9/2)/a^3/(1-1/a/x)^(9/2)/x^2+16/21*(1+1/a/x)^(5/2)*(-a *c*x+c)^(9/2)/a^2/(1-1/a/x)^(9/2)/x+2/11*(a-1/x)^3*(1+1/a/x)^(5/2)*x*(-a*c *x+c)^(9/2)/a^3/(1-1/a/x)^(9/2)
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.39 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 c^4 \sqrt {1+\frac {1}{a x}} (1+a x)^2 \sqrt {c-a c x} \left (-533+755 a x-455 a^2 x^2+105 a^3 x^3\right )}{1155 a \sqrt {1-\frac {1}{a x}}} \]
(2*c^4*Sqrt[1 + 1/(a*x)]*(1 + a*x)^2*Sqrt[c - a*c*x]*(-533 + 755*a*x - 455 *a^2*x^2 + 105*a^3*x^3))/(1155*a*Sqrt[1 - 1/(a*x)])
Time = 0.29 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.80, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6727, 27, 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 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \int \frac {\left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2}}{a^3 \left (\frac {1}{x}\right )^{13/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \int \frac {\left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2}}{\left (\frac {1}{x}\right )^{13/2}}d\frac {1}{x}}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {12}{11} \int \frac {\left (a-\frac {1}{x}\right )^2 \left (1+\frac {1}{a x}\right )^{3/2}}{\left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}-\frac {2 \left (\frac {1}{a x}+1\right )^{5/2} \left (a-\frac {1}{x}\right )^3}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {12}{11} \left (\frac {2}{9} \int -\frac {\left (22 a-\frac {9}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}{2 \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{5/2}}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{5/2} \left (a-\frac {1}{x}\right )^3}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {12}{11} \left (-\frac {1}{9} \int \frac {\left (22 a-\frac {9}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}{\left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{5/2}}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{5/2} \left (a-\frac {1}{x}\right )^3}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {12}{11} \left (\frac {1}{9} \left (\frac {107}{7} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2}}{\left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}+\frac {44 a \left (\frac {1}{a x}+1\right )^{5/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{5/2}}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{5/2} \left (a-\frac {1}{x}\right )^3}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} \left (-\frac {12}{11} \left (\frac {1}{9} \left (\frac {44 a \left (\frac {1}{a x}+1\right )^{5/2}}{7 \left (\frac {1}{x}\right )^{7/2}}-\frac {214 \left (\frac {1}{a x}+1\right )^{5/2}}{35 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{5/2}}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{5/2} \left (a-\frac {1}{x}\right )^3}{11 \left (\frac {1}{x}\right )^{11/2}}\right ) (c-a c x)^{9/2}}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
-((((-12*(((44*a*(1 + 1/(a*x))^(5/2))/(7*(x^(-1))^(7/2)) - (214*(1 + 1/(a* x))^(5/2))/(35*(x^(-1))^(5/2)))/9 - (2*a^2*(1 + 1/(a*x))^(5/2))/(9*(x^(-1) )^(9/2))))/11 - (2*(a - x^(-1))^3*(1 + 1/(a*x))^(5/2))/(11*(x^(-1))^(11/2) ))*(x^(-1))^(9/2)*(c - a*c*x)^(9/2))/(a^3*(1 - 1/(a*x))^(9/2)))
3.3.43.3.1 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 = 0.42 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.32
| method | result | size |
| gosper | \(\frac {2 \left (a x +1\right ) \left (105 a^{3} x^{3}-455 a^{2} x^{2}+755 a x -533\right ) \left (-a c x +c \right )^{\frac {9}{2}}}{1155 a \left (a x -1\right )^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(64\) |
| default | \(\frac {2 \left (a x -1\right ) \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{4} \left (105 a^{3} x^{3}-455 a^{2} x^{2}+755 a x -533\right )}{1155 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(66\) |
| risch | \(-\frac {2 c^{5} \left (a x -1\right ) \left (105 a^{5} x^{5}-245 a^{4} x^{4}-50 a^{3} x^{3}+522 a^{2} x^{2}-311 a x -533\right )}{1155 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\) | \(77\) |
2/1155*(a*x+1)*(105*a^3*x^3-455*a^2*x^2+755*a*x-533)*(-a*c*x+c)^(9/2)/a/(a *x-1)^3/((a*x-1)/(a*x+1))^(3/2)
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.53 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (105 \, a^{6} c^{4} x^{6} - 140 \, a^{5} c^{4} x^{5} - 295 \, a^{4} c^{4} x^{4} + 472 \, a^{3} c^{4} x^{3} + 211 \, a^{2} c^{4} x^{2} - 844 \, a c^{4} x - 533 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{1155 \, {\left (a^{2} x - a\right )}} \]
2/1155*(105*a^6*c^4*x^6 - 140*a^5*c^4*x^5 - 295*a^4*c^4*x^4 + 472*a^3*c^4* x^3 + 211*a^2*c^4*x^2 - 844*a*c^4*x - 533*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 = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.54 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (105 \, a^{5} \sqrt {-c} c^{4} x^{5} - 455 \, a^{4} \sqrt {-c} c^{4} x^{4} + 650 \, a^{3} \sqrt {-c} c^{4} x^{3} - 78 \, a^{2} \sqrt {-c} c^{4} x^{2} - 755 \, a \sqrt {-c} c^{4} x + 533 \, \sqrt {-c} c^{4}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}}{1155 \, {\left (a x - 1\right )} a} \]
2/1155*(105*a^5*sqrt(-c)*c^4*x^5 - 455*a^4*sqrt(-c)*c^4*x^4 + 650*a^3*sqrt (-c)*c^4*x^3 - 78*a^2*sqrt(-c)*c^4*x^2 - 755*a*sqrt(-c)*c^4*x + 533*sqrt(- c)*c^4)*(a*x + 1)^(3/2)/((a*x - 1)*a)
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.66 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=-\frac {2 \, {\left (512 \, \sqrt {2} \sqrt {-c} c^{3} + \frac {105 \, {\left (a c x + c\right )}^{5} \sqrt {-a c x - c} - 770 \, {\left (a c x + c\right )}^{4} \sqrt {-a c x - c} c + 1980 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} c^{2} - 1848 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c^{3}}{c^{2}}\right )} c^{2}}{1155 \, a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \]
-2/1155*(512*sqrt(2)*sqrt(-c)*c^3 + (105*(a*c*x + c)^5*sqrt(-a*c*x - c) - 770*(a*c*x + c)^4*sqrt(-a*c*x - c)*c + 1980*(a*c*x + c)^3*sqrt(-a*c*x - c) *c^2 - 1848*(a*c*x + c)^2*sqrt(-a*c*x - c)*c^3)/c^2)*c^2/(a*abs(c)*sgn(a*x + 1))
Time = 4.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.56 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (105\,a^5\,x^5-35\,a^4\,x^4-330\,a^3\,x^3+142\,a^2\,x^2+353\,a\,x-491\right )}{1155\,a}-\frac {2048\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{1155\,a\,\left (a\,x-1\right )} \]