Integrand size = 18, antiderivative size = 254 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=-\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9088 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 \left (1-\frac {1}{a x}\right )^{9/2} x^3}-\frac {768 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {128 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}} \]
-32/99*(a-1/x)^3*(1+1/a/x)^(3/2)*(-a*c*x+c)^(9/2)/a^4/(1-1/a/x)^(9/2)+9088 /3465*(1+1/a/x)^(3/2)*(-a*c*x+c)^(9/2)/a^4/(1-1/a/x)^(9/2)/x^3-768/385*(1+ 1/a/x)^(3/2)*(-a*c*x+c)^(9/2)/a^3/(1-1/a/x)^(9/2)/x^2+128/231*(1+1/a/x)^(3 /2)*(-a*c*x+c)^(9/2)/a^2/(1-1/a/x)^(9/2)/x+2/11*(a-1/x)^4*(1+1/a/x)^(3/2)* x*(-a*c*x+c)^(9/2)/a^4/(1-1/a/x)^(9/2)
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.34 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 c^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (5419-977 a x-1866 a^2 x^2+2710 a^3 x^3-1505 a^4 x^4+315 a^5 x^5\right )}{3465 a \sqrt {1-\frac {1}{a x}}} \]
(2*c^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(5419 - 977*a*x - 1866*a^2*x^2 + 2710*a^3*x^3 - 1505*a^4*x^4 + 315*a^5*x^5))/(3465*a*Sqrt[1 - 1/(a*x)])
Time = 0.31 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.77, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6727, 27, 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.
| \(\displaystyle \int (c-a c x)^{9/2} e^{\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6727 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \int \frac {\left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}}}{a^4 \left (\frac {1}{x}\right )^{13/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \int \frac {\left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{13/2}}d\frac {1}{x}}{a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {16}{11} \int \frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^4}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {16}{11} \left (-\frac {4}{3} \int \frac {\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^4}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {16}{11} \left (-\frac {4}{3} \left (\frac {2}{7} \int -\frac {\left (18 a-\frac {7}{x}\right ) \sqrt {1+\frac {1}{a x}}}{2 \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{3/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^4}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {16}{11} \left (-\frac {4}{3} \left (-\frac {1}{7} \int \frac {\left (18 a-\frac {7}{x}\right ) \sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{3/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^4}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {16}{11} \left (-\frac {4}{3} \left (\frac {1}{7} \left (\frac {71}{5} \int \frac {\sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}+\frac {36 a \left (\frac {1}{a x}+1\right )^{3/2}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{3/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^4}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} \left (-\frac {16}{11} \left (-\frac {4}{3} \left (\frac {1}{7} \left (\frac {36 a \left (\frac {1}{a x}+1\right )^{3/2}}{5 \left (\frac {1}{x}\right )^{5/2}}-\frac {142 \left (\frac {1}{a x}+1\right )^{3/2}}{15 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{3/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^4}{11 \left (\frac {1}{x}\right )^{11/2}}\right ) (c-a c x)^{9/2}}{a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
-((((-16*((-4*(((36*a*(1 + 1/(a*x))^(3/2))/(5*(x^(-1))^(5/2)) - (142*(1 + 1/(a*x))^(3/2))/(15*(x^(-1))^(3/2)))/7 - (2*a^2*(1 + 1/(a*x))^(3/2))/(7*(x ^(-1))^(7/2))))/3 - (2*(a - x^(-1))^3*(1 + 1/(a*x))^(3/2))/(9*(x^(-1))^(9/ 2))))/11 - (2*(a - x^(-1))^4*(1 + 1/(a*x))^(3/2))/(11*(x^(-1))^(11/2)))*(x ^(-1))^(9/2)*(c - a*c*x)^(9/2))/(a^4*(1 - 1/(a*x))^(9/2)))
3.3.26.3.1 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 = 0.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.27
| method | result | size |
| default | \(\frac {2 \sqrt {-c \left (a x -1\right )}\, c^{4} \left (a x +1\right ) \left (315 a^{4} x^{4}-1820 a^{3} x^{3}+4530 a^{2} x^{2}-6396 a x +5419\right )}{3465 \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(69\) |
| gosper | \(\frac {2 \left (a x +1\right ) \left (315 a^{4} x^{4}-1820 a^{3} x^{3}+4530 a^{2} x^{2}-6396 a x +5419\right ) \left (-a c x +c \right )^{\frac {9}{2}}}{3465 a \left (a x -1\right )^{4} \sqrt {\frac {a x -1}{a x +1}}}\) | \(72\) |
| risch | \(-\frac {2 c^{5} \left (a x -1\right ) \left (315 a^{5} x^{5}-1505 a^{4} x^{4}+2710 a^{3} x^{3}-1866 a^{2} x^{2}-977 a x +5419\right )}{3465 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\) | \(77\) |
2/3465/((a*x-1)/(a*x+1))^(1/2)*(-c*(a*x-1))^(1/2)*c^4*(a*x+1)*(315*a^4*x^4 -1820*a^3*x^3+4530*a^2*x^2-6396*a*x+5419)/a
Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.41 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (315 \, a^{6} c^{4} x^{6} - 1190 \, a^{5} c^{4} x^{5} + 1205 \, a^{4} c^{4} x^{4} + 844 \, a^{3} c^{4} x^{3} - 2843 \, a^{2} c^{4} x^{2} + 4442 \, a c^{4} x + 5419 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3465 \, {\left (a^{2} x - a\right )}} \]
2/3465*(315*a^6*c^4*x^6 - 1190*a^5*c^4*x^5 + 1205*a^4*c^4*x^4 + 844*a^3*c^ 4*x^3 - 2843*a^2*c^4*x^2 + 4442*a*c^4*x + 5419*c^4)*sqrt(-a*c*x + c)*sqrt( (a*x - 1)/(a*x + 1))/(a^2*x - a)
Timed out. \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.39 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (315 \, a^{5} \sqrt {-c} c^{4} x^{5} - 1505 \, a^{4} \sqrt {-c} c^{4} x^{4} + 2710 \, a^{3} \sqrt {-c} c^{4} x^{3} - 1866 \, a^{2} \sqrt {-c} c^{4} x^{2} - 977 \, a \sqrt {-c} c^{4} x + 5419 \, \sqrt {-c} c^{4}\right )} \sqrt {a x + 1}}{3465 \, a} \]
2/3465*(315*a^5*sqrt(-c)*c^4*x^5 - 1505*a^4*sqrt(-c)*c^4*x^4 + 2710*a^3*sq rt(-c)*c^4*x^3 - 1866*a^2*sqrt(-c)*c^4*x^2 - 977*a*sqrt(-c)*c^4*x + 5419*s qrt(-c)*c^4)*sqrt(a*x + 1)/a
Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.58 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (4096 \, \sqrt {2} \sqrt {-c} c^{3} - \frac {315 \, {\left (a c x + c\right )}^{5} \sqrt {-a c x - c} - 3080 \, {\left (a c x + c\right )}^{4} \sqrt {-a c x - c} c + 11880 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} c^{2} - 22176 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c^{3} - 18480 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{4}}{c^{2}}\right )} c^{2}}{3465 \, a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \]
2/3465*(4096*sqrt(2)*sqrt(-c)*c^3 - (315*(a*c*x + c)^5*sqrt(-a*c*x - c) - 3080*(a*c*x + c)^4*sqrt(-a*c*x - c)*c + 11880*(a*c*x + c)^3*sqrt(-a*c*x - c)*c^2 - 22176*(a*c*x + c)^2*sqrt(-a*c*x - c)*c^3 - 18480*(-a*c*x - c)^(3/ 2)*c^4)/c^2)*c^2/(a*abs(c)*sgn(a*x + 1))
Time = 4.53 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.30 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2\,c^4\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (315\,a^4\,x^4-1820\,a^3\,x^3+4530\,a^2\,x^2-6396\,a\,x+5419\right )}{3465\,a\,\left (a\,x-1\right )} \]