Integrand size = 27, antiderivative size = 623 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {a^2 \left (5+4 n+n^2\right ) (1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{2 c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{2 c^2 x^2 \sqrt {c-a^2 c x^2}}-\frac {a n (1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{2 c^2 x \sqrt {c-a^2 c x^2}}+\frac {a^2 \left (30+17 n+6 n^2+n^3\right ) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{2 c^2 (1+n) (3+n) \sqrt {c-a^2 c x^2}}-\frac {a^2 \left (75+54 n+20 n^2+6 n^3+n^4\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{2 c^2 (3+n) \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {a^2 \left (90+59 n+8 n^2+2 n^3-2 n^4-n^5\right ) (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{2 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}}+\frac {a^2 \left (5+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {1+a x}{1-a x}\right )}{c^2 (1-n) \sqrt {c-a^2 c x^2}} \]
1/2*a^2*(n^2+4*n+5)*(-a*x+1)^(-3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1 )^(1/2)/c^2/(3+n)/(-a^2*c*x^2+c)^(1/2)-1/2*(-a*x+1)^(-3/2-1/2*n)*(a*x+1)^( -3/2+1/2*n)*(-a^2*x^2+1)^(1/2)/c^2/x^2/(-a^2*c*x^2+c)^(1/2)-1/2*a*n*(-a*x+ 1)^(-3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(1/2)/c^2/x/(-a^2*c*x^2+ c)^(1/2)+1/2*a^2*(n^3+6*n^2+17*n+30)*(-a*x+1)^(-1/2-1/2*n)*(a*x+1)^(-3/2+1 /2*n)*(-a^2*x^2+1)^(1/2)/c^2/(n^2+4*n+3)/(-a^2*c*x^2+c)^(1/2)-1/2*a^2*(n^4 +6*n^3+20*n^2+54*n+75)*(-a*x+1)^(1/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2 +1)^(1/2)/c^2/(-n^3-3*n^2+n+3)/(-a^2*c*x^2+c)^(1/2)+1/2*a^2*(-n^5-2*n^4+2* n^3+8*n^2+59*n+90)*(-a*x+1)^(3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^ (1/2)/c^2/(n^4-10*n^2+9)/(-a^2*c*x^2+c)^(1/2)+a^2*(n^2+5)*(-a*x+1)^(1/2-1/ 2*n)*(a*x+1)^(-1/2+1/2*n)*hypergeom([1, -1/2+1/2*n],[1/2+1/2*n],(a*x+1)/(- a*x+1))*(-a^2*x^2+1)^(1/2)/c^2/(1-n)/(-a^2*c*x^2+c)^(1/2)
Time = 0.53 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.43 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \left (\frac {a^2 \left (5+4 n+n^2\right )}{3+n}-\frac {1}{x^2}-\frac {a n}{x}-\frac {a^2 (1-a x) \left (-(-3+n)^2 (-1+n) \left (30+17 n+6 n^2+n^3\right )+(-3+n)^2 \left (75+54 n+20 n^2+6 n^3+n^4\right ) (-1+a x)-(-1+a x)^2 \left (-270-87 n+35 n^2+2 n^3+8 n^4+n^5-n^6+2 \left (45-41 n^2-5 n^4+n^6\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}-\frac {n}{2},\frac {5}{2}-\frac {n}{2},\frac {1-a x}{1+a x}\right )\right )\right )}{(-3+n)^2 (-1+n) (1+n) (3+n)}\right )}{2 c^2 \sqrt {c-a^2 c x^2}} \]
((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*((a^2*(5 + 4*n + n^2))/(3 + n) - x^(-2) - (a*n)/x - (a^2*(1 - a*x)*(-((-3 + n)^2*(- 1 + n)*(30 + 17*n + 6*n^2 + n^3)) + (-3 + n)^2*(75 + 54*n + 20*n^2 + 6*n^3 + n^4)*(-1 + a*x) - (-1 + a*x)^2*(-270 - 87*n + 35*n^2 + 2*n^3 + 8*n^4 + n^5 - n^6 + 2*(45 - 41*n^2 - 5*n^4 + n^6)*Hypergeometric2F1[1, 3/2 - n/2, 5/2 - n/2, (1 - a*x)/(1 + a*x)])))/((-3 + n)^2*(-1 + n)*(1 + n)*(3 + n)))) /(2*c^2*Sqrt[c - a^2*c*x^2])
Time = 0.93 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.67, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {6703, 6700, 144, 25, 27, 168, 25, 27, 172, 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.
| \(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6703 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \left (1-a^2 x^2\right )^{5/2}}dx}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 6700 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}}}{x^3}dx}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 144 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (-\frac {1}{2} \int -\frac {a (1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}} (n+5 a x)}{x^2}dx-\frac {(a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} \int \frac {a (1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}} (n+5 a x)}{x^2}dx-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \int \frac {(1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}} (n+5 a x)}{x^2}dx-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (-\int -\frac {a (1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}} \left (n^2+4 a x n+5\right )}{x}dx-\frac {n (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (\int \frac {a (1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}} \left (n^2+4 a x n+5\right )}{x}dx-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \int \frac {(1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}} \left (n^2+4 a x n+5\right )}{x}dx-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\left (n^2+4 n+5\right ) (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{n+3}-\frac {\int -\frac {a (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}} \left ((n+3) \left (n^2+5\right )+3 a \left (n^2+4 n+5\right ) x\right )}{x}dx}{a (n+3)}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\int \frac {a (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}} \left ((n+3) \left (n^2+5\right )+3 a \left (n^2+4 n+5\right ) x\right )}{x}dx}{a (n+3)}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\int \frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-5}{2}} \left ((n+3) \left (n^2+5\right )+3 a \left (n^2+4 n+5\right ) x\right )}{x}dx}{n+3}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\frac {\left (n^3+6 n^2+17 n+30\right ) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{n+1}-\frac {\int -\frac {a (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-5}{2}} \left ((n+1) (n+3) \left (n^2+5\right )+2 a \left (n^3+6 n^2+17 n+30\right ) x\right )}{x}dx}{a (n+1)}}{n+3}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\frac {\int \frac {a (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-5}{2}} \left ((n+1) (n+3) \left (n^2+5\right )+2 a \left (n^3+6 n^2+17 n+30\right ) x\right )}{x}dx}{a (n+1)}+\frac {\left (n^3+6 n^2+17 n+30\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\frac {\int \frac {(1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-5}{2}} \left ((n+1) (n+3) \left (n^2+5\right )+2 a \left (n^3+6 n^2+17 n+30\right ) x\right )}{x}dx}{n+1}+\frac {\left (n^3+6 n^2+17 n+30\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\frac {\frac {\int \frac {a (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-5}{2}} \left ((1-n) (n+1) (n+3) \left (n^2+5\right )-a \left (n^4+6 n^3+20 n^2+54 n+75\right ) x\right )}{x}dx}{a (1-n)}-\frac {\left (n^4+6 n^3+20 n^2+54 n+75\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^3+6 n^2+17 n+30\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\frac {\frac {\int \frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-5}{2}} \left ((1-n) (n+1) (n+3) \left (n^2+5\right )-a \left (n^4+6 n^3+20 n^2+54 n+75\right ) x\right )}{x}dx}{1-n}-\frac {\left (n^4+6 n^3+20 n^2+54 n+75\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^3+6 n^2+17 n+30\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\frac {\frac {\frac {\int \frac {a (1-n) (3-n) (n+1) (n+3) \left (n^2+5\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{x}dx}{a (3-n)}+\frac {\left (-n^5-2 n^4+2 n^3+8 n^2+59 n+90\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}}}{3-n}}{1-n}-\frac {\left (n^4+6 n^3+20 n^2+54 n+75\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^3+6 n^2+17 n+30\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\frac {\frac {(1-n) (n+1) (n+3) \left (n^2+5\right ) \int \frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{x}dx+\frac {\left (-n^5-2 n^4+2 n^3+8 n^2+59 n+90\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}}}{3-n}}{1-n}-\frac {\left (n^4+6 n^3+20 n^2+54 n+75\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^3+6 n^2+17 n+30\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 141 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2} a \left (a \left (\frac {\frac {\frac {2 (n+1) (n+3) \left (n^2+5\right ) (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-1}{2},\frac {n+1}{2},\frac {a x+1}{1-a x}\right )+\frac {\left (-n^5-2 n^4+2 n^3+8 n^2+59 n+90\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}}}{3-n}}{1-n}-\frac {\left (n^4+6 n^3+20 n^2+54 n+75\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^3+6 n^2+17 n+30\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {\left (n^2+4 n+5\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {n (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{2 x^2}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
(Sqrt[1 - a^2*x^2]*(-1/2*((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2))/x ^2 + (a*(-((n*(1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2))/x) + a*(((5 + 4*n + n^2)*(1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2))/(3 + n) + (((30 + 17*n + 6*n^2 + n^3)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2))/(1 + n) + (-(((75 + 54*n + 20*n^2 + 6*n^3 + n^4)*(1 - a*x)^((1 - n)/2)*(1 + a* x)^((-3 + n)/2))/(1 - n)) + (((90 + 59*n + 8*n^2 + 2*n^3 - 2*n^4 - n^5)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((-3 + n)/2))/(3 - n) + 2*(1 + n)*(3 + n)*(5 + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Hypergeometric2F1[1, (-1 + n)/2, (1 + n)/2, (1 + a*x)/(1 - a*x)])/(1 - n))/(1 + n))/(3 + n))))/ 2))/(c^2*Sqrt[c - a^2*c*x^2])
3.14.54.3.1 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_] :> 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] + S imp[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)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[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^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar t[p]) Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !I ntegerQ[n/2]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{x^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
integral(-sqrt(-a^2*c*x^2 + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^6*c^3*x^9 - 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 - c^3*x^3), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{3} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^3\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]