\[ \frac {n x Q_n(x)-n Q_{n-1}(x)}{x^2-1}-n (n+1) y(x)+\left (x^2-1\right ) y''(x)=0 \] ✓ Mathematica : cpu = 0.544659 (sec), leaf count = 6628
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};x^2\right )+\int _1^x\left (\frac {3 \left (n \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_{n-1}(K[1])-n \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1])\right )}{4 \left (\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n^2+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n+3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )\right ) (K[1]-1)^2}-\frac {3 \left (n \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_{n-1}(K[1])+n \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1])\right )}{4 \left (\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n^2+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n+3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )\right ) (K[1]+1)^2}+\frac {3 \left (-2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_{n-1}(K[1]) n^3+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^3-2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_{n-1}(K[1]) n^2+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^2-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^2+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^2+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n\right )}{4 \left (-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n^2-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )\right ){}^2 (K[1]-1)}-\frac {3 \left (2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_{n-1}(K[1]) n^3+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^3+2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_{n-1}(K[1]) n^2+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^2-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^2+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^2+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n\right )}{4 \left (-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n^2-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )\right ){}^2 (K[1]+1)}+\frac {3 \left (\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ){}^2 \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ){}^2 \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) K[1] Q_{n-1}(K[1]) n^5+2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ){}^2 \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ){}^2 \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) K[1] Q_{n-1}(K[1]) n^4+3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ){}^2 \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^4-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^4+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ){}^2 \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ){}^2 \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) K[1] Q_{n-1}(K[1]) n^3+3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ){}^2 \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^3-6 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^3-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) Q_n(K[1]) n^2\right )}{\left (-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n^2-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) n-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )\right ){}^2 \left (n^2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) K[1]^2+n \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) K[1]^2+3 n \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )-3 n \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )\right )}\right )dK[1] \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};x^2\right )+i x c_2 \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};x^2\right )+i x \, _2F_1\left (\frac {n}{2}+\frac {1}{2},-\frac {n}{2};\frac {3}{2};x^2\right ) \int _1^x\left (\frac {3 i \left (n \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) Q_{n-1}(K[2])-n \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) Q_n(K[2])\right )}{4 \left (\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n^2+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n+3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )\right ) (K[2]-1)^2}+\frac {3 i \left (n \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) Q_{n-1}(K[2])+n \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) Q_n(K[2])\right )}{4 \left (\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n^2+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n+3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )\right ) (K[2]+1)^2}-\frac {3 i \left (-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) Q_{n-1}(K[2]) n^3-2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) Q_n(K[2]) n^3-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) Q_{n-1}(K[2]) n^2-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) Q_{n-1}(K[2]) n^2+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) Q_{n-1}(K[2]) n^2-2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) Q_n(K[2]) n^2+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) Q_{n-1}(K[2]) n\right )}{4 \left (-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n^2-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )\right ){}^2 (K[2]+1)}+\frac {3 i \left (-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) Q_{n-1}(K[2]) n^3+2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) Q_n(K[2]) n^3-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) Q_{n-1}(K[2]) n^2-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) Q_{n-1}(K[2]) n^2+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) Q_{n-1}(K[2]) n^2+2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) Q_n(K[2]) n^2+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) Q_{n-1}(K[2]) n\right )}{4 \left (-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n^2-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )\right ){}^2 (K[2]-1)}+\frac {3 i \left (\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ){}^2 \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ){}^2 Q_{n-1}(K[2]) n^5-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ){}^2 \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ){}^2 K[2] Q_n(K[2]) n^5+2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ){}^2 \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ){}^2 Q_{n-1}(K[2]) n^4-2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ){}^2 \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ){}^2 K[2] Q_n(K[2]) n^4+\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ){}^2 \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ){}^2 Q_{n-1}(K[2]) n^3-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ){}^2 \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ){}^2 K[2] Q_n(K[2]) n^3\right )}{\left (-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n^2-\, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) n-3 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right ) n+3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )\right ){}^2 \left (n^2 \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) K[2]^2+n \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) K[2]^2+3 n \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )-3 n \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )-3 \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )\right )}\right )dK[2]\right \}\right \}\] ✓ Maple : cpu = 0.334 (sec), leaf count = 409
\[ \left \{ y \left ( x \right ) =3\, \left ( 1+x \right ) \left ( -{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} \left ( n+1 \right ) \int \!1/3\,{\frac {x{\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})} \left ( x{\it LegendreQ} \left ( n,x \right ) -{\it LegendreQ} \left ( n+1,x \right ) \right ) }{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3} \left ( \left ( {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}+ \left ( {n}^{2}+n-2 \right ) {x}^{2}{\mbox {$_2$F$_1$}(n/2+2,3/2-n/2;\,3/2;\,{x}^{2})} \right ) {\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}-1/3\,{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}{\mbox {$_2$F$_1$}(-n/2+2,n/2+5/2;\,5/2;\,{x}^{2})}{x}^{2} \left ( n+3 \right ) \left ( n-2 \right ) \right ) }}\,{\rm d}x+x{\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})} \left ( n+1 \right ) \int \!1/3\,{\frac {{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} \left ( x{\it LegendreQ} \left ( n,x \right ) -{\it LegendreQ} \left ( n+1,x \right ) \right ) }{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3} \left ( \left ( {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}+ \left ( {n}^{2}+n-2 \right ) {x}^{2}{\mbox {$_2$F$_1$}(n/2+2,3/2-n/2;\,3/2;\,{x}^{2})} \right ) {\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}-1/3\,{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}{\mbox {$_2$F$_1$}(-n/2+2,n/2+5/2;\,5/2;\,{x}^{2})}{x}^{2} \left ( n+3 \right ) \left ( n-2 \right ) \right ) }}\,{\rm d}x-1/3\,{\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}{\it \_C1}\,x-1/3\,{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}{\it \_C2} \right ) \left ( x-1 \right ) \right \} \]