Integrand size = 88, antiderivative size = 92 \[ \int \frac {\left (1+2 x+x^2\right ) \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d}+\sqrt [4]{d} x}{\sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d}+\sqrt [4]{d} x}{\sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{3/4}} \]
arctan((d^(1/4)+d^(1/4)*x)/(1+(-k^2-1)*x^2+k^2*x^4)^(1/4))/d^(3/4)-arctanh ((d^(1/4)+d^(1/4)*x)/(1+(-k^2-1)*x^2+k^2*x^4)^(1/4))/d^(3/4)
\[ \int \frac {\left (1+2 x+x^2\right ) \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx=\int \frac {\left (1+2 x+x^2\right ) \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx \]
Integrate[((1 + 2*x + x^2)*(-2 - (-1 + k)*(1 + k)*x + 2*k^2*x^2))/(((1 - x ^2)*(1 - k^2*x^2))^(3/4)*(1 - d - (1 + 3*d)*x - (3*d + k^2)*x^2 + (-d + k^ 2)*x^3)),x]
Integrate[((1 + 2*x + x^2)*(-2 - (-1 + k)*(1 + k)*x + 2*k^2*x^2))/(((1 - x ^2)*(1 - k^2*x^2))^(3/4)*(1 - d - (1 + 3*d)*x - (3*d + k^2)*x^2 + (-d + k^ 2)*x^3)), x]
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.
| \(\displaystyle \int \frac {\left (x^2+2 x+1\right ) \left (2 k^2 x^2-(k-1) (k+1) x-2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (x^3 \left (k^2-d\right )-x^2 \left (3 d+k^2\right )-(3 d+1) x-d+1\right )} \, dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {(x+1)^2 \left (2 k^2 x^2-(k-1) (k+1) x-2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (x^3 \left (k^2-d\right )-x^2 \left (3 d+k^2\right )-(3 d+1) x-d+1\right )}dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {(x+1)^2 \left (2 k^2 x^2-(k-1) (k+1) x-2\right )}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4} \left (x^3 \left (k^2-d\right )-x^2 \left (3 d+k^2\right )-(3 d+1) x-d+1\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {(x+1)^2 \left (2 k^2 x^2+(1-k) (k+1) x-2\right )}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4} \left (-\left (x^3 \left (d-k^2\right )\right )-x^2 \left (3 d+k^2\right )-(3 d+1) x-d+1\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-3 d^2 \left (1-k^2\right )+x^2 \left (-3 d^2 \left (1-k^2\right )+24 d k^4+\left (5 k^2+3\right ) k^4\right )+x \left (-6 d^2 \left (1-k^2\right )-d \left (-19 k^4-14 k^2+1\right )-k^6+k^2\right )+d \left (5 k^4+2 k^2+1\right )-k^2 \left (7 k^2+1\right )}{\left (d-k^2\right )^2 \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4} \left (-\left (x^3 \left (d-k^2\right )\right )-x^2 \left (3 d+k^2\right )-(3 d+1) x-d+1\right )}-\frac {2 k^2 x}{\left (d-k^2\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}+\frac {-d \left (1-3 k^2\right )+5 k^4+k^2}{\left (d-k^2\right )^2 \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (3 d^2 \left (1-k^2\right )-d \left (5 k^4+2 k^2+1\right )+7 k^4+k^2\right ) \int \frac {1}{\left (-\left (\left (d-k^2\right ) x^3\right )-\left (k^2+3 d\right ) x^2-(3 d+1) x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}dx}{\left (d-k^2\right )^2}+\frac {\left (-3 d^2 \left (1-k^2\right )+24 d k^4+\left (5 k^2+3\right ) k^4\right ) \int \frac {x^2}{\left (-\left (\left (d-k^2\right ) x^3\right )-\left (k^2+3 d\right ) x^2-(3 d+1) x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}dx}{\left (d-k^2\right )^2}+\frac {\left (-6 d^2 \left (1-k^2\right )-d \left (-19 k^4-14 k^2+1\right )-k^6+k^2\right ) \int \frac {x}{\left (-\left (\left (d-k^2\right ) x^3\right )-\left (k^2+3 d\right ) x^2-(3 d+1) x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{3/4}}dx}{\left (d-k^2\right )^2}+\frac {\sqrt {2} \sqrt {k^2-1} k^{3/2} \sqrt {\left (2 k^2 x^2-k^2-1\right )^2} \sqrt {\frac {\left (k^2 \left (1-2 x^2\right )+1\right )^2}{\left (1-k^2\right )^2 \left (1-\frac {2 k \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{1-k^2}\right )^2}} \left (1-\frac {2 k \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{1-k^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt {k} \sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{\sqrt {k^2-1}}\right ),\frac {1}{2}\right )}{\left (d-k^2\right ) \left (-2 k^2 x^2+k^2+1\right ) \sqrt {\left (-\left (k^2 \left (1-2 x^2\right )\right )-1\right )^2}}+\frac {x \left (-d \left (1-3 k^2\right )+5 k^4+k^2\right ) \left (\frac {1-x^2}{1-k^2 x^2}\right )^{3/4} \left (1-k^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {\left (1-k^2\right ) x^2}{1-k^2 x^2}\right )}{\left (d-k^2\right )^2 \left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{3/4}}\) |
Int[((1 + 2*x + x^2)*(-2 - (-1 + k)*(1 + k)*x + 2*k^2*x^2))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(1 - d - (1 + 3*d)*x - (3*d + k^2)*x^2 + (-d + k^2)*x^3 )),x]
3.13.66.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) , x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F reeQ[{a, b, c, d, e, n, p}, x]
\[\int \frac {\left (x^{2}+2 x +1\right ) \left (-2-\left (-1+k \right ) \left (1+k \right ) x +2 k^{2} x^{2}\right )}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {3}{4}} \left (1-d -\left (1+3 d \right ) x -\left (k^{2}+3 d \right ) x^{2}+\left (k^{2}-d \right ) x^{3}\right )}d x\]
int((x^2+2*x+1)*(-2-(-1+k)*(1+k)*x+2*k^2*x^2)/((-x^2+1)*(-k^2*x^2+1))^(3/4 )/(1-d-(1+3*d)*x-(k^2+3*d)*x^2+(k^2-d)*x^3),x)
int((x^2+2*x+1)*(-2-(-1+k)*(1+k)*x+2*k^2*x^2)/((-x^2+1)*(-k^2*x^2+1))^(3/4 )/(1-d-(1+3*d)*x-(k^2+3*d)*x^2+(k^2-d)*x^3),x)
Timed out. \[ \int \frac {\left (1+2 x+x^2\right ) \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx=\text {Timed out} \]
integrate((x^2+2*x+1)*(-2-(-1+k)*(1+k)*x+2*k^2*x^2)/((-x^2+1)*(-k^2*x^2+1) )^(3/4)/(1-d-(1+3*d)*x-(k^2+3*d)*x^2+(k^2-d)*x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {\left (1+2 x+x^2\right ) \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx=\text {Timed out} \]
integrate((x**2+2*x+1)*(-2-(-1+k)*(1+k)*x+2*k**2*x**2)/((-x**2+1)*(-k**2*x **2+1))**(3/4)/(1-d-(1+3*d)*x-(k**2+3*d)*x**2+(k**2-d)*x**3),x)
\[ \int \frac {\left (1+2 x+x^2\right ) \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx=\int { \frac {{\left (2 \, k^{2} x^{2} - {\left (k + 1\right )} {\left (k - 1\right )} x - 2\right )} {\left (x^{2} + 2 \, x + 1\right )}}{{\left ({\left (k^{2} - d\right )} x^{3} - {\left (k^{2} + 3 \, d\right )} x^{2} - {\left (3 \, d + 1\right )} x - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {3}{4}}} \,d x } \]
integrate((x^2+2*x+1)*(-2-(-1+k)*(1+k)*x+2*k^2*x^2)/((-x^2+1)*(-k^2*x^2+1) )^(3/4)/(1-d-(1+3*d)*x-(k^2+3*d)*x^2+(k^2-d)*x^3),x, algorithm="maxima")
integrate((2*k^2*x^2 - (k + 1)*(k - 1)*x - 2)*(x^2 + 2*x + 1)/(((k^2 - d)* x^3 - (k^2 + 3*d)*x^2 - (3*d + 1)*x - d + 1)*((k^2*x^2 - 1)*(x^2 - 1))^(3/ 4)), x)
\[ \int \frac {\left (1+2 x+x^2\right ) \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx=\int { \frac {{\left (2 \, k^{2} x^{2} - {\left (k + 1\right )} {\left (k - 1\right )} x - 2\right )} {\left (x^{2} + 2 \, x + 1\right )}}{{\left ({\left (k^{2} - d\right )} x^{3} - {\left (k^{2} + 3 \, d\right )} x^{2} - {\left (3 \, d + 1\right )} x - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {3}{4}}} \,d x } \]
integrate((x^2+2*x+1)*(-2-(-1+k)*(1+k)*x+2*k^2*x^2)/((-x^2+1)*(-k^2*x^2+1) )^(3/4)/(1-d-(1+3*d)*x-(k^2+3*d)*x^2+(k^2-d)*x^3),x, algorithm="giac")
integrate((2*k^2*x^2 - (k + 1)*(k - 1)*x - 2)*(x^2 + 2*x + 1)/(((k^2 - d)* x^3 - (k^2 + 3*d)*x^2 - (3*d + 1)*x - d + 1)*((k^2*x^2 - 1)*(x^2 - 1))^(3/ 4)), x)
Timed out. \[ \int \frac {\left (1+2 x+x^2\right ) \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx=\int \frac {\left (x^2+2\,x+1\right )\,\left (x\,\left (k-1\right )\,\left (k+1\right )-2\,k^2\,x^2+2\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{3/4}\,\left (\left (d-k^2\right )\,x^3+\left (k^2+3\,d\right )\,x^2+\left (3\,d+1\right )\,x+d-1\right )} \,d x \]
int(((2*x + x^2 + 1)*(x*(k - 1)*(k + 1) - 2*k^2*x^2 + 2))/(((x^2 - 1)*(k^2 *x^2 - 1))^(3/4)*(d + x^3*(d - k^2) + x^2*(3*d + k^2) + x*(3*d + 1) - 1)), x)