Integrand size = 95, antiderivative size = 231 \[ \int \frac {3 k+2 \left (1+k^2\right ) x-k \left (1+k^2\right ) x^2-4 k^2 x^3-k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d}+\sqrt {3} \sqrt [3]{d} k x}{\sqrt [3]{d}+\sqrt [3]{d} k x+2 \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{2/3}}-\frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{d} k x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{d^{2/3}}+\frac {\log \left (d^{2/3}+2 d^{2/3} k x+d^{2/3} k^2 x^2+\left (\sqrt [3]{d}+\sqrt [3]{d} k x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 d^{2/3}} \]
-3^(1/2)*arctan((3^(1/2)*d^(1/3)+3^(1/2)*d^(1/3)*k*x)/(d^(1/3)+d^(1/3)*k*x +2*(1+(-k^2-1)*x^2+k^2*x^4)^(1/3)))/d^(2/3)-ln(-d^(1/3)-d^(1/3)*k*x+(1+(-k ^2-1)*x^2+k^2*x^4)^(1/3))/d^(2/3)+1/2*ln(d^(2/3)+2*d^(2/3)*k*x+d^(2/3)*k^2 *x^2+(d^(1/3)+d^(1/3)*k*x)*(1+(-k^2-1)*x^2+k^2*x^4)^(1/3)+(1+(-k^2-1)*x^2+ k^2*x^4)^(2/3))/d^(2/3)
\[ \int \frac {3 k+2 \left (1+k^2\right ) x-k \left (1+k^2\right ) x^2-4 k^2 x^3-k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {3 k+2 \left (1+k^2\right ) x-k \left (1+k^2\right ) x^2-4 k^2 x^3-k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right )} \, dx \]
Integrate[(3*k + 2*(1 + k^2)*x - k*(1 + k^2)*x^2 - 4*k^2*x^3 - k^3*x^4)/(( (1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d - (1 + 2*d)*k*x - (1 + d*k^2)*x^2 + k*x^3)),x]
Integrate[(3*k + 2*(1 + k^2)*x - k*(1 + k^2)*x^2 - 4*k^2*x^3 - k^3*x^4)/(( (1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d - (1 + 2*d)*k*x - (1 + d*k^2)*x^2 + k*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.
| \(\displaystyle \int \frac {-k^3 x^4-4 k^2 x^3-k \left (k^2+1\right ) x^2+2 \left (k^2+1\right ) x+3 k}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-x^2 \left (d k^2+1\right )-(2 d+1) k x-d+k x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {-k^3 x^4-4 k^2 x^3-k \left (k^2+1\right ) x^2+2 \left (k^2+1\right ) x+3 k}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (-x^2 \left (d k^2+1\right )-(2 d+1) k x-d+k x^3+1\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {k \left (-d^2 k^2-d \left (5-k^2\right )+8\right )-k x^2 \left (d^2 k^4+(8 d+2) k^2+6\right )+x \left (-d (2 d+1) k^4-(11 d+2) k^2+2\right )}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (-x^2 \left (d k^2+1\right )-(2 d+1) k x-d+k x^3+1\right )}-\frac {k \left (d k^2+5\right )}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}-\frac {k^2 x}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {\left (2 x^2 k^2-k^2-1\right )^2} \left (2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (k^2-1\right )^{2/3}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} k^{4/3}-2^{2/3} \left (k^2-1\right )^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (k^2-1\right )^{4/3}}{\left (2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (1+\sqrt {3}\right ) \left (k^2-1\right )^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (1-\sqrt {3}\right ) \left (k^2-1\right )^{2/3}}{2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (1+\sqrt {3}\right ) \left (k^2-1\right )^{2/3}}\right ),-7-4 \sqrt {3}\right ) k^{4/3}}{2^{2/3} \left (-2 x^2 k^2+k^2+1\right ) \sqrt {\left (-\left (\left (1-2 x^2\right ) k^2\right )-1\right )^2} \sqrt {\frac {\left (k^2-1\right )^{2/3} \left (2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (k^2-1\right )^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (1+\sqrt {3}\right ) \left (k^2-1\right )^{2/3}\right )^2}}}-\frac {\left (d k^2+5\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},\frac {2}{3},\frac {3}{2},x^2,k^2 x^2\right ) k}{\left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{2/3}}+\left (-d^2 k^2-d \left (5-k^2\right )+8\right ) \int \frac {1}{\left (k x^3-\left (d k^2+1\right ) x^2-(2 d+1) k x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx k-\left (d^2 k^4+(8 d+2) k^2+6\right ) \int \frac {x^2}{\left (k x^3-\left (d k^2+1\right ) x^2-(2 d+1) k x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx k+\left (-d (2 d+1) k^4-(11 d+2) k^2+2\right ) \int \frac {x}{\left (k x^3-\left (d k^2+1\right ) x^2-(2 d+1) k x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx\) |
Int[(3*k + 2*(1 + k^2)*x - k*(1 + k^2)*x^2 - 4*k^2*x^3 - k^3*x^4)/(((1 - x ^2)*(1 - k^2*x^2))^(2/3)*(1 - d - (1 + 2*d)*k*x - (1 + d*k^2)*x^2 + k*x^3) ),x]
3.27.28.3.1 Defintions of rubi rules used
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]
\[\int \frac {3 k +2 \left (k^{2}+1\right ) x -k \left (k^{2}+1\right ) x^{2}-4 k^{2} x^{3}-k^{3} x^{4}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {2}{3}} \left (1-d -\left (1+2 d \right ) k x -\left (d \,k^{2}+1\right ) x^{2}+k \,x^{3}\right )}d x\]
int((3*k+2*(k^2+1)*x-k*(k^2+1)*x^2-4*k^2*x^3-k^3*x^4)/((-x^2+1)*(-k^2*x^2+ 1))^(2/3)/(1-d-(1+2*d)*k*x-(d*k^2+1)*x^2+k*x^3),x)
int((3*k+2*(k^2+1)*x-k*(k^2+1)*x^2-4*k^2*x^3-k^3*x^4)/((-x^2+1)*(-k^2*x^2+ 1))^(2/3)/(1-d-(1+2*d)*k*x-(d*k^2+1)*x^2+k*x^3),x)
Timed out. \[ \int \frac {3 k+2 \left (1+k^2\right ) x-k \left (1+k^2\right ) x^2-4 k^2 x^3-k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\text {Timed out} \]
integrate((3*k+2*(k^2+1)*x-k*(k^2+1)*x^2-4*k^2*x^3-k^3*x^4)/((-x^2+1)*(-k^ 2*x^2+1))^(2/3)/(1-d-(1+2*d)*k*x-(d*k^2+1)*x^2+k*x^3),x, algorithm="fricas ")
Timed out. \[ \int \frac {3 k+2 \left (1+k^2\right ) x-k \left (1+k^2\right ) x^2-4 k^2 x^3-k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\text {Timed out} \]
integrate((3*k+2*(k**2+1)*x-k*(k**2+1)*x**2-4*k**2*x**3-k**3*x**4)/((-x**2 +1)*(-k**2*x**2+1))**(2/3)/(1-d-(1+2*d)*k*x-(d*k**2+1)*x**2+k*x**3),x)
\[ \int \frac {3 k+2 \left (1+k^2\right ) x-k \left (1+k^2\right ) x^2-4 k^2 x^3-k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int { -\frac {k^{3} x^{4} + 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} k x^{2} - 2 \, {\left (k^{2} + 1\right )} x - 3 \, k}{{\left (k x^{3} - {\left (2 \, d + 1\right )} k x - {\left (d k^{2} + 1\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate((3*k+2*(k^2+1)*x-k*(k^2+1)*x^2-4*k^2*x^3-k^3*x^4)/((-x^2+1)*(-k^ 2*x^2+1))^(2/3)/(1-d-(1+2*d)*k*x-(d*k^2+1)*x^2+k*x^3),x, algorithm="maxima ")
-integrate((k^3*x^4 + 4*k^2*x^3 + (k^2 + 1)*k*x^2 - 2*(k^2 + 1)*x - 3*k)/( (k*x^3 - (2*d + 1)*k*x - (d*k^2 + 1)*x^2 - d + 1)*((k^2*x^2 - 1)*(x^2 - 1) )^(2/3)), x)
\[ \int \frac {3 k+2 \left (1+k^2\right ) x-k \left (1+k^2\right ) x^2-4 k^2 x^3-k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int { -\frac {k^{3} x^{4} + 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} k x^{2} - 2 \, {\left (k^{2} + 1\right )} x - 3 \, k}{{\left (k x^{3} - {\left (2 \, d + 1\right )} k x - {\left (d k^{2} + 1\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate((3*k+2*(k^2+1)*x-k*(k^2+1)*x^2-4*k^2*x^3-k^3*x^4)/((-x^2+1)*(-k^ 2*x^2+1))^(2/3)/(1-d-(1+2*d)*k*x-(d*k^2+1)*x^2+k*x^3),x, algorithm="giac")
integrate(-(k^3*x^4 + 4*k^2*x^3 + (k^2 + 1)*k*x^2 - 2*(k^2 + 1)*x - 3*k)/( (k*x^3 - (2*d + 1)*k*x - (d*k^2 + 1)*x^2 - d + 1)*((k^2*x^2 - 1)*(x^2 - 1) )^(2/3)), x)
Timed out. \[ \int \frac {3 k+2 \left (1+k^2\right ) x-k \left (1+k^2\right ) x^2-4 k^2 x^3-k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d-(1+2 d) k x-\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {4\,k^2\,x^3-2\,x\,\left (k^2+1\right )-3\,k+k^3\,x^4+k\,x^2\,\left (k^2+1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (-k\,x^3+\left (d\,k^2+1\right )\,x^2+k\,\left (2\,d+1\right )\,x+d-1\right )} \,d x \]
int((4*k^2*x^3 - 2*x*(k^2 + 1) - 3*k + k^3*x^4 + k*x^2*(k^2 + 1))/(((x^2 - 1)*(k^2*x^2 - 1))^(2/3)*(d + x^2*(d*k^2 + 1) - k*x^3 + k*x*(2*d + 1) - 1) ),x)