\(\int \frac {3 k+(-2+k^2) x-3 k x^2+k^2 x^3}{\sqrt [3]{(1-x^2) (1-k^2 x^2)} (-1+d-(1+2 d) k x+(1+d k^2) x^2+k x^3)} \, dx\) [2650]

Optimal result

Integrand size = 77, antiderivative size = 236 \[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2 \sqrt [3]{d}-2 \sqrt [3]{d} k x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{\sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} k x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{\sqrt [3]{d}}+\frac {\log \left (d^{2/3}-2 d^{2/3} k x+d^{2/3} k^2 x^2+\left (\sqrt [3]{d}-\sqrt [3]{d} k x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

-3^(1/2)*arctan(3^(1/2)*(1+(-k^2-1)*x^2+k^2*x^4)^(1/3)/(2*d^(1/3)-2*d^(1/3)*k*x+(1+(-k^2-1)*x^2+k^2*x^4)^(1/3)
))/d^(1/3)-ln(-d^(1/3)+d^(1/3)*k*x+(1+(-k^2-1)*x^2+k^2*x^4)^(1/3))/d^(1/3)+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^(1/3)

Rubi [F]

\[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx \]

[In]

Int[(3*k + (-2 + k^2)*x - 3*k*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d - (1 + 2*d)*k*x + (1 + d
*k^2)*x^2 + k*x^3)),x]

[Out]

(k*x*(1 - x^2)^(1/3)*(1 - k^2*x^2)^(1/3)*AppellF1[1/2, 1/3, 1/3, 3/2, x^2, k^2*x^2])/(1 - (1 + k^2)*x^2 + k^2*
x^4)^(1/3) - (4 - d)*k*Defer[Int][1/((1 - d + (1 + 2*d)*k*x - (1 + d*k^2)*x^2 - k*x^3)*(1 + (-1 - k^2)*x^2 + k
^2*x^4)^(1/3)), x] + 2*(1 - (1 + d)*k^2)*Defer[Int][x/((1 - d + (1 + 2*d)*k*x - (1 + d*k^2)*x^2 - k*x^3)*(1 +
(-1 - k^2)*x^2 + k^2*x^4)^(1/3)), x] + k*(4 + d*k^2)*Defer[Int][x^2/((1 - d + (1 + 2*d)*k*x - (1 + d*k^2)*x^2
- k*x^3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \frac {-3 k+\left (2-k^2\right ) x+3 k x^2-k^2 x^3}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \left (\frac {k}{\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}-\frac {(4-d) k-2 \left (1-(1+d) k^2\right ) x-k \left (4+d k^2\right ) x^2}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = k \int \frac {1}{\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\int \frac {(4-d) k-2 \left (1-(1+d) k^2\right ) x-k \left (4+d k^2\right ) x^2}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (k \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2}\right ) \int \frac {1}{\sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2}} \, dx}{\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}-\int \left (\frac {(4-d) k}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {2 \left (-1+(1+d) k^2\right ) x}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {k \left (-4-d k^2\right ) x^2}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = \frac {k x \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},\frac {1}{3},\frac {3}{2},x^2,k^2 x^2\right )}{\sqrt [3]{1-\left (1+k^2\right ) x^2+k^2 x^4}}-((4-d) k) \int \frac {1}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\left (k \left (4+d k^2\right )\right ) \int \frac {x^2}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\left (2 \left (1-(1+d) k^2\right )\right ) \int \frac {x}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx \]

[In]

Integrate[(3*k + (-2 + k^2)*x - 3*k*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d - (1 + 2*d)*k*x +
(1 + d*k^2)*x^2 + k*x^3)),x]

[Out]

Integrate[(3*k + (-2 + k^2)*x - 3*k*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d - (1 + 2*d)*k*x +
(1 + d*k^2)*x^2 + k*x^3)), x]

Maple [F]

\[\int \frac {3 k +\left (k^{2}-2\right ) x -3 k \,x^{2}+k^{2} x^{3}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {1}{3}} \left (-1+d -\left (1+2 d \right ) k x +\left (d \,k^{2}+1\right ) x^{2}+k \,x^{3}\right )}d x\]

[In]

int((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d-(1+2*d)*k*x+(d*k^2+1)*x^2+k*x^3),x)

[Out]

int((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d-(1+2*d)*k*x+(d*k^2+1)*x^2+k*x^3),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d-(1+2*d)*k*x+(d*k^2+1)*x^2+k*x^3)
,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate((3*k+(k**2-2)*x-3*k*x**2+k**2*x**3)/((-x**2+1)*(-k**2*x**2+1))**(1/3)/(-1+d-(1+2*d)*k*x+(d*k**2+1)*x
**2+k*x**3),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int { \frac {k^{2} x^{3} - 3 \, k x^{2} + {\left (k^{2} - 2\right )} x + 3 \, k}{{\left (k x^{3} - {\left (2 \, d + 1\right )} k x + {\left (d k^{2} + 1\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d-(1+2*d)*k*x+(d*k^2+1)*x^2+k*x^3)
,x, algorithm="maxima")

[Out]

integrate((k^2*x^3 - 3*k*x^2 + (k^2 - 2)*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))^(1/3)), x)

Giac [F]

\[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int { \frac {k^{2} x^{3} - 3 \, k x^{2} + {\left (k^{2} - 2\right )} x + 3 \, k}{{\left (k x^{3} - {\left (2 \, d + 1\right )} k x + {\left (d k^{2} + 1\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d-(1+2*d)*k*x+(d*k^2+1)*x^2+k*x^3)
,x, algorithm="giac")

[Out]

integrate((k^2*x^3 - 3*k*x^2 + (k^2 - 2)*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))^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {3\,k+x\,\left (k^2-2\right )+k^2\,x^3-3\,k\,x^2}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/3}\,\left (k\,x^3+\left (d\,k^2+1\right )\,x^2-k\,\left (2\,d+1\right )\,x+d-1\right )} \,d x \]

[In]

int((3*k + x*(k^2 - 2) + k^2*x^3 - 3*k*x^2)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/3)*(d + x^2*(d*k^2 + 1) + k*x^3 - k*
x*(2*d + 1) - 1)),x)

[Out]

int((3*k + x*(k^2 - 2) + k^2*x^3 - 3*k*x^2)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/3)*(d + x^2*(d*k^2 + 1) + k*x^3 - k*
x*(2*d + 1) - 1)), x)