\(\int \frac {-3 k+2 (1+k^2) x+k (1+k^2) x^2-4 k^2 x^3+k^3 x^4}{((1-x^2) (1-k^2 x^2))^{2/3} (-1+d-(1+2 d) k x+(1+d k^2) x^2+k x^3)} \, dx\) [2632]

Optimal result

Integrand size = 90, antiderivative size = 232 \[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {-\sqrt {3} \sqrt [3]{d}+\sqrt {3} \sqrt [3]{d} k x}{-\sqrt [3]{d}+\sqrt [3]{d} k x-2 \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{2/3}}+\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} k x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3}-2 d^{2/3} k x+d^{2/3} k^2 x^2+\left (\sqrt [3]{d}-\sqrt [3]{d} k x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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)

Rubi [F]

\[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx \]

[In]

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]

[Out]

-((k*(5 + d*k^2)*x*(1 - x^2)^(2/3)*(1 - k^2*x^2)^(2/3)*AppellF1[1/2, 2/3, 2/3, 3/2, x^2, k^2*x^2])/(1 - (1 + k
^2)*x^2 + k^2*x^4)^(2/3)) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*k^(4/3)*Sqrt[(-1 - k^2 + 2*k^2*x^2)^2]*((-1 + k^2)^(2/3
) + 2^(2/3)*k^(2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3))*Sqrt[((-1 + k^2)^(4/3) - 2^(2/3)*k^(2/3)*(-1 + k^2)^(2/3)
*((1 - x^2)*(1 - k^2*x^2))^(1/3) + 2*2^(1/3)*k^(4/3)*((1 - x^2)*(1 - k^2*x^2))^(2/3))/((1 + Sqrt[3])*(-1 + k^2
)^(2/3) + 2^(2/3)*k^(2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*(-1 + k^2)^(2/3)
+ 2^(2/3)*k^(2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3))/((1 + Sqrt[3])*(-1 + k^2)^(2/3) + 2^(2/3)*k^(2/3)*((1 - x^
2)*(1 - k^2*x^2))^(1/3))], -7 - 4*Sqrt[3]])/(2^(2/3)*(1 + k^2 - 2*k^2*x^2)*Sqrt[(-1 - k^2*(1 - 2*x^2))^2]*Sqrt
[((-1 + k^2)^(2/3)*((-1 + k^2)^(2/3) + 2^(2/3)*k^(2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3)))/((1 + Sqrt[3])*(-1 +
k^2)^(2/3) + 2^(2/3)*k^(2/3)*((1 - x^2)*(1 - k^2*x^2))^(1/3))^2]) + k*(8 - d^2*k^2 - d*(5 - k^2))*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)^(2/3)), x] - (2 - (2 + 11*d
)*k^2 - d*(1 + 2*d)*k^4)*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)^(2/3)), x] - k*(6 + (2 + 8*d)*k^2 + d^2*k^4)*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)^(2/3)), x]

Rubi steps \begin{align*} \text {integral}& = \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 (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx \\ & = \int \left (-\frac {k \left (5+d k^2\right )}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {k^2 x}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}-\frac {k \left (8-d^2 k^2-d \left (5-k^2\right )\right )-\left (2-(2+11 d) k^2-d (1+2 d) k^4\right ) x-k \left (6+(2+8 d) k^2+d^2 k^4\right ) x^2}{\left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right ) \, dx \\ & = k^2 \int \frac {x}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx-\left (k \left (5+d k^2\right )\right ) \int \frac {1}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx-\int \frac {k \left (8-d^2 k^2-d \left (5-k^2\right )\right )-\left (2-(2+11 d) k^2-d (1+2 d) k^4\right ) x-k \left (6+(2+8 d) k^2+d^2 k^4\right ) x^2}{\left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx \\ & = \frac {1}{2} k^2 \text {Subst}\left (\int \frac {1}{\left (1+\left (-1-k^2\right ) x+k^2 x^2\right )^{2/3}} \, dx,x,x^2\right )-\frac {\left (k \left (5+d k^2\right ) \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3}} \, dx}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}-\int \left (\frac {k \left (-8+d^2 k^2+d \left (5-k^2\right )\right )}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {\left (2-(2+11 d) k^2-d (1+2 d) k^4\right ) x}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}+\frac {k \left (6+(2+8 d) k^2+d^2 k^4\right ) x^2}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right ) \, dx \\ & = -\frac {k \left (5+d k^2\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 )}{\left (1-\left (1+k^2\right ) x^2+k^2 x^4\right )^{2/3}}-\left (k \left (6+(2+8 d) k^2+d^2 k^4\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 ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx-\left (2-(2+11 d) k^2-d (1+2 d) k^4\right ) \int \frac {x}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx+\left (k \left (8-d^2 k^2-d \left (5-k^2\right )\right )\right ) \int \frac {1}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx+\frac {\left (3 k^2 \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4 k^2+\left (-1-k^2\right )^2+4 k^2 x^3}} \, dx,x,\sqrt [3]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}\right )}{2 \left (-1-k^2+2 k^2 x^2\right )} \\ & = -\frac {k \left (5+d k^2\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 )}{\left (1-\left (1+k^2\right ) x^2+k^2 x^4\right )^{2/3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} k^{4/3} \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2} \left (\left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right ) \sqrt {\frac {\left (-1+k^2\right )^{4/3}-2^{2/3} k^{2/3} \left (-1+k^2\right )^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}+2 \sqrt [3]{2} k^{4/3} \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{\left (1+\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\right ),-7-4 \sqrt {3}\right )}{2^{2/3} \left (1+k^2-2 k^2 x^2\right ) \sqrt {\left (-1-k^2 \left (1-2 x^2\right )\right )^2} \sqrt {\frac {\left (-1+k^2\right )^{2/3} \left (\left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )}{\left (\left (1+\sqrt {3}\right ) \left (-1+k^2\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}}}-\left (k \left (6+(2+8 d) k^2+d^2 k^4\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 ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx-\left (2-(2+11 d) k^2-d (1+2 d) k^4\right ) \int \frac {x}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx+\left (k \left (8-d^2 k^2-d \left (5-k^2\right )\right )\right ) \int \frac {1}{\left (1-d+(1+2 d) k x-\left (1+d k^2\right ) x^2-k x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx \]

[In]

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]

[Out]

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]

Maple [F]

\[\int \frac {-3 k +2 \left (k^{2}+1\right ) x +k \left (k^{2}+1\right ) x^{2}-4 k^{2} x^{3}+k^{3} x^{4}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {2}{3}} \left (-1+d -\left (1+2 d \right ) k x +\left (d \,k^{2}+1\right ) x^{2}+k \,x^{3}\right )}d x\]

[In]

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)

[Out]

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)

Fricas [F(-1)]

Timed out. \[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\text {Timed out} \]

[In]

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")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\text {Timed out} \]

[In]

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)

[Out]

Timed out

Maxima [F]

\[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int { \frac {k^{3} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} k x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3 \, k}{{\left (k x^{3} - {\left (2 \, d + 1\right )} k x + {\left (d k^{2} + 1\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]

[In]

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")

[Out]

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)

Giac [F]

\[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int { \frac {k^{3} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} k x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3 \, k}{{\left (k x^{3} - {\left (2 \, d + 1\right )} k x + {\left (d k^{2} + 1\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]

[In]

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")

[Out]

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)

Mupad [F(-1)]

Timed out. \[ \int \frac {-3 k+2 \left (1+k^2\right ) x+k \left (1+k^2\right ) x^2-4 k^2 x^3+k^3 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(1+2 d) k x+\left (1+d k^2\right ) x^2+k x^3\right )} \, dx=\int \frac {2\,x\,\left (k^2+1\right )-3\,k-4\,k^2\,x^3+k^3\,x^4+k\,x^2\,\left (k^2+1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (k\,x^3+\left (d\,k^2+1\right )\,x^2-k\,\left (2\,d+1\right )\,x+d-1\right )} \,d x \]

[In]

int((2*x*(k^2 + 1) - 3*k - 4*k^2*x^3 + 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)

[Out]

int((2*x*(k^2 + 1) - 3*k - 4*k^2*x^3 + 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)