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

3.27.50.1 Optimal result

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

output

-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)
 

3.27.50.2 Mathematica [F]

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

input

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]
 

output

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]
 

3.27.50.3 Rubi [F]

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.

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -(4-d) k \int \frac {1}{\left (-k x^3-\left (d k^2+1\right ) x^2+(2 d+1) k x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx+2 \left (1-(d+1) k^2\right ) \int \frac {x}{\left (-k x^3-\left (d k^2+1\right ) x^2+(2 d+1) k x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx+k \left (d k^2+4\right ) \int \frac {x^2}{\left (-k x^3-\left (d k^2+1\right ) x^2+(2 d+1) k x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx+\frac {k x \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},\frac {1}{3},\frac {3}{2},x^2,k^2 x^2\right )}{\sqrt [3]{k^2 x^4-\left (k^2+1\right ) x^2+1}}\)

input

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]
 

output

$Aborted
 

3.27.50.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2048

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]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.27.50.4 Maple [F]

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

input

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)
 

output

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)
 

3.27.50.5 Fricas [F(-1)]

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

input

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")
 

output

Timed out
 

3.27.50.6 Sympy [F(-1)]

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

input

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)
 

output

Timed out
 

3.27.50.7 Maxima [F]

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

input

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")
 

output

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)
 

3.27.50.8 Giac [F]

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

input

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")
 

output

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)
 

3.27.50.9 Mupad [F(-1)]

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

input

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)
 

output

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)