\(\int \frac {x^2 (8-7 (1+k) x+6 k x^2)}{\sqrt [3]{(1-x) x (1-k x)} (-b+b (1+k) x-b k x^2+x^8)} \, dx\) [2345]

Optimal result

Integrand size = 57, antiderivative size = 185 \[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2 x^3+\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {\log \left (x^3-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}-\frac {\log \left (x^6+\sqrt [3]{b} x^3 \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \]

[Out]

3^(1/2)*arctan(3^(1/2)*b^(1/3)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(2*x^3+b^(1/3)*(x+(-1-k)*x^2+k*x^3)^(1/3)))/b^(2/3)+
ln(x^3-b^(1/3)*(x+(-1-k)*x^2+k*x^3)^(1/3))/b^(2/3)-1/2*ln(x^6+b^(1/3)*x^3*(x+(-1-k)*x^2+k*x^3)^(1/3)+b^(2/3)*(
x+(-1-k)*x^2+k*x^3)^(2/3))/b^(2/3)

Rubi [F]

\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx \]

[In]

Int[(x^2*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + b*(1 + k)*x - b*k*x^2 + x^8)),x]

[Out]

(21*(1 + k)*(1 - x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defer[Subst][Defer[Int][x^10/((1 - x^3)^(1/3)*(1 - k*x^3)^(1
/3)*(b - b*(1 + k)*x^3 + b*k*x^6 - x^24)), x], x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(1/3) + (24*(1 - x)^(1/3)*x^
(1/3)*(1 - k*x)^(1/3)*Defer[Subst][Defer[Int][x^7/((1 - x^3)^(1/3)*(1 - k*x^3)^(1/3)*(-b + b*(1 + k)*x^3 - b*k
*x^6 + x^24)), x], x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(1/3) + (18*k*(1 - x)^(1/3)*x^(1/3)*(1 - k*x)^(1/3)*Defe
r[Subst][Defer[Int][x^13/((1 - x^3)^(1/3)*(1 - k*x^3)^(1/3)*(-b + b*(1 + k)*x^3 - b*k*x^6 + x^24)), x], x, x^(
1/3)])/((1 - x)*x*(1 - k*x))^(1/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {x^{5/3} \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{1-x} \sqrt [3]{1-k x} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^7 \left (8-7 (1+k) x^3+6 k x^6\right )}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \left (\frac {7 (1+k) x^{10}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (b-b (1+k) x^3+b k x^6-x^{24}\right )}+\frac {8 x^7}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )}+\frac {6 k x^{13}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (24 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (18 k \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^{13}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (21 (1+k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (b-b (1+k) x^3+b k x^6-x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx \]

[In]

Integrate[(x^2*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + b*(1 + k)*x - b*k*x^2 + x^8)),x
]

[Out]

Integrate[(x^2*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + b*(1 + k)*x - b*k*x^2 + x^8)),
x]

Maple [F]

\[\int \frac {x^{2} \left (8-7 \left (1+k \right ) x +6 k \,x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-b +b \left (1+k \right ) x -b k \,x^{2}+x^{8}\right )}d x\]

[In]

int(x^2*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+b*(1+k)*x-b*k*x^2+x^8),x)

[Out]

int(x^2*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+b*(1+k)*x-b*k*x^2+x^8),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate(x^2*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+b*(1+k)*x-b*k*x^2+x^8),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\int \frac {x^{2} \cdot \left (6 k x^{2} - 7 k x - 7 x + 8\right )}{\sqrt [3]{x \left (x - 1\right ) \left (k x - 1\right )} \left (- b k x^{2} + b k x + b x - b + x^{8}\right )}\, dx \]

[In]

integrate(x**2*(8-7*(1+k)*x+6*k*x**2)/((1-x)*x*(-k*x+1))**(1/3)/(-b+b*(1+k)*x-b*k*x**2+x**8),x)

[Out]

Integral(x**2*(6*k*x**2 - 7*k*x - 7*x + 8)/((x*(x - 1)*(k*x - 1))**(1/3)*(-b*k*x**2 + b*k*x + b*x - b + x**8))
, x)

Maxima [F]

\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\int { \frac {{\left (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{2}}{{\left (x^{8} - b k x^{2} + b {\left (k + 1\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x^2*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+b*(1+k)*x-b*k*x^2+x^8),x, algorithm="maxima")

[Out]

integrate((6*k*x^2 - 7*(k + 1)*x + 8)*x^2/((x^8 - b*k*x^2 + b*(k + 1)*x - b)*((k*x - 1)*(x - 1)*x)^(1/3)), x)

Giac [F]

\[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\int { \frac {{\left (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{2}}{{\left (x^{8} - b k x^{2} + b {\left (k + 1\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x^2*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+b*(1+k)*x-b*k*x^2+x^8),x, algorithm="giac")

[Out]

integrate((6*k*x^2 - 7*(k + 1)*x + 8)*x^2/((x^8 - b*k*x^2 + b*(k + 1)*x - b)*((k*x - 1)*(x - 1)*x)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (8-7 (1+k) x+6 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=-\int \frac {x^2\,\left (6\,k\,x^2-7\,x\,\left (k+1\right )+8\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (-x^8+b\,k\,x^2-b\,\left (k+1\right )\,x+b\right )} \,d x \]

[In]

int(-(x^2*(6*k*x^2 - 7*x*(k + 1) + 8))/((x*(k*x - 1)*(x - 1))^(1/3)*(b - x^8 - b*x*(k + 1) + b*k*x^2)),x)

[Out]

-int((x^2*(6*k*x^2 - 7*x*(k + 1) + 8))/((x*(k*x - 1)*(x - 1))^(1/3)*(b - x^8 - b*x*(k + 1) + b*k*x^2)), x)