\(\int \frac {x (5-4 (1+k) x+3 k x^2)}{\sqrt [3]{(1-x) x (1-k x)} (-b+(b+b k) x-b k x^2+x^5)} \, dx\) [2340]

Optimal result

Integrand size = 56, antiderivative size = 185 \[ \int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2 x^2+\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {\log \left (x^2-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}-\frac {\log \left (x^4+\sqrt [3]{b} x^2 \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \]

[Out]

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

\[ \int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx=\int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx \]

[In]

Int[(x*(5 - 4*(1 + k)*x + 3*k*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + (b + b*k)*x - b*k*x^2 + x^5)),x]

[Out]

(12*(1 + k)*(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^15)), x], x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(1/3) + (15*(1 - x)^(1/3)*x^(
1/3)*(1 - k*x)^(1/3)*Defer[Subst][Defer[Int][x^4/((1 - x^3)^(1/3)*(1 - k*x^3)^(1/3)*(-b + b*(1 + k)*x^3 - b*k*
x^6 + x^15)), x], x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(1/3) + (9*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^15)), 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^{2/3} \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{1-x} \sqrt [3]{1-k x} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {x^{2/3} \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{1-x} \sqrt [3]{1-k x} \left (x^5-b (-1+x) (-1+k x)\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^4 \left (5-4 (1+k) x^3+3 k x^6\right )}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (x^{15}-b \left (-1+x^3\right ) \left (-1+k x^3\right )\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 {4 (1+k) 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^{15}\right )}+\frac {5 x^4}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-b+b (1+k) x^3-b k x^6+x^{15}\right )}+\frac {3 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^{15}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (15 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-b+b (1+k) x^3-b k x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (9 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^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (12 (1+k) \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^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx=\int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx \]

[In]

Integrate[(x*(5 - 4*(1 + k)*x + 3*k*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + (b + b*k)*x - b*k*x^2 + x^5)),x]

[Out]

Integrate[(x*(5 - 4*(1 + k)*x + 3*k*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + (b + b*k)*x - b*k*x^2 + x^5)), x]

Maple [F]

\[\int \frac {x \left (5-4 \left (1+k \right ) x +3 k \,x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-b +\left (b k +b \right ) x -b k \,x^{2}+x^{5}\right )}d x\]

[In]

int(x*(5-4*(1+k)*x+3*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(b*k+b)*x-b*k*x^2+x^5),x)

[Out]

int(x*(5-4*(1+k)*x+3*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(b*k+b)*x-b*k*x^2+x^5),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx=\text {Timed out} \]

[In]

integrate(x*(5-4*(1+k)*x+3*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(b*k+b)*x-b*k*x^2+x^5),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx=\int \frac {x \left (3 k x^{2} - 4 k x - 4 x + 5\right )}{\sqrt [3]{x \left (x - 1\right ) \left (k x - 1\right )} \left (- b k x^{2} + b k x + b x - b + x^{5}\right )}\, dx \]

[In]

integrate(x*(5-4*(1+k)*x+3*k*x**2)/((1-x)*x*(-k*x+1))**(1/3)/(-b+(b*k+b)*x-b*k*x**2+x**5),x)

[Out]

Integral(x*(3*k*x**2 - 4*k*x - 4*x + 5)/((x*(x - 1)*(k*x - 1))**(1/3)*(-b*k*x**2 + b*k*x + b*x - b + x**5)), x
)

Maxima [F]

\[ \int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx=\int { \frac {{\left (3 \, k x^{2} - 4 \, {\left (k + 1\right )} x + 5\right )} x}{{\left (x^{5} - b k x^{2} + {\left (b k + b\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x*(5-4*(1+k)*x+3*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(b*k+b)*x-b*k*x^2+x^5),x, algorithm="maxima")

[Out]

integrate((3*k*x^2 - 4*(k + 1)*x + 5)*x/((x^5 - b*k*x^2 + (b*k + b)*x - b)*((k*x - 1)*(x - 1)*x)^(1/3)), x)

Giac [F]

\[ \int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx=\int { \frac {{\left (3 \, k x^{2} - 4 \, {\left (k + 1\right )} x + 5\right )} x}{{\left (x^{5} - b k x^{2} + {\left (b k + b\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x*(5-4*(1+k)*x+3*k*x^2)/((1-x)*x*(-k*x+1))^(1/3)/(-b+(b*k+b)*x-b*k*x^2+x^5),x, algorithm="giac")

[Out]

integrate((3*k*x^2 - 4*(k + 1)*x + 5)*x/((x^5 - b*k*x^2 + (b*k + b)*x - b)*((k*x - 1)*(x - 1)*x)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (5-4 (1+k) x+3 k x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(b+b k) x-b k x^2+x^5\right )} \, dx=\int -\frac {x\,\left (3\,k\,x^2-4\,x\,\left (k+1\right )+5\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (-x^5+b\,k\,x^2+\left (-b-b\,k\right )\,x+b\right )} \,d x \]

[In]

int(-(x*(3*k*x^2 - 4*x*(k + 1) + 5))/((x*(k*x - 1)*(x - 1))^(1/3)*(b - x*(b + b*k) - x^5 + b*k*x^2)),x)

[Out]

int(-(x*(3*k*x^2 - 4*x*(k + 1) + 5))/((x*(k*x - 1)*(x - 1))^(1/3)*(b - x*(b + b*k) - x^5 + b*k*x^2)), x)