3.22.97 \(\int \frac {x^5 (8-7 (1+k) x+6 k x^2)}{((1-x) x (1-k x))^{2/3} (-b+b (1+k) x-b k x^2+x^8)} \, dx\)

Optimal. Leaf size=162 \[ -\frac {\log \left (b^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+\sqrt [3]{b} x^3 \sqrt [3]{k x^3+(-k-1) x^2+x}+x^6\right )}{2 \sqrt [3]{b}}+\frac {\log \left (x^3-\sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}\right )}{\sqrt [3]{b}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^3}{2 \sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}+x^3}\right )}{\sqrt [3]{b}} \]

________________________________________________________________________________________

Rubi [F]  time = 27.47, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^5*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(-b + b*(1 + k)*x - b*k*x^2 + x^8)),x]

[Out]

(21*(1 + k)*(1 - x)^(2/3)*x^(2/3)*(1 - k*x)^(2/3)*Defer[Subst][Defer[Int][x^18/((1 - x^3)^(2/3)*(1 - k*x^3)^(2
/3)*(b - b*(1 + k)*x^3 + b*k*x^6 - x^24)), x], x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(2/3) + (24*(1 - x)^(2/3)*x^
(2/3)*(1 - k*x)^(2/3)*Defer[Subst][Defer[Int][x^15/((1 - x^3)^(2/3)*(1 - k*x^3)^(2/3)*(-b + b*(1 + k)*x^3 - b*
k*x^6 + x^24)), x], x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(2/3) + (18*k*(1 - x)^(2/3)*x^(2/3)*(1 - k*x)^(2/3)*Def
er[Subst][Defer[Int][x^21/((1 - x^3)^(2/3)*(1 - k*x^3)^(2/3)*(-b + b*(1 + k)*x^3 - b*k*x^6 + x^24)), x], x, x^
(1/3)])/((1 - x)*x*(1 - k*x))^(2/3)

Rubi steps

\begin {align*} \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx &=\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {x^{13/3} \left (8-7 (1+k) x+6 k x^2\right )}{(1-x)^{2/3} (1-k x)^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left (3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{15} \left (8-7 (1+k) x^3+6 k x^6\right )}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left (3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {7 (1+k) x^{18}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (b-b (1+k) x^3+b k x^6-x^{24}\right )}+\frac {8 x^{15}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )}+\frac {6 k x^{21}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left (24 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{15}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}+\frac {\left (18 k (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{21}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}+\frac {\left (21 (1+k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{18}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (b-b (1+k) x^3+b k x^6-x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^5*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(-b + b*(1 + k)*x - b*k*x^2 + x^8)),x
]

[Out]

Integrate[(x^5*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(-b + b*(1 + k)*x - b*k*x^2 + x^8)),
x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 8.30, size = 162, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^3}{x^3+2 \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\log \left (x^3-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}-\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 \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^5*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(-b + b*(1 + k)*x - b*k*x^2
+ x^8)),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*x^3)/(x^3 + 2*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[x^3 - b^(1/3)
*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[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)]/(2*b^(1/3))

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-b+b*(1+k)*x-b*k*x^2+x^8),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{5}}{{\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 {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-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^5/((x^8 - b*k*x^2 + b*(k + 1)*x - b)*((k*x - 1)*(x - 1)*x)^(2/3)), x)

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (8-7 \left (1+k \right ) x +6 k \,x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}} \left (-b +b \left (1+k \right ) x -b k \,x^{2}+x^{8}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-b+b*(1+k)*x-b*k*x^2+x^8),x)

[Out]

int(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-b+b*(1+k)*x-b*k*x^2+x^8),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{5}}{{\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 {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-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^5/((x^8 - b*k*x^2 + b*(k + 1)*x - b)*((k*x - 1)*(x - 1)*x)^(2/3)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^5\,\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 )}^{2/3}\,\left (-x^8+b\,k\,x^2-b\,\left (k+1\right )\,x+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^5*(6*k*x^2 - 7*x*(k + 1) + 8))/((x*(k*x - 1)*(x - 1))^(2/3)*(b - x^8 - b*x*(k + 1) + b*k*x^2)),x)

[Out]

-int((x^5*(6*k*x^2 - 7*x*(k + 1) + 8))/((x*(k*x - 1)*(x - 1))^(2/3)*(b - x^8 - b*x*(k + 1) + b*k*x^2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (6 k x^{2} - 7 k x - 7 x + 8\right )}{\left (x \left (x - 1\right ) \left (k x - 1\right )\right )^{\frac {2}{3}} \left (- b k x^{2} + b k x + b x - b + x^{8}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(8-7*(1+k)*x+6*k*x**2)/((1-x)*x*(-k*x+1))**(2/3)/(-b+b*(1+k)*x-b*k*x**2+x**8),x)

[Out]

Integral(x**5*(6*k*x**2 - 7*k*x - 7*x + 8)/((x*(x - 1)*(k*x - 1))**(2/3)*(-b*k*x**2 + b*k*x + b*x - b + x**8))
, x)

________________________________________________________________________________________