3.28.32 \(\int -\frac {(-1+x) x (-1+2 x+(-2 k+k^2) x^2)}{((1-x) x (1-k x))^{2/3} (1-4 k x+(-b+6 k^2) x^2+(2 b-4 k^3) x^3+(-b+k^4) x^4)} \, dx\)

Optimal. Leaf size=252 \[ \frac {\log \left (-\sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+k^2 x^2-2 k x+1\right )}{2 b^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}}{\sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+2 k^2 x^2-4 k x+2}\right )}{2 b^{2/3}}-\frac {\log \left (b^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{4/3}+\left (k x^3+(-k-1) x^2+x\right )^{2/3} \left (\sqrt [3]{b} k^2 x^2-2 \sqrt [3]{b} k x+\sqrt [3]{b}\right )+k^4 x^4-4 k^3 x^3+6 k^2 x^2-4 k x+1\right )}{4 b^{2/3}} \]

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Rubi [F]  time = 10.81, 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 {(-1+x) x \left (-1+2 x+\left (-2 k+k^2\right ) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[-(((-1 + x)*x*(-1 + 2*x + (-2*k + k^2)*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - 4*k*x + (-b + 6*k^2)*x^2 +
(2*b - 4*k^3)*x^3 + (-b + k^4)*x^4))),x]

[Out]

(3*(2 - k)*(1 - x)^(2/3)*x^(2/3)*(1 - k*x)^(2/3)*Defer[Subst][Defer[Int][(x^6*(1 - x^3)^(1/3)*(1 - k*x^3)^(1/3
))/(1 - 4*k*x^3 - b*(1 - (6*k^2)/b)*x^6 + 2*b*(1 - (2*k^3)/b)*x^9 - b*(1 - k^4/b)*x^12), x], x, x^(1/3)])/((1
- x)*x*(1 - k*x))^(2/3) + (3*(1 - x)^(2/3)*x^(2/3)*(1 - k*x)^(2/3)*Defer[Subst][Defer[Int][(x^3*(1 - x^3)^(1/3
)*(1 - k*x^3)^(1/3))/(-1 + 4*k*x^3 + b*(1 - (6*k^2)/b)*x^6 - 2*b*(1 - (2*k^3)/b)*x^9 + b*(1 - k^4/b)*x^12), x]
, x, x^(1/3)])/((1 - x)*x*(1 - k*x))^(2/3)

Rubi steps

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

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Mathematica [F]  time = 3.62, size = 87, normalized size = 0.35 \begin {gather*} -\int \frac {(x-1) x \left (\left (k^2-2 k\right ) x^2+2 x-1\right )}{((1-x) x (1-k x))^{2/3} \left (x^4 \left (k^4-b\right )+x^3 \left (2 b-4 k^3\right )+x^2 \left (6 k^2-b\right )-4 k x+1\right )} \, dx \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-(((-1 + x)*x*(-1 + 2*x + (-2*k + k^2)*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - 4*k*x + (-b + 6*k^2)*
x^2 + (2*b - 4*k^3)*x^3 + (-b + k^4)*x^4))),x]

[Out]

-Integrate[((-1 + x)*x*(-1 + 2*x + (-2*k + k^2)*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - 4*k*x + (-b + 6*k^2)*x
^2 + (2*b - 4*k^3)*x^3 + (-b + k^4)*x^4)), x]

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IntegrateAlgebraic [A]  time = 3.19, size = 252, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}{2-4 k x+2 k^2 x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}\right )}{2 b^{2/3}}+\frac {\log \left (1-2 k x+k^2 x^2-\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}}-\frac {\log \left (1-4 k x+6 k^2 x^2-4 k^3 x^3+k^4 x^4+\left (\sqrt [3]{b}-2 \sqrt [3]{b} k x+\sqrt [3]{b} k^2 x^2\right ) \left (x+(-1-k) x^2+k x^3\right )^{2/3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{4/3}\right )}{4 b^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[-(((-1 + x)*x*(-1 + 2*x + (-2*k + k^2)*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - 4*k*x + (-b
+ 6*k^2)*x^2 + (2*b - 4*k^3)*x^3 + (-b + k^4)*x^4))),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(2 - 4*k*x + 2*k^2*x^2 + b^(1/3)*(x + (-1 -
k)*x^2 + k*x^3)^(2/3))])/(2*b^(2/3)) + Log[1 - 2*k*x + k^2*x^2 - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2
*b^(2/3)) - Log[1 - 4*k*x + 6*k^2*x^2 - 4*k^3*x^3 + k^4*x^4 + (b^(1/3) - 2*b^(1/3)*k*x + b^(1/3)*k^2*x^2)*(x +
(-1 - k)*x^2 + k*x^3)^(2/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(4/3)]/(4*b^(2/3))

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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(-(-1+x)*x*(-1+2*x+(k^2-2*k)*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(1-4*k*x+(6*k^2-b)*x^2+(-4*k^3+2*b)*x^3+(k
^4-b)*x^4),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left ({\left (k^{2} - 2 \, k\right )} x^{2} + 2 \, x - 1\right )} {\left (x - 1\right )} x}{{\left ({\left (k^{4} - b\right )} x^{4} - 2 \, {\left (2 \, k^{3} - b\right )} x^{3} + {\left (6 \, k^{2} - b\right )} x^{2} - 4 \, k x + 1\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(-(-1+x)*x*(-1+2*x+(k^2-2*k)*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(1-4*k*x+(6*k^2-b)*x^2+(-4*k^3+2*b)*x^3+(k
^4-b)*x^4),x, algorithm="giac")

[Out]

integrate(-((k^2 - 2*k)*x^2 + 2*x - 1)*(x - 1)*x/(((k^4 - b)*x^4 - 2*(2*k^3 - b)*x^3 + (6*k^2 - b)*x^2 - 4*k*x
+ 1)*((k*x - 1)*(x - 1)*x)^(2/3)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int -\frac {\left (-1+x \right ) x \left (-1+2 x +\left (k^{2}-2 k \right ) x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}} \left (1-4 k x +\left (6 k^{2}-b \right ) x^{2}+\left (-4 k^{3}+2 b \right ) x^{3}+\left (k^{4}-b \right ) x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(-1+x)*x*(-1+2*x+(k^2-2*k)*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(1-4*k*x+(6*k^2-b)*x^2+(-4*k^3+2*b)*x^3+(k^4-b)*
x^4),x)

[Out]

int(-(-1+x)*x*(-1+2*x+(k^2-2*k)*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(1-4*k*x+(6*k^2-b)*x^2+(-4*k^3+2*b)*x^3+(k^4-b)*
x^4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left ({\left (k^{2} - 2 \, k\right )} x^{2} + 2 \, x - 1\right )} {\left (x - 1\right )} x}{{\left ({\left (k^{4} - b\right )} x^{4} - 2 \, {\left (2 \, k^{3} - b\right )} x^{3} + {\left (6 \, k^{2} - b\right )} x^{2} - 4 \, k x + 1\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(-(-1+x)*x*(-1+2*x+(k^2-2*k)*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(1-4*k*x+(6*k^2-b)*x^2+(-4*k^3+2*b)*x^3+(k
^4-b)*x^4),x, algorithm="maxima")

[Out]

-integrate(((k^2 - 2*k)*x^2 + 2*x - 1)*(x - 1)*x/(((k^4 - b)*x^4 - 2*(2*k^3 - b)*x^3 + (6*k^2 - b)*x^2 - 4*k*x
+ 1)*((k*x - 1)*(x - 1)*x)^(2/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x\,\left (x-1\right )\,\left (\left (2\,k-k^2\right )\,x^2-2\,x+1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}\,\left (\left (b-k^4\right )\,x^4+\left (4\,k^3-2\,b\right )\,x^3+\left (b-6\,k^2\right )\,x^2+4\,k\,x-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*(x - 1)*(x^2*(2*k - k^2) - 2*x + 1))/((x*(k*x - 1)*(x - 1))^(2/3)*(x^4*(b - k^4) + x^2*(b - 6*k^2) + 4
*k*x - x^3*(2*b - 4*k^3) - 1)),x)

[Out]

-int((x*(x - 1)*(x^2*(2*k - k^2) - 2*x + 1))/((x*(k*x - 1)*(x - 1))^(2/3)*(x^4*(b - k^4) + x^2*(b - 6*k^2) + 4
*k*x - x^3*(2*b - 4*k^3) - 1)), x)

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sympy [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(-(-1+x)*x*(-1+2*x+(k**2-2*k)*x**2)/((1-x)*x*(-k*x+1))**(2/3)/(1-4*k*x+(6*k**2-b)*x**2+(-4*k**3+2*b)*
x**3+(k**4-b)*x**4),x)

[Out]

Timed out

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