3.25.89 \(\int \frac {(-2+(1+k) x) (1-(1+k) x+(a+k) x^2)}{x ((1-x) x (1-k x))^{2/3} (1-(1+k) x+(-b+k) x^2)} \, dx\)

Optimal. Leaf size=205 \[ \frac {(a+b) \log \left (\sqrt [3]{k x^3+(-k-1) x^2+x}-\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {(-a-b) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{k x^3+(-k-1) x^2+x}+\left (k x^3+(-k-1) x^2+x\right )^{2/3}\right )}{2 b^{2/3}}+\frac {\left (\sqrt {3} a+\sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{k x^3+(-k-1) x^2+x}}\right )}{b^{2/3}}+\frac {3 \sqrt [3]{k x^3+(-k-1) x^2+x}}{x} \]

________________________________________________________________________________________

Rubi [F]  time = 25.00, 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 {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{x ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*(2*k^2 - a*(1 + 2*b + k^2) - b*(1 + 4*k + k^2))*(1 - x)*(1 - k*x))/(2*(b - k)^2*((1 - x)*x*(1 - k*x))^(2/3)
) + (3*(1 + k)*(a + k)*(1 - x)*(((1 - k)*x)/(1 - k*x))^(2/3)*(1 - k*x)*Hypergeometric2F1[1/3, 2/3, 4/3, (1 - x
)/(1 - k*x)])/((1 - k)*(b - k)*((1 - x)*x*(1 - k*x))^(2/3)) + (3*(1 + k)*(2*k^2 - a*(1 + 2*b + k^2) - b*(1 + 4
*k + k^2))*(1 - x)*(((1 - k)*x)/(1 - k*x))^(2/3)*(1 - k*x)*Hypergeometric2F1[1/3, 2/3, 4/3, (1 - x)/(1 - k*x)]
)/(2*(1 - k)*(b - k)^2*((1 - x)*x*(1 - k*x))^(2/3)) + ((a + b)*(1 + k^3 + 3*b*(1 + k) + (4*b^2 + (1 - k)^2*(1
+ k + k^2) + b*(5 + 2*k + 5*k^2))/Sqrt[4*b + (-1 + k)^2])*(1 - x)^(2/3)*x^(2/3)*(1 - k*x)^(2/3)*Defer[Int][1/(
(1 - x)^(2/3)*x^(5/3)*(1 - k*x)^(2/3)*(-1 - k - Sqrt[1 + 4*b - 2*k + k^2] + 2*(-b + k)*x)), x])/((b - k)^2*((1
- x)*x*(1 - k*x))^(2/3)) + ((a + b)*(1 + k^3 + 3*b*(1 + k) - (1 + 4*b^2 - k - k^3 + k^4 + b*(5 + 2*k + 5*k^2)
)/Sqrt[4*b + (-1 + k)^2])*(1 - x)^(2/3)*x^(2/3)*(1 - k*x)^(2/3)*Defer[Int][1/((1 - x)^(2/3)*x^(5/3)*(1 - k*x)^
(2/3)*(-1 - k + Sqrt[1 + 4*b - 2*k + k^2] + 2*(-b + k)*x)), x])/((b - k)^2*((1 - x)*x*(1 - k*x))^(2/3))

Rubi steps

\begin {align*} \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{x ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx &=\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{(1-x)^{2/3} x^{5/3} (1-k x)^{2/3} \left (1-(1+k) x+(-b+k) x^2\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 \left (-\frac {2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )}{(b-k)^2 (1-x)^{2/3} x^{5/3} (1-k x)^{2/3}}-\frac {(1+k) (a+k)}{(b-k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}}-\frac {(a+b) \left (1+2 b+k^2\right )-(a+b) (1+k) \left (1+3 b-k+k^2\right ) x}{(b-k)^2 (1-x)^{2/3} x^{5/3} (1-k x)^{2/3} \left (1+(-1-k) x+(-b+k) x^2\right )}\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 {(a+b) \left (1+2 b+k^2\right )-(a+b) (1+k) \left (1+3 b-k+k^2\right ) x}{(1-x)^{2/3} x^{5/3} (1-k x)^{2/3} \left (1+(-1-k) x+(-b+k) x^2\right )} \, dx}{(b-k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {\left ((1+k) (a+k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {1}{(1-x)^{2/3} x^{2/3} (1-k x)^{2/3}} \, dx}{(b-k) ((1-x) x (1-k x))^{2/3}}-\frac {\left (\left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {1}{(1-x)^{2/3} x^{5/3} (1-k x)^{2/3}} \, dx}{(b-k)^2 ((1-x) x (1-k x))^{2/3}}\\ &=\frac {3 \left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) (1-x) (1-k x)}{2 (b-k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {3 (1+k) (a+k) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{2/3} (1-k x) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {1-x}{1-k x}\right )}{(1-k) (b-k) ((1-x) x (1-k x))^{2/3}}-\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \left (\frac {-\left ((a+b) (1+k) \left (1+3 b-k+k^2\right )\right )-\frac {(a+b) \left (1+5 b+4 b^2-k+2 b k+5 b k^2-k^3+k^4\right )}{\sqrt {1+4 b-2 k+k^2}}}{(1-x)^{2/3} x^{5/3} (1-k x)^{2/3} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )}+\frac {-\left ((a+b) (1+k) \left (1+3 b-k+k^2\right )\right )+\frac {(a+b) \left (1+5 b+4 b^2-k+2 b k+5 b k^2-k^3+k^4\right )}{\sqrt {1+4 b-2 k+k^2}}}{(1-x)^{2/3} x^{5/3} (1-k x)^{2/3} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )}\right ) \, dx}{(b-k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left ((-1-k) \left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {1}{(1-x)^{2/3} x^{2/3} (1-k x)^{2/3}} \, dx}{2 (b-k)^2 ((1-x) x (1-k x))^{2/3}}\\ &=\frac {3 \left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) (1-x) (1-k x)}{2 (b-k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {3 (1+k) (a+k) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{2/3} (1-k x) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {1-x}{1-k x}\right )}{(1-k) (b-k) ((1-x) x (1-k x))^{2/3}}+\frac {3 (1+k) \left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{2/3} (1-k x) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {1-x}{1-k x}\right )}{2 (1-k) (b-k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {\left (\left (-\left ((a+b) (1+k) \left (1+3 b-k+k^2\right )\right )+\frac {(a+b) \left (1+5 b+4 b^2-k+2 b k+5 b k^2-k^3+k^4\right )}{\sqrt {1+4 b-2 k+k^2}}\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {1}{(1-x)^{2/3} x^{5/3} (1-k x)^{2/3} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left ((a+b) \left (1+k^3+3 b (1+k)+\frac {4 b^2+(1-k)^2 \left (1+k+k^2\right )+b \left (5+2 k+5 k^2\right )}{\sqrt {4 b+(-1+k)^2}}\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {1}{(1-x)^{2/3} x^{5/3} (1-k x)^{2/3} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^2 ((1-x) x (1-k x))^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 14.01, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{x ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.42, size = 205, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{x+(-1-k) x^2+k x^3}}{x}+\frac {\left (\sqrt {3} a+\sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {(a+b) \log \left (-\sqrt [3]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}+\frac {(-a-b) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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((-2+(1+k)*x)*(1-(1+k)*x+(a+k)*x^2)/x/((1-x)*x*(-k*x+1))^(2/3)/(1-(1+k)*x+(-b+k)*x^2),x, algorithm="f
ricas")

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+(1+k)*x)*(1-(1+k)*x+(a+k)*x^2)/x/((1-x)*x*(-k*x+1))^(2/3)/(1-(1+k)*x+(-b+k)*x^2),x, algorithm="g
iac")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+(1+k)*x)*(1-(1+k)*x+(a+k)*x^2)/x/((1-x)*x*(-k*x+1))^(2/3)/(1-(1+k)*x+(-b+k)*x^2),x, algorithm="m
axima")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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((-2+(1+k)*x)*(1-(1+k)*x+(a+k)*x**2)/x/((1-x)*x*(-k*x+1))**(2/3)/(1-(1+k)*x+(-b+k)*x**2),x)

[Out]

Timed out

________________________________________________________________________________________