\(\int \frac {(-2+(1+k) x) (1-(1+k) x+(a+k) x^2)}{x ((1-x) x (1-k x))^{2/3} (1-(1+k) x+(-b+k) x^2)} \, dx\) [2489]

Optimal result

Integrand size = 62, antiderivative size = 205 \[ \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{x ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\frac {3 \sqrt [3]{x+(-1-k) x^2+k x^3}}{x}+\frac {\left (\sqrt {3} a+\sqrt {3} b\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {(a+b) \log \left (-\sqrt [3]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}+\frac {(-a-b) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \]

[Out]

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

Rubi [F]

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

[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*} \text {integral}& = \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) \operatorname {Hypergeometric2F1}\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) \operatorname {Hypergeometric2F1}\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) \operatorname {Hypergeometric2F1}\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 [A] (verified)

Time = 15.39 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.81 \[ \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{x ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\frac {6 b^{2/3} \sqrt [3]{(-1+x) x (-1+k x)}+2 \sqrt {3} (a+b) x \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{(-1+x) x (-1+k x)}}\right )+2 (a+b) x \log \left (-\sqrt [3]{b} x+\sqrt [3]{(-1+x) x (-1+k x)}\right )-(a+b) x \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{(-1+x) x (-1+k x)}+((-1+x) x (-1+k x))^{2/3}\right )}{2 b^{2/3} x} \]

[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]

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

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.67

method	result	size
pseudoelliptic	\(-\frac {-6 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} b^{\frac {2}{3}}+\left (2 \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}}{x}\right )\right ) x \left (a +b \right )}{2 b^{\frac {2}{3}} x}\)	\(138\)

[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,method=_RETURNVERBO
SE)

[Out]

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

Fricas [F(-1)]

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

[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

Sympy [F(-1)]

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

[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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)