\(\int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} (b+a^2 d-(1+2 a d) x+d x^2)} \, dx\) [2468]

Optimal result

Integrand size = 55, antiderivative size = 201 \[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (b+a^2 d-(1+2 a d) x+d x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}{-2 b+2 x+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (b-x+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (b^2-2 b x+x^2+\left (-b \sqrt [3]{d}+\sqrt [3]{d} x\right ) \left (a b+(-a-b) x+x^2\right )^{2/3}+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{4/3}\right )}{2 d^{2/3}} \]

[Out]

3^(1/2)*arctan(3^(1/2)*d^(1/3)*(a*b+(-a-b)*x+x^2)^(2/3)/(-2*b+2*x+d^(1/3)*(a*b+(-a-b)*x+x^2)^(2/3)))/d^(2/3)+l
n(b-x+d^(1/3)*(a*b+(-a-b)*x+x^2)^(2/3))/d^(2/3)-1/2*ln(b^2-2*b*x+x^2+(-b*d^(1/3)+d^(1/3)*x)*(a*b+(-a-b)*x+x^2)
^(2/3)+d^(2/3)*(a*b+(-a-b)*x+x^2)^(4/3))/d^(2/3)

Rubi [F]

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

[In]

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

[Out]

-((2^(1/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*((a - b)^(2/3) + 2^(2/3)*((a - x)*(b - x))^(1/3))*Sqrt[(-a - b + 2*x)^2]*
Sqrt[((a - b)^(4/3) - 2^(2/3)*(a - b)^(2/3)*(a*b - (a + b)*x + x^2)^(1/3) + 2*2^(1/3)*(a*b - (a + b)*x + x^2)^
(2/3))/((1 + Sqrt[3])*(a - b)^(2/3) + 2^(2/3)*((a - x)*(b - x))^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*(a -
b)^(2/3) + 2^(2/3)*((a - x)*(b - x))^(1/3))/((1 + Sqrt[3])*(a - b)^(2/3) + 2^(2/3)*((a - x)*(b - x))^(1/3))],
-7 - 4*Sqrt[3]])/(d*Sqrt[((a - b)^(2/3)*((a - b)^(2/3) + 2^(2/3)*((a - x)*(b - x))^(1/3)))/((1 + Sqrt[3])*(a
- b)^(2/3) + 2^(2/3)*((a - x)*(b - x))^(1/3))^2]*(a + b - 2*x)*Sqrt[(a + b - 2*x)^2])) - Defer[Int][(b + 2*a^2
*d - 2*a*b*d + (-1 - 2*a*d + 2*b*d)*x)/((a*b + (-a - b)*x + x^2)^(2/3)*(b + a^2*d + (-1 - 2*a*d)*x + d*x^2)),
x]/d

Rubi steps \begin{align*} \text {integral}& = \int \frac {-a (a-2 b)-2 b x+x^2}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (b+a^2 d-(1+2 a d) x+d x^2\right )} \, dx \\ & = \int \left (\frac {1}{d \left (a b+(-a-b) x+x^2\right )^{2/3}}-\frac {b+2 a^2 d-2 a b d-(1+2 a d-2 b d) x}{d \left (a b+(-a-b) x+x^2\right )^{2/3} \left (b+a^2 d+(-1-2 a d) x+d x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {1}{\left (a b+(-a-b) x+x^2\right )^{2/3}} \, dx}{d}-\frac {\int \frac {b+2 a^2 d-2 a b d+(-1-2 a d+2 b d) x}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (b+a^2 d+(-1-2 a d) x+d x^2\right )} \, dx}{d} \\ & = -\frac {\int \frac {b+2 a^2 d-2 a b d+(-1-2 a d+2 b d) x}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (b+a^2 d+(-1-2 a d) x+d x^2\right )} \, dx}{d}+\frac {\left (3 \sqrt {(-a-b+2 x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {(-a-b)^2-4 a b+4 x^3}} \, dx,x,\sqrt [3]{a b+(-a-b) x+x^2}\right )}{d (-a-b+2 x)} \\ & = -\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2+\sqrt {3}} \left ((a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}\right ) \sqrt {(-a-b+2 x)^2} \sqrt {\frac {(a-b)^{4/3}-2^{2/3} (a-b)^{2/3} \sqrt [3]{a b-(a+b) x+x^2}+2 \sqrt [3]{2} \left (a b-(a+b) x+x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) (a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}}{\left (1+\sqrt {3}\right ) (a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}}\right ),-7-4 \sqrt {3}\right )}{d \sqrt {\frac {(a-b)^{2/3} \left ((a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}\right )}{\left (\left (1+\sqrt {3}\right ) (a-b)^{2/3}+2^{2/3} \sqrt [3]{(a-x) (b-x)}\right )^2}} (a+b-2 x) \sqrt {(a+b-2 x)^2}}-\frac {\int \frac {b+2 a^2 d-2 a b d+(-1-2 a d+2 b d) x}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (b+a^2 d+(-1-2 a d) x+d x^2\right )} \, dx}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.35 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.89 \[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (b+a^2 d-(1+2 a d) x+d x^2\right )} \, dx=-\frac {(a-x)^{2/3} (b-x)^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} (a-x)^{2/3}}{\sqrt [3]{d} (a-x)^{2/3}-2 \sqrt [3]{b-x}}\right )-2 \log \left (\sqrt [3]{d} (a-x)^{2/3}+\sqrt [3]{b-x}\right )+\log \left (d^{2/3} (a-x)^{4/3}-\sqrt [3]{d} (a-x)^{2/3} \sqrt [3]{b-x}+(b-x)^{2/3}\right )\right )}{2 d^{2/3} ((-a+x) (-b+x))^{2/3}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

integrate((-a*(a-2*b)-2*b*x+x^2)/((-a+x)*(-b+x))^(2/3)/(b+a^2*d-(2*a*d+1)*x+d*x^2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate((-a*(a-2*b)-2*b*x+x^2)/((-a+x)*(-b+x))^(2/3)/(b+a^2*d-(2*a*d+1)*x+d*x^2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((-a*(a-2*b)-2*b*x+x^2)/((-a+x)*(-b+x))^(2/3)/(b+a^2*d-(2*a*d+1)*x+d*x^2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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