\(\int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} (a^2+b d-(2 a+d) x+x^2)} \, dx\) [2467]

Optimal result

Integrand size = 42, antiderivative size = 201 \[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 4.83 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.90 \[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx=\frac {\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-b+x}}{2 (-a+x)^{2/3}+\sqrt [3]{d} \sqrt [3]{-b+x}}\right )+2 \log \left ((-a+x)^{2/3}-\sqrt [3]{d} \sqrt [3]{-b+x}\right )-\log \left ((-a+x)^{4/3}+\sqrt [3]{d} (-a+x)^{2/3} \sqrt [3]{-b+x}+d^{2/3} (-b+x)^{2/3}\right )\right )}{2 d^{2/3} \sqrt [3]{(-a+x) (-b+x)}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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