\(\int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} (b+a^2 d+(-1-2 a d) x+d x^2)} \, dx\) [2559]

Optimal result

Integrand size = 44, antiderivative size = 216 \[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (b+a^2 d+(-1-2 a d) x+d x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a b+(-a-b) x+x^2}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{a b+(-a-b) x+x^2}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{a b+(-a-b) x+x^2}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+\left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

Defer[Int][(a - 2*b + x)/((a*b + (-a - b)*x + x^2)^(1/3)*(b + a^2*d + (-1 - 2*a*d)*x + d*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 (b+a^2 d+(-1-2 a d) x+d x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.41 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.81 \[ \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (b+a^2 d+(-1-2 a d) x+d x^2\right )} \, dx=\frac {\sqrt [3]{a-x} \sqrt [3]{b-x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b-x}}{-2 \sqrt [3]{d} (a-x)^{2/3}+\sqrt [3]{b-x}}\right )+2 \log \left (\sqrt [3]{d} (a-x)^{2/3}+\sqrt [3]{b-x}\right )-\log \left (d^{2/3} (a-x)^{4/3}-\sqrt [3]{d} (a-x)^{2/3} \sqrt [3]{b-x}+(b-x)^{2/3}\right )\right )}{2 \sqrt [3]{d} \sqrt [3]{(-a+x) (-b+x)}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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