\(\int \frac {a b-2 b x+x^2}{\sqrt [3]{x (-a+x) (-b+x)^2} (b-(1+a d) x+d x^2)} \, dx\) [2842]

Optimal result

Integrand size = 47, antiderivative size = 291 \[ \int \frac {a b-2 b x+x^2}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (b-(1+a d) x+d x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{-2 b+2 x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (b-x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (b^2-2 b x+x^2+\left (-b \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(3*x*(1 - x/a)^(1/3)*(1 - x/b)^(2/3)*AppellF1[2/3, 1/3, 2/3, 5/3, x/a, x/b])/(2*d*(-((a - x)*(b - x)^2*x))^(1/
3)) + ((1 + a*d - 2*b*d + Sqrt[1 + 2*a*d - 4*b*d + a^2*d^2])*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(2/3)*Defer[Int][
1/(x^(1/3)*(-a + x)^(1/3)*(-b + x)^(2/3)*(-1 - a*d - Sqrt[1 + 2*a*d - 4*b*d + a^2*d^2] + 2*d*x)), x])/(d*(-((a
- x)*(b - x)^2*x))^(1/3)) + ((1 + a*d - 2*b*d - Sqrt[1 + 2*a*d - 4*b*d + a^2*d^2])*x^(1/3)*(-a + x)^(1/3)*(-b
+ x)^(2/3)*Defer[Int][1/(x^(1/3)*(-a + x)^(1/3)*(-b + x)^(2/3)*(-1 - a*d + Sqrt[1 + 2*a*d - 4*b*d + a^2*d^2]
+ 2*d*x)), x])/(d*(-((a - x)*(b - x)^2*x))^(1/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {a b-2 b x+x^2}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (b-(1+a d) x+d x^2\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {1}{d \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}}-\frac {b-a b d-(1+a d-2 b d) x}{d \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (b+(-1-a d) x+d x^2\right )}\right ) \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}} \, dx}{d \sqrt [3]{x (-a+x) (-b+x)^2}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {b-a b d-(1+a d-2 b d) x}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (b+(-1-a d) x+d x^2\right )} \, dx}{d \sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = -\frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {-1-a d+2 b d-\sqrt {1+2 a d-4 b d+a^2 d^2}}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-1-a d-\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )}+\frac {-1-a d+2 b d+\sqrt {1+2 a d-4 b d+a^2 d^2}}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-1-a d+\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )}\right ) \, dx}{d \sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (\sqrt [3]{x} (-b+x)^{2/3} \sqrt [3]{1-\frac {x}{a}}\right ) \int \frac {1}{\sqrt [3]{x} (-b+x)^{2/3} \sqrt [3]{1-\frac {x}{a}}} \, dx}{d \sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = -\frac {\left (\left (-1-a d+2 b d-\sqrt {1+2 a d-4 b d+a^2 d^2}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-1-a d-\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x (-a+x) (-b+x)^2}}-\frac {\left (\left (-1-a d+2 b d+\sqrt {1+2 a d-4 b d+a^2 d^2}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-1-a d+\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1-\frac {x}{a}} \left (1-\frac {x}{b}\right )^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{1-\frac {x}{a}} \left (1-\frac {x}{b}\right )^{2/3}} \, dx}{d \sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = \frac {3 x \sqrt [3]{1-\frac {x}{a}} \left (1-\frac {x}{b}\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {x}{a},\frac {x}{b}\right )}{2 d \sqrt [3]{-\left ((a-x) (b-x)^2 x\right )}}-\frac {\left (\left (-1-a d+2 b d-\sqrt {1+2 a d-4 b d+a^2 d^2}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-1-a d-\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x (-a+x) (-b+x)^2}}-\frac {\left (\left (-1-a d+2 b d+\sqrt {1+2 a d-4 b d+a^2 d^2}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-1-a d+\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )} \, dx}{d \sqrt [3]{x (-a+x) (-b+x)^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.53 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.69 \[ \int \frac {a b-2 b x+x^2}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (b-(1+a d) x+d x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{(b-x)^2 x (-a+x)}}{-2 b+2 x+\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}}\right )+2 \log \left (-b+x-\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}\right )-\log \left (b^2-2 b x+x^2-b \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)^2}+d^{2/3} \left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{2 d^{2/3}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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