\(\int \frac {(-b+x) (a b-2 b x+x^2)}{(x (-a+x) (-b+x)^2)^{2/3} (b-(1+a d) x+d x^2)} \, dx\) [2756]

Optimal result

Integrand size = 52, antiderivative size = 258 \[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b-(1+a d) x+d x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} b-\sqrt {3} x}{b-x-2 \sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (b-x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (b^2-2 b x+x^2+\left (-b \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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