\(\int \frac {(-4 a+b+3 x) (-b^3+3 b^2 x-3 b x^2+x^3)}{((-a+x) (-b+x)^2)^{2/3} (a+b^4 d-(1+4 b^3 d) x+6 b^2 d x^2-4 b d x^3+d x^4)} \, dx\) [2812]

Optimal result

Integrand size = 88, antiderivative size = 276 \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}{-2 a+2 x+\sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{4/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

\[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx \]

[In]

Int[((-4*a + b + 3*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(((-a + x)*(-b + x)^2)^(2/3)*(a + b^4*d - (1 + 4*b^3*d
)*x + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4)),x]

[Out]

(9*a*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[Subst][Defer[Int][(a - b + x^3)^(5/3)/(a^4*(1 + (b*(-4*a^3 + 6*a^2*b
- 4*a*b^2 + b^3))/a^4)*d - (1 - 4*(a - b)^3*d)*x^3 + 6*a^2*(1 + (b*(-2*a + b))/a^2)*d*x^6 + 4*a*(1 - b/a)*d*x^
9 + d*x^12), x], x, (-a + x)^(1/3)])/(-((a - x)*(b - x)^2))^(2/3) + (9*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[Sub
st][Defer[Int][(x^3*(a - b + x^3)^(5/3))/(a^4*(1 + (b*(-4*a^3 + 6*a^2*b - 4*a*b^2 + b^3))/a^4)*d - (1 - 4*(a -
b)^3*d)*x^3 + 6*a^2*(1 + (b*(-2*a + b))/a^2)*d*x^6 + 4*a*(1 - b/a)*d*x^9 + d*x^12), x], x, (-a + x)^(1/3)])/(
-((a - x)*(b - x)^2))^(2/3) - (3*(4*a - b)*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[Subst][Defer[Int][(a - b + x^3)
^(5/3)/(a*(1 + (b^4*d)/a) - (1 + 4*b^3*d)*(a + x^3) + 6*b^2*d*(a + x^3)^2 - 4*b*d*(a + x^3)^3 + d*(a + x^3)^4)
, x], x, (-a + x)^(1/3)])/(-((a - x)*(b - x)^2))^(2/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{(-a+x)^{2/3} (-b+x)^{4/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-4 a+b+3 x) \left (b^2-2 b x+x^2\right )}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-b+x)^{5/3} (-4 a+b+3 x)}{(-a+x)^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \left (\frac {4 a \left (1-\frac {b}{4 a}\right ) (-b+x)^{5/3}}{(-a+x)^{2/3} \left (-a-b^4 d+\left (1+4 b^3 d\right ) x-6 b^2 d x^2+4 b d x^3-d x^4\right )}+\frac {3 x (-b+x)^{5/3}}{(-a+x)^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )}\right ) \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (3 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {x (-b+x)^{5/3}}{(-a+x)^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left ((4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-b+x)^{5/3}}{(-a+x)^{2/3} \left (-a-b^4 d+\left (1+4 b^3 d\right ) x-6 b^2 d x^2+4 b d x^3-d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (9 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {\left (a+x^3\right ) \left (a-b+x^3\right )^{5/3}}{a+b^4 d-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left (3 (4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a+b^4 d-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (9 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {\left (a+x^3\right ) \left (a-b+x^3\right )^{5/3}}{a \left (1+\frac {b^4 d}{a}\right )-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left (3 (4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a \left (1+\frac {b^4 d}{a}\right )-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (9 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \left (\frac {a \left (a-b+x^3\right )^{5/3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) d-\left (1-4 (a-b)^3 d\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^6+4 a \left (1-\frac {b}{a}\right ) d x^9+d x^{12}}+\frac {x^3 \left (a-b+x^3\right )^{5/3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) d-\left (1-4 (a-b)^3 d\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^6+4 a \left (1-\frac {b}{a}\right ) d x^9+d x^{12}}\right ) \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left (3 (4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a \left (1+\frac {b^4 d}{a}\right )-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (9 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^3 \left (a-b+x^3\right )^{5/3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) d-\left (1-4 (a-b)^3 d\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^6+4 a \left (1-\frac {b}{a}\right ) d x^9+d x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (9 a (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) d-\left (1-4 (a-b)^3 d\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^6+4 a \left (1-\frac {b}{a}\right ) d x^9+d x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left (3 (4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a \left (1+\frac {b^4 d}{a}\right )-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00 \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=-\frac {(a-x)^{2/3} (b-x)^{4/3} \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a-x}}{\sqrt [3]{d} (b-x)^{4/3}}}{\sqrt {3}}\right )-2 \log \left (a \sqrt [3]{d}-b \sqrt [3]{d}+\frac {(a-x)^{4/3}}{(b-x)^{4/3}}-\frac {\sqrt [3]{a-x}}{\sqrt [3]{b-x}}\right )+\log \left (\frac {(a-b)^2 \left ((a-x)^{2/3}+b^2 d^{2/3} (b-x)^{2/3}+\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{b-x} x+d^{2/3} (b-x)^{2/3} x^2-b \sqrt [3]{d} \sqrt [3]{b-x} \left (\sqrt [3]{a-x}+2 \sqrt [3]{d} \sqrt [3]{b-x} x\right )\right )}{(b-x)^{8/3}}\right )\right )}{2 d^{2/3} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]

[In]

Integrate[((-4*a + b + 3*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(((-a + x)*(-b + x)^2)^(2/3)*(a + b^4*d - (1 + 4
*b^3*d)*x + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4)),x]

[Out]

-1/2*((a - x)^(2/3)*(b - x)^(4/3)*(2*Sqrt[3]*ArcTan[(1 - (2*(a - x)^(1/3))/(d^(1/3)*(b - x)^(4/3)))/Sqrt[3]] -
2*Log[a*d^(1/3) - b*d^(1/3) + (a - x)^(4/3)/(b - x)^(4/3) - (a - x)^(1/3)/(b - x)^(1/3)] + Log[((a - b)^2*((a
- x)^(2/3) + b^2*d^(2/3)*(b - x)^(2/3) + d^(1/3)*(a - x)^(1/3)*(b - x)^(1/3)*x + d^(2/3)*(b - x)^(2/3)*x^2 -
b*d^(1/3)*(b - x)^(1/3)*((a - x)^(1/3) + 2*d^(1/3)*(b - x)^(1/3)*x)))/(b - x)^(8/3)]))/(d^(2/3)*((b - x)^2*(-a
+ x))^(2/3))

Maple [F]

\[\int \frac {\left (-4 a +b +3 x \right ) \left (-b^{3}+3 b^{2} x -3 b \,x^{2}+x^{3}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (a +b^{4} d -\left (4 b^{3} d +1\right ) x +6 b^{2} d \,x^{2}-4 b d \,x^{3}+d \,x^{4}\right )}d x\]

[In]

int((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/(a+b^4*d-(4*b^3*d+1)*x+6*b^2*d*x^2-4*b*d*x
^3+d*x^4),x)

[Out]

int((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/(a+b^4*d-(4*b^3*d+1)*x+6*b^2*d*x^2-4*b*d*x
^3+d*x^4),x)

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.27 \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=-\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left ({\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, {\left (b^{2} d^{2} - 2 \, b d^{2} x + d^{2} x^{2}\right )}}\right ) + {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {2}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d + {\left (b^{4} d - 4 \, b^{3} d x + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4}\right )} {\left (d^{2}\right )}^{\frac {1}{3}}}{b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}}\right ) - 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b^{2} - 2 \, b x + x^{2}}\right )}{2 \, d^{2}} \]

[In]

integrate((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/(a+b^4*d-(4*b^3*d+1)*x+6*b^2*d*x^2-4
*b*d*x^3+d*x^4),x, algorithm="fricas")

[Out]

-1/2*(2*sqrt(3)*(d^2)^(1/6)*d*arctan(1/3*sqrt(3)*(d^2)^(1/6)*((b^2*d - 2*b*d*x + d*x^2)*(d^2)^(1/3) + 2*(-a*b^
2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(d^2)^(2/3))/(b^2*d^2 - 2*b*d^2*x + d^2*x^2)) + (d^2)^(2/3)*l
og(((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b^2 - 2*b*x + x^2)*(d^2)^(2/3) + (-a*b^2 - (a + 2*
b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d + (b^4*d - 4*b^3*d*x + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4)*(d^2)^(1/3))/(
b^4 - 4*b^3*x + 6*b^2*x^2 - 4*b*x^3 + x^4)) - 2*(d^2)^(2/3)*log(-((b^2 - 2*b*x + x^2)*(d^2)^(2/3) - (-a*b^2 -
(a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*d)/(b^2 - 2*b*x + x^2)))/d^2

Sympy [F(-1)]

Timed out. \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((-4*a+b+3*x)*(-b**3+3*b**2*x-3*b*x**2+x**3)/((-a+x)*(-b+x)**2)**(2/3)/(a+b**4*d-(4*b**3*d+1)*x+6*b**
2*d*x**2-4*b*d*x**3+d*x**4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\int { \frac {{\left (b^{3} - 3 \, b^{2} x + 3 \, b x^{2} - x^{3}\right )} {\left (4 \, a - b - 3 \, x\right )}}{{\left (b^{4} d + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4} - {\left (4 \, b^{3} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}}} \,d x } \]

[In]

integrate((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/(a+b^4*d-(4*b^3*d+1)*x+6*b^2*d*x^2-4
*b*d*x^3+d*x^4),x, algorithm="maxima")

[Out]

integrate((b^3 - 3*b^2*x + 3*b*x^2 - x^3)*(4*a - b - 3*x)/((b^4*d + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4 - (4*b^3*d
+ 1)*x + a)*(-(a - x)*(b - x)^2)^(2/3)), x)

Giac [F]

\[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\int { \frac {{\left (b^{3} - 3 \, b^{2} x + 3 \, b x^{2} - x^{3}\right )} {\left (4 \, a - b - 3 \, x\right )}}{{\left (b^{4} d + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4} - {\left (4 \, b^{3} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}}} \,d x } \]

[In]

integrate((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/(a+b^4*d-(4*b^3*d+1)*x+6*b^2*d*x^2-4
*b*d*x^3+d*x^4),x, algorithm="giac")

[Out]

integrate((b^3 - 3*b^2*x + 3*b*x^2 - x^3)*(4*a - b - 3*x)/((b^4*d + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4 - (4*b^3*d
+ 1)*x + a)*(-(a - x)*(b - x)^2)^(2/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\int -\frac {\left (b-4\,a+3\,x\right )\,\left (b^3-3\,b^2\,x+3\,b\,x^2-x^3\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (a+b^4\,d+d\,x^4-x\,\left (4\,d\,b^3+1\right )+6\,b^2\,d\,x^2-4\,b\,d\,x^3\right )} \,d x \]

[In]

int(-((b - 4*a + 3*x)*(3*b*x^2 - 3*b^2*x + b^3 - x^3))/((-(a - x)*(b - x)^2)^(2/3)*(a + b^4*d + d*x^4 - x*(4*b
^3*d + 1) + 6*b^2*d*x^2 - 4*b*d*x^3)),x)

[Out]

int(-((b - 4*a + 3*x)*(3*b*x^2 - 3*b^2*x + b^3 - x^3))/((-(a - x)*(b - x)^2)^(2/3)*(a + b^4*d + d*x^4 - x*(4*b
^3*d + 1) + 6*b^2*d*x^2 - 4*b*d*x^3)), x)