\(\int \frac {(-4 a+b+3 x) (-b^3+3 b^2 x-3 b x^2+x^3)}{((-a+x) (-b+x)^2)^{2/3} (b^4+a d-(4 b^3+d) x+6 b^2 x^2-4 b x^3+x^4)} \, dx\) [2844]

Optimal result

Integrand size = 83, antiderivative size = 291 \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt {d}-\sqrt {d} x+\sqrt [6]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d-2 a d x+d x^2+\left (-a d^{2/3}+d^{2/3} x\right ) \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}+\sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{4/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

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

Rubi [F]

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

[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)*(b^4 + a*d - (4*b^3 + d)*
x + 6*b^2*x^2 - 4*b*x^3 + 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) + 4*a^3*(1 - (12*a^2*b - 12*a*b^2 + 4*b^3 + d)/(4*a^3))*x^3 + 6*a^2*(1 + (b*(-2*a + b))
/a^2)*x^6 + 4*a*(1 - b/a)*x^9 + 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[Subst][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) + 4*a^3*(1 - (12*a^2*b - 12*a*b^2 + 4*b^3 + d)/(4*a^3))*x^3 + 6*a^2*(1 + (b*(-2*a + b))/a^2)*x^6 +
4*a*(1 - b/a)*x^9 + 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)/(b^4*(1 + (a*d)/b^4) - (4*b^3 + d)*(a + x^3) + 6*b^
2*(a + x^3)^2 - 4*b*(a + x^3)^3 + (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 (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+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 (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+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 (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+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 (-b^4-a d+\left (4 b^3+d\right ) x-6 b^2 x^2+4 b x^3-x^4\right )}+\frac {3 x (-b+x)^{5/3}}{(-a+x)^{2/3} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+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 (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+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 (-b^4-a d+\left (4 b^3+d\right ) x-6 b^2 x^2+4 b x^3-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}}{b^4+a d-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\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}}{b^4+a d-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\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}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\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}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\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 )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+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 )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+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}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\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 )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+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 )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+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}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\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 = 0.48 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.68 \[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=-\frac {(a-x)^{2/3} (b-x)^{4/3} \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a-x}}{(b-x)^{4/3}}}{\sqrt {3}}\right )+\log \left (\frac {(a-b)^2 \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} (b-x)^{4/3}+(b-x)^{8/3}\right )}{(b-x)^{8/3}}\right )-2 \log \left (\frac {(a-b) \left (\sqrt [3]{d} \sqrt [3]{a-x}+(b-x)^{4/3}\right )}{(b-x)^{4/3}}\right )\right )}{2 \sqrt [3]{d} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]

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

[Out]

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

Maple [F]

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

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

Fricas [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.74 \[ \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 (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {b^{4} + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4} - 2 \, a d - 2 \, {\left (2 \, b^{3} - d\right )} x + \sqrt {3} {\left ({\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 )} d^{\frac {2}{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 {2}{3}} d + {\left (b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\right )} d^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} - 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}} {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\frac {1}{3}}}{b^{4} + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4} + a d - {\left (4 \, b^{3} + d\right )} x}\right ) - d^{\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 )} d^{\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} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\right )} d^{\frac {1}{3}}}{b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\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}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left ({\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\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}} d^{\frac {2}{3}}\right )}}{3 \, {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\frac {1}{3}}}\right ) + d^{\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 )} d^{\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} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\right )} d^{\frac {1}{3}}}{b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}}\right ) - 2 \, d^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\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}\right ] \]

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

[Out]

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

Sympy [F(-1)]

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

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

[Out]

Timed out

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

[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*d - 4*b*x^3 - x*(d + 4*b
^3) + b^4 + x^4 + 6*b^2*x^2)),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*d - 4*b*x^3 - x*(d + 4*b
^3) + b^4 + x^4 + 6*b^2*x^2)), x)