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

3.29.44.1 Optimal result

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

output

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)
 

3.29.44.2 Mathematica [A] (verified)

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

input

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]
 

output

-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]] + Log[((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))
 

3.29.44.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2006

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

\(\Big \downarrow \) 7270

\(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int -\frac {(4 a-b-3 x) (x-b)^{5/3}}{(x-a)^{2/3} \left (b^4+6 x^2 b^2-4 x^3 b+x^4+a d-\left (4 b^3+d\right ) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \int \frac {(4 a-b-3 x) (x-b)^{5/3}}{(x-a)^{2/3} \left (b^4+6 x^2 b^2-4 x^3 b+x^4+a d-\left (4 b^3+d\right ) x\right )}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \int \left (\frac {\left (1-\frac {4 a}{b}\right ) b (x-b)^{5/3}}{(x-a)^{2/3} \left (-b^4-6 x^2 b^2+4 x^3 b-x^4-a d+\left (4 b^3+d\right ) x\right )}+\frac {3 x (x-b)^{5/3}}{(x-a)^{2/3} \left (-b^4-6 x^2 b^2+4 x^3 b-x^4-a d+\left (4 b^3+d\right ) x\right )}\right )dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \left (-9 a \text {Subst}\left (\int \frac {\left (x^3+a-b\right )^{5/3}}{x^{12}+4 a \left (1-\frac {b}{a}\right ) x^9+6 a^2 \left (\frac {b (b-2 a)}{a^2}+1\right ) x^6+4 a^3 \left (1-\frac {4 b^3-12 a b^2+12 a^2 b+d}{4 a^3}\right ) x^3+a^4 \left (\frac {b \left (-4 a^3+6 b a^2-4 b^2 a+b^3\right )}{a^4}+1\right )}dx,x,\sqrt [3]{x-a}\right )-9 \text {Subst}\left (\int \frac {x^3 \left (x^3+a-b\right )^{5/3}}{x^{12}+4 a \left (1-\frac {b}{a}\right ) x^9+6 a^2 \left (\frac {b (b-2 a)}{a^2}+1\right ) x^6+4 a^3 \left (1-\frac {4 b^3-12 a b^2+12 a^2 b+d}{4 a^3}\right ) x^3+a^4 \left (\frac {b \left (-4 a^3+6 b a^2-4 b^2 a+b^3\right )}{a^4}+1\right )}dx,x,\sqrt [3]{x-a}\right )+3 (4 a-b) \text {Subst}\left (\int \frac {\left (x^3+a-b\right )^{5/3}}{\left (\frac {a d}{b^4}+1\right ) b^4+6 \left (x^3+a\right )^2 b^2-4 \left (x^3+a\right )^3 b+\left (x^3+a\right )^4-\left (4 b^3+d\right ) \left (x^3+a\right )}dx,x,\sqrt [3]{x-a}\right )\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\)

input

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]
 

output

$Aborted
 

3.29.44.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2006

Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], 
b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px 
, x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P 
x, x], 1] && NeQ[Coeff[Px, x, 0], 0] &&  !MatchQ[Px, (a_.)*(v_)^Expon[Px, x 
] /; FreeQ[a, x] && LinearQ[v, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7270

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p 
]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])))   Int[u*v 
^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !Free 
Q[v, x] &&  !FreeQ[w, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.29.44.4 Maple [F]

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

input

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)
 

output

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)
 

3.29.44.5 Fricas [A] (verification not implemented)

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 ] \]

input

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")
 

output

[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)*lo 
g(-((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]
 

3.29.44.6 Sympy [F(-1)]

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

input

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)
 

output

Timed out
 

3.29.44.7 Maxima [F]

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

input

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")
 

output

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)
 

3.29.44.8 Giac [F]

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

input

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")
 

output

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)
 

3.29.44.9 Mupad [F(-1)]

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

input

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)
 

output

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)