3.31.8 \(\int \frac {(a-2 b+x) (a^2-2 a x+x^2)}{\sqrt [3]{(-a+x) (-b+x)} (-b^2+a^4 d+2 (b-2 a^3 d) x+(-1+6 a^2 d) x^2-4 a d x^3+d x^4)} \, dx\) [3008]

3.31.8.1 Optimal result

Integrand size = 81, antiderivative size = 405 \[ \int \frac {(a-2 b+x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{(-a+x) (-b+x)} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {a^2}{\sqrt {3}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}+\frac {2 \left (a b+(-a-b) x+x^2\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{(a-x)^2}\right )}{2 d^{2/3}}+\frac {\log \left (a^2 \sqrt [6]{d}-a \sqrt [6]{d} x-a \sqrt [3]{a b+(-a-b) x+x^2}\right )}{2 d^{2/3}}+\frac {\log \left (a^2 \sqrt [6]{d}-a \sqrt [6]{d} x+a \sqrt [3]{a b+(-a-b) x+x^2}\right )}{2 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2+\left (a^3 \sqrt [6]{d}-a^2 \sqrt [6]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+a^2 \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2+\left (-a^3 \sqrt [6]{d}+a^2 \sqrt [6]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+a^2 \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{4 d^{2/3}} \]

output

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

3.31.8.2 Mathematica [F]

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

input

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

output

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

3.31.8.3 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.

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.31.8.3.1 Defintions of rubi rules used

rule 2009

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

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

rule 7292

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

rule 7293

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

3.31.8.4 Maple [F]

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

input

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

output

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

3.31.8.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.31.8.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.31.8.7 Maxima [F]

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

input

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

output

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

3.31.8.8 Giac [F]

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

input

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

output

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

3.31.8.9 Mupad [F(-1)]

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

input

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

output

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