3.29.36 \(\int \frac {(a-2 b+x) (-b+x)}{\sqrt [3]{(-a+x) (-b+x)} (a^4-b^2 d-2 (2 a^3-b d) x+(6 a^2-d) x^2-4 a x^3+x^4)} \, dx\) [2836]

3.29.36.1 Optimal result

Integrand size = 74, antiderivative size = 289 \[ \int \frac {(a-2 b+x) (-b+x)}{\sqrt [3]{(-a+x) (-b+x)} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{2 a-2 x+\sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{-2 a+2 x+\sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{2 d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{a-x}\right )}{d^{5/6}}+\frac {\text {arctanh}\left (\frac {\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}}{a^2-2 a x+x^2+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}\right )}{2 d^{5/6}} \]

output

-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*(a*b+(-a-b)*x+x^2)^(1/3)/(2*a-2*x+d^(1 
/6)*(a*b+(-a-b)*x+x^2)^(1/3)))/d^(5/6)+1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)* 
(a*b+(-a-b)*x+x^2)^(1/3)/(-2*a+2*x+d^(1/6)*(a*b+(-a-b)*x+x^2)^(1/3)))/d^(5 
/6)+arctanh(d^(1/6)*(a*b+(-a-b)*x+x^2)^(1/3)/(a-x))/d^(5/6)+1/2*arctanh((a 
*d^(1/6)-d^(1/6)*x)*(a*b+(-a-b)*x+x^2)^(1/3)/(a^2-2*a*x+x^2+d^(1/3)*(a*b+( 
-a-b)*x+x^2)^(2/3)))/d^(5/6)
 

3.29.36.2 Mathematica [A] (verified)

Time = 11.93 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.78 \[ \int \frac {(a-2 b+x) (-b+x)}{\sqrt [3]{(-a+x) (-b+x)} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx=\frac {\sqrt {3} \left (-\arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{(-a+x) (-b+x)}}{2 a-2 x+\sqrt [6]{d} \sqrt [3]{(-a+x) (-b+x)}}\right )+\arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{(-a+x) (-b+x)}}{-2 a+2 x+\sqrt [6]{d} \sqrt [3]{(-a+x) (-b+x)}}\right )\right )-2 \text {arctanh}\left (\frac {\sqrt [6]{d} (-b+x)}{((-a+x) (-b+x))^{2/3}}\right )-\text {arctanh}\left (\frac {\sqrt [6]{d} (-a+x) \sqrt [3]{(-a+x) (-b+x)}}{a^2-2 a x+x^2+\sqrt [3]{d} ((-a+x) (-b+x))^{2/3}}\right )}{2 d^{5/6}} \]

input

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

output

(Sqrt[3]*(-ArcTan[(Sqrt[3]*d^(1/6)*((-a + x)*(-b + x))^(1/3))/(2*a - 2*x + 
 d^(1/6)*((-a + x)*(-b + x))^(1/3))] + ArcTan[(Sqrt[3]*d^(1/6)*((-a + x)*( 
-b + x))^(1/3))/(-2*a + 2*x + d^(1/6)*((-a + x)*(-b + x))^(1/3))]) - 2*Arc 
Tanh[(d^(1/6)*(-b + x))/((-a + x)*(-b + x))^(2/3)] - ArcTanh[(d^(1/6)*(-a 
+ x)*((-a + x)*(-b + x))^(1/3))/(a^2 - 2*a*x + x^2 + d^(1/3)*((-a + x)*(-b 
 + x))^(2/3))])/(2*d^(5/6))
 

3.29.36.3 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.

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.29.36.3.1 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] 
]
 

3.29.36.4 Maple [F]

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

input

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

output

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

3.29.36.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.29.36.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.29.36.7 Maxima [F]

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

input

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

output

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

3.29.36.8 Giac [F]

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

input

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

output

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

3.29.36.9 Mupad [F(-1)]

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

input

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

output

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