3.29.86 \(\int \frac {(-b+x) (a b-2 b x+x^2)}{\sqrt [3]{x (-a+x) (-b+x)^2} (-b^2 d+2 b d x-(-a^2+d) x^2-2 a x^3+x^4)} \, dx\) [2886]

3.29.86.1 Optimal result

Integrand size = 70, antiderivative size = 313 \[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2 d+2 b d x-\left (-a^2+d\right ) x^2-2 a x^3+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{-2 a x+2 x^2+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{2 d^{2/3}}+\frac {\log \left (a x-x^2+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{2 d^{2/3}}-\frac {\log \left (a^2 x^2-2 a x^3+x^4+\left (-a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{4 d^{2/3}} \]

output

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

3.29.86.2 Mathematica [A] (verified)

Time = 15.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.69 \[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-b^2 d+2 b d x-\left (-a^2+d\right ) x^2-2 a x^3+x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{(b-x)^2 x (-a+x)}}{-2 a x+2 x^2+\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}}\right )+2 \log \left (-a x+x^2-\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}\right )-\log \left (a^2 x^2-2 a x^3+x^4-a \sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)^2}+\sqrt [3]{d} x^2 \sqrt [3]{x (-a+x) (-b+x)^2}+d^{2/3} \left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{4 d^{2/3}} \]

input

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

output

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

3.29.86.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {\sqrt [3]{x-b} x^{7/3}}{\sqrt [3]{a-x} \left (-x^4+2 a x^3-a^2 \left (1-\frac {d}{a^2}\right ) x^2-2 b d x+b^2 d\right )}+\frac {2 b \sqrt [3]{x-b} x^{4/3}}{\sqrt [3]{a-x} \left (x^4-2 a x^3+a^2 \left (1-\frac {d}{a^2}\right ) x^2+2 b d x-b^2 d\right )}+\frac {a b \sqrt [3]{x-b} \sqrt [3]{x}}{\sqrt [3]{a-x} \left (-x^4+2 a x^3-a^2 \left (1-\frac {d}{a^2}\right ) x^2-2 b d x+b^2 d\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \left (a b \int \frac {\sqrt [3]{x} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (-x^4+2 a x^3-a^2 \left (1-\frac {d}{a^2}\right ) x^2-2 b d x+b^2 d\right )}d\sqrt [3]{x}+\int \frac {x^{7/3} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (-x^4+2 a x^3-a^2 \left (1-\frac {d}{a^2}\right ) x^2-2 b d x+b^2 d\right )}d\sqrt [3]{x}+2 b \int \frac {x^{4/3} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (x^4-2 a x^3+a^2 \left (1-\frac {d}{a^2}\right ) x^2+2 b d x-b^2 d\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

input

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

output

$Aborted
 

3.29.86.3.1 Defintions of rubi rules used

rule 281

Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ 
Symbol] :> Simp[(b/d)^p   Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, 
 c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] &&  !(IntegerQ[q] & 
& SimplerQ[a + b*x^n, c + d*x^n])
 

rule 2009

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

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, 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] 
]
 

3.29.86.4 Maple [F]

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

input

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

output

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

3.29.86.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.29.86.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.29.86.7 Maxima [F]

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

input

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

output

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

3.29.86.8 Giac [F]

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

input

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

output

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

3.29.86.9 Mupad [F(-1)]

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

input

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

output

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