3.30.28 \(\int \frac {x (-a+x) (-b+x) (a b-2 b x+x^2)}{(x (-a+x) (-b+x)^2)^{2/3} (-b^2+2 b x-(1-a^2 d) x^2-2 a d x^3+d x^4)} \, dx\) [2928]

3.30.28.1 Optimal result

Integrand size = 78, antiderivative size = 340 \[ \int \frac {x (-a+x) (-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+2 b x-\left (1-a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}}{2 b^2-4 b x+2 x^2+\sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (b^2-2 b x+x^2-\sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}}-\frac {\log \left (b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left (b^2 \sqrt [3]{d}-2 b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{4/3}\right )}{4 d^{2/3}} \]

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

3.30.28.2 Mathematica [A] (verified)

Time = 15.69 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.62 \[ \int \frac {x (-a+x) (-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+2 b x-\left (1-a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (x (-a+x) (-b+x)^2\right )^{2/3}}{2 b^2-4 b x+2 x^2+\sqrt [3]{d} \left (x (-a+x) (-b+x)^2\right )^{2/3}}\right )-\log \left (b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\sqrt [3]{d} (b-x)^2 \left ((b-x)^2 x (-a+x)\right )^{2/3}+d^{2/3} \left ((b-x)^2 x (-a+x)\right )^{4/3}\right )+2 \log \left (b^2-2 b x+x^2-\sqrt [3]{d} \left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{4 d^{2/3}} \]

input

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

output

(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(x*(-a + x)*(-b + x)^2)^(2/3))/(2*b^2 - 
 4*b*x + 2*x^2 + d^(1/3)*(x*(-a + x)*(-b + x)^2)^(2/3))] - Log[b^4 - 4*b^3 
*x + 6*b^2*x^2 - 4*b*x^3 + x^4 + d^(1/3)*(b - x)^2*((b - x)^2*x*(-a + x))^ 
(2/3) + d^(2/3)*((b - x)^2*x*(-a + x))^(4/3)] + 2*Log[b^2 - 2*b*x + x^2 - 
d^(1/3)*(x*(-a + x)*(-b + x)^2)^(2/3)])/(4*d^(2/3))
 

3.30.28.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.30.28.3.1 Defintions of rubi rules used

rule 25

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

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] 
]
 

3.30.28.4 Maple [F]

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

input

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

output

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

3.30.28.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.30.28.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.30.28.7 Maxima [F]

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

input

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

output

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

3.30.28.8 Giac [F]

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

input

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

output

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

3.30.28.9 Mupad [F(-1)]

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

input

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

output

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