3.26.15 \(\int \frac {x (4 a b-3 (a+b) x+2 x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (-a b d+(a+b) d x-d x^2+x^4)} \, dx\) [2515]

3.26.15.1 Optimal result

Integrand size = 59, antiderivative size = 210 \[ \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{2 x^2+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (x^2-\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (x^4+\sqrt [3]{d} x^2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

output

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

3.26.15.2 Mathematica [A] (verified)

Time = 11.94 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.80 \[ \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}}{2 x^2+\sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}}\right )+2 \log \left (x^2-\sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}\right )-\log \left (x^4+\sqrt [3]{d} x^2 \sqrt [3]{x^2 (-a+x) (-b+x)}+d^{2/3} \left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 d^{2/3}} \]

input

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

output

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

3.26.15.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.26.15.3.1 Defintions of rubi rules used

rule 25

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

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 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 7293

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

3.26.15.4 Maple [F]

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

input

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

output

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

3.26.15.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.26.15.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.26.15.7 Maxima [F]

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

input

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

output

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

3.26.15.8 Giac [F]

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

input

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

output

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

3.26.15.9 Mupad [F(-1)]

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

input

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

output

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