3.30.52 \(\int \frac {x^3 (4 a b-3 (a+b) x+2 x^2)}{(x^2 (-a+x) (-b+x))^{2/3} (-a b d+(a+b) d x-d x^2+x^4)} \, dx\) [2952]

3.30.52.1 Optimal result

Integrand size = 61, antiderivative size = 355 \[ \int \frac {x^3 \left (4 a b-3 (a+b) x+2 x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}{2 a b \sqrt [3]{d}+\left (-2 a \sqrt [3]{d}-2 b \sqrt [3]{d}\right ) x+2 \sqrt [3]{d} x^2+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (-a b \sqrt {d}+a \sqrt {d} x+b \sqrt {d} x-\sqrt {d} x^2+\sqrt [6]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 b^2 d-2 a^2 b d x-2 a b^2 d x+a^2 d x^2+4 a b d x^2+b^2 d x^2-2 a d x^3-2 b d x^3+d x^4+\left (a b d^{2/3}-a d^{2/3} x-b d^{2/3} x+d^{2/3} x^2\right ) \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}+\sqrt [3]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{4/3}\right )}{2 \sqrt [3]{d}} \]

output

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

3.30.52.2 Mathematica [A] (verified)

Time = 13.72 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.70 \[ \int \frac {x^3 \left (4 a b-3 (a+b) x+2 x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \left (x^2 (-a+x) (-b+x)\right )^{2/3}}{2 a \sqrt [3]{d} (b-x)-2 b \sqrt [3]{d} x+2 \sqrt [3]{d} x^2+\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\right )+2 \log \left (-\sqrt [6]{d} \left (b \sqrt [3]{d} x-\sqrt [3]{d} x^2+a \sqrt [3]{d} (-b+x)+\left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )\right )-\log \left (\sqrt [3]{d} (-a+x) (-b+x) \left (a d^{2/3} (b-x)-b d^{2/3} x+d^{2/3} x^2+x^2 \sqrt [3]{x^2 (-a+x) (-b+x)}+\sqrt [3]{d} \left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )\right )}{2 \sqrt [3]{d}} \]

input

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

output

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

3.30.52.3 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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{4/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int -\frac {x^{5/3} \left (2 x^2-3 (a+b) x+4 a b\right )}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-x^4+d x^2-(a+b) d x+a b d\right )}dx}{\left (x^2 (a-x) (b-x)\right )^{2/3}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {x^{4/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {x^{5/3} \left (2 x^2-3 (a+b) x+4 a b\right )}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-x^4+d x^2-(a+b) d x+a b d\right )}dx}{\left (x^2 (a-x) (b-x)\right )^{2/3}}\)

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 x^{4/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \left (2 a b d \int \frac {\sqrt [3]{x}}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-x^4+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt [3]{x}-2 d (a+b) \int \frac {x^{4/3}}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-x^4+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt [3]{x}+2 (2 a b+d) \int \frac {x^{7/3}}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-x^4+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt [3]{x}-3 (a+b) \int \frac {x^{10/3}}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-x^4+d x^2-a \left (\frac {b}{a}+1\right ) d x+a b d\right )}d\sqrt [3]{x}-\frac {x^{2/3} \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {2}{3},\frac {2}{3},\frac {5}{3},\frac {2 x}{a+b+\sqrt {a^2-2 b a+b^2}},\frac {2 x}{a+b-\sqrt {a^2-2 b a+b^2}}\right )}{\left (-x (a+b)+a b+x^2\right )^{2/3}}\right )}{\left (x^2 (a-x) (b-x)\right )^{2/3}}\)

input

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

output

$Aborted
 

3.30.52.3.1 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] 
]
 

3.30.52.4 Maple [F]

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

input

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

output

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

3.30.52.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.30.52.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.30.52.7 Maxima [F]

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

input

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

3.30.52.8 Giac [F]

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

input

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

3.30.52.9 Mupad [F(-1)]

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

input

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

output

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