3.27.5 \(\int \frac {2 a b x-3 a x^2+x^3}{\sqrt [3]{x^2 (-a+x) (-b+x)} (-a^2+2 a x-(1+b d) x^2+d x^3)} \, dx\) [2605]

3.27.5.1 Optimal result

Integrand size = 61, antiderivative size = 226 \[ \int \frac {2 a b x-3 a x^2+x^3}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

output

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

3.27.5.2 Mathematica [A] (verified)

Time = 13.34 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.80 \[ \int \frac {2 a b x-3 a x^2+x^3}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}}\right )+2 \log \left (-a+x-\sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}\right )-\log \left (a^2-2 a x+x^2+\sqrt [3]{d} (-a+x) \sqrt [3]{x^2 (-a+x) (-b+x)}+d^{2/3} \left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 d^{2/3}} \]

input

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

output

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

3.27.5.3 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.

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

\(\Big \downarrow \) 2028

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.27.5.3.1 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 2028

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), 
x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ 
{a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] &&  !(E 
qQ[p, 1] && EqQ[u, 1])
 

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] 
]
 

3.27.5.4 Maple [F]

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

input

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

output

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

3.27.5.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.27.5.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.27.5.7 Maxima [F]

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

input

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

output

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

3.27.5.8 Giac [F]

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

input

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

output

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

3.27.5.9 Mupad [F(-1)]

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

input

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

output

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