3.25.90 \(\int \frac {-2 a^2 b x+a (3 a+2 b) x^2-4 a x^3+x^4}{(x^2 (-a+x) (-b+x))^{2/3} (-a^2+2 a x-(1+b d) x^2+d x^3)} \, dx\) [2490]

3.25.90.1 Optimal result

Integrand size = 75, antiderivative size = 206 \[ \int \frac {-2 a^2 b x+a (3 a+2 b) x^2-4 a x^3+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a^2+2 a x-(1+b d) x^2+d x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a-\sqrt {3} x}{a-x-2 \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

output

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

3.25.90.2 Mathematica [F]

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

input

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

output

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

3.25.90.3 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.

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

\(\Big \downarrow \) 2029

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.25.90.3.1 Defintions of rubi rules used

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

rule 2009

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

rule 2029

Int[(Fx_.)*((d_.)*(x_)^(q_.) + (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) 
+ d*x^(q - r))^p*Fx, x] /; FreeQ[{a, b, c, d, r, s, t, q}, x] && IntegerQ[p 
] && PosQ[s - r] && PosQ[t - r] && PosQ[q - r] &&  !(EqQ[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 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] 
]
 

3.25.90.4 Maple [F]

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

input

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

output

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

3.25.90.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.25.90.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.25.90.7 Maxima [F]

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

input

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

output

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

3.25.90.8 Giac [F]

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

input

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

output

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

3.25.90.9 Mupad [F(-1)]

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

input

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

output

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