3.28.88 \(\int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3)} \, dx\) [2788]

3.28.88.1 Optimal result

Integrand size = 76, antiderivative size = 269 \[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{2 x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

output

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

3.28.88.2 Mathematica [A] (verified)

Time = 13.98 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.61 \[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}}{2 x+\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}}\right )+2 \log \left (x-\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}\right )-\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)^2}+d^{2/3} \left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{2 d^{2/3}} \]

input

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

output

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

3.28.88.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.28.88.3.1 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] 
]
 

3.28.88.4 Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.55

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x +2 \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 \left (\frac {1}{d}\right )^{\frac {1}{3}} d}\)	\(149\)

input

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

output

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

3.28.88.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.28.88.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.28.88.7 Maxima [F]

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

input

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

output

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

3.28.88.8 Giac [F]

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

input

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

output

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

3.28.88.9 Mupad [F(-1)]

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

input

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

output

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