3.29.72 \(\int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} (-a^2 d+(b^2+2 a d) x-(2 b+d) x^2+x^3)} \, dx\) [2872]

3.29.72.1 Optimal result

Integrand size = 77, antiderivative size = 306 \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2 d+\left (b^2+2 a d\right ) x-(2 b+d) x^2+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

output

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

3.29.72.2 Mathematica [A] (verified)

Time = 10.95 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.70 \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2 d+\left (b^2+2 a d\right ) x-(2 b+d) x^2+x^3\right )} \, dx=-\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x (-a+x) (-b+x)^2}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)^2}}\right )-2 \log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)^2}\right )+\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2-a \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)^2}+\left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

input

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

output

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

3.29.72.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.29.72.3.1 Defintions of rubi rules used

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] 
]
 

3.29.72.4 Maple [F]

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

input

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

output

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

3.29.72.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.29.72.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.29.72.7 Maxima [F]

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

input

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

output

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

3.29.72.8 Giac [F]

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

input

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

output

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

3.29.72.9 Mupad [F(-1)]

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

input

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

output

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