\(\int \frac {1}{\sqrt [3]{-3 x+x^2}} \, dx\) [326]

Optimal result

Integrand size = 11, antiderivative size = 688 \[ \int \frac {1}{\sqrt [3]{-3 x+x^2}} \, dx=\frac {3 \sqrt [3]{\frac {3}{2}} (3-2 x) \sqrt [3]{3 x-x^2}}{\left (3 \left (1-\sqrt {3}\right )-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}\right ) \sqrt [3]{-3 x+x^2}}+\frac {3 \sqrt [3]{\frac {3}{2}} \sqrt {2+\sqrt {3}} \left (3^{3/4}-\sqrt [12]{3} \sqrt [3]{9-(3-2 x)^2}\right ) \sqrt {\frac {9+3 \sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}+3^{2/3} \left (9-(3-2 x)^2\right )^{2/3}}{\left (3 \left (1-\sqrt {3}\right )-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}\right )^2}} \sqrt [3]{3 x-x^2} E\left (\arcsin \left (\frac {3 \left (1+\sqrt {3}\right )-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}}{3 \left (1-\sqrt {3}\right )-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {3-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}}{\left (3 \left (1-\sqrt {3}\right )-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}\right )^2}} (3-2 x) \sqrt [3]{-3 x+x^2}}-\frac {\sqrt [6]{2} \left (3\ 3^{7/12}-3^{11/12} \sqrt [3]{9-(3-2 x)^2}\right ) \sqrt {\frac {9+3 \sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}+3^{2/3} \left (9-(3-2 x)^2\right )^{2/3}}{\left (3 \left (1-\sqrt {3}\right )-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}\right )^2}} \sqrt [3]{3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {3 \left (1+\sqrt {3}\right )-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}}{3 \left (1-\sqrt {3}\right )-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {3-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}}{\left (3 \left (1-\sqrt {3}\right )-\sqrt [3]{3} \sqrt [3]{9-(3-2 x)^2}\right )^2}} (3-2 x) \sqrt [3]{-3 x+x^2}} \] Output:

3/2*3^(1/3)*2^(2/3)*(3-2*x)*(-x^2+3*x)^(1/3)/(3-3*3^(1/2)-3^(1/3)*(9-(3-2* 
x)^2)^(1/3))/(x^2-3*x)^(1/3)+3/4*3^(1/3)*2^(2/3)*(1/2*6^(1/2)+1/2*2^(1/2)) 
*(3^(3/4)-3^(1/12)*(9-(3-2*x)^2)^(1/3))*((9+3*3^(1/3)*(9-(3-2*x)^2)^(1/3)+ 
3^(2/3)*(9-(3-2*x)^2)^(2/3))/(3-3*3^(1/2)-3^(1/3)*(9-(3-2*x)^2)^(1/3))^2)^ 
(1/2)*(-x^2+3*x)^(1/3)*EllipticE((3+3*3^(1/2)-3^(1/3)*(9-(3-2*x)^2)^(1/3)) 
/(3-3*3^(1/2)-3^(1/3)*(9-(3-2*x)^2)^(1/3)),2*I-I*3^(1/2))/(-(3-3^(1/3)*(9- 
(3-2*x)^2)^(1/3))/(3-3*3^(1/2)-3^(1/3)*(9-(3-2*x)^2)^(1/3))^2)^(1/2)/(3-2* 
x)/(x^2-3*x)^(1/3)-2^(1/6)*(3*3^(7/12)-3^(11/12)*(9-(3-2*x)^2)^(1/3))*((9+ 
3*3^(1/3)*(9-(3-2*x)^2)^(1/3)+3^(2/3)*(9-(3-2*x)^2)^(2/3))/(3-3*3^(1/2)-3^ 
(1/3)*(9-(3-2*x)^2)^(1/3))^2)^(1/2)*(-x^2+3*x)^(1/3)*EllipticF((3+3*3^(1/2 
)-3^(1/3)*(9-(3-2*x)^2)^(1/3))/(3-3*3^(1/2)-3^(1/3)*(9-(3-2*x)^2)^(1/3)),2 
*I-I*3^(1/2))/(-(3-3^(1/3)*(9-(3-2*x)^2)^(1/3))/(3-3*3^(1/2)-3^(1/3)*(9-(3 
-2*x)^2)^(1/3))^2)^(1/2)/(3-2*x)/(x^2-3*x)^(1/3)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.06 \[ \int \frac {1}{\sqrt [3]{-3 x+x^2}} \, dx=\frac {3^{2/3} ((-3+x) x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},1-\frac {x}{3}\right )}{2 x^{2/3}} \] Input:

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

Output:

(3^(2/3)*((-3 + x)*x)^(2/3)*Hypergeometric2F1[1/3, 2/3, 5/3, 1 - x/3])/(2* 
x^(2/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.71 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.66, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1093, 1090, 233, 833, 760, 2418}

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.

\(\displaystyle \int \frac {1}{\sqrt [3]{x^2-3 x}} \, dx\)

\(\Big \downarrow \) 1093

\(\displaystyle \frac {\sqrt [3]{3 x-x^2} \int \frac {1}{\sqrt [3]{\frac {x}{3}-\frac {x^2}{9}}}dx}{3^{2/3} \sqrt [3]{x^2-3 x}}\)

\(\Big \downarrow \) 1090

\(\displaystyle -\frac {3 \sqrt [3]{\frac {3}{2}} \sqrt [3]{3 x-x^2} \int \frac {1}{\sqrt [3]{1-9 \left (\frac {1}{3}-\frac {2 x}{9}\right )^2}}d\left (\frac {1}{3}-\frac {2 x}{9}\right )}{\sqrt [3]{x^2-3 x}}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3 \sqrt [3]{\frac {3}{2}} \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2} \sqrt [3]{3 x-x^2} \int \frac {\sqrt [3]{1-9 \left (\frac {1}{3}-\frac {2 x}{9}\right )^2}}{3 \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2}}d\sqrt [3]{1-9 \left (\frac {1}{3}-\frac {2 x}{9}\right )^2}}{2 \left (\frac {1}{3}-\frac {2 x}{9}\right ) \sqrt [3]{x^2-3 x}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3 \sqrt [3]{\frac {3}{2}} \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2} \sqrt [3]{3 x-x^2} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{3 \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2}}d\sqrt [3]{1-9 \left (\frac {1}{3}-\frac {2 x}{9}\right )^2}-\int \frac {\frac {2 x}{9}+\sqrt {3}+\frac {2}{3}}{3 \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2}}d\sqrt [3]{1-9 \left (\frac {1}{3}-\frac {2 x}{9}\right )^2}\right )}{2 \left (\frac {1}{3}-\frac {2 x}{9}\right ) \sqrt [3]{x^2-3 x}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 \sqrt [3]{\frac {3}{2}} \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2} \sqrt [3]{3 x-x^2} \left (-\int \frac {\frac {2 x}{9}+\sqrt {3}+\frac {2}{3}}{3 \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2}}d\sqrt [3]{1-9 \left (\frac {1}{3}-\frac {2 x}{9}\right )^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (\frac {2 x}{9}+\frac {2}{3}\right ) \sqrt {\frac {-\frac {2 x}{9}+\left (1-9 \left (\frac {1}{3}-\frac {2 x}{9}\right )^2\right )^{2/3}+\frac {4}{3}}{\left (\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {2 x}{9}+\sqrt {3}+\frac {2}{3}}{\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2} \sqrt {-\frac {\frac {2 x}{9}+\frac {2}{3}}{\left (\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}\right )^2}}}\right )}{2 \left (\frac {1}{3}-\frac {2 x}{9}\right ) \sqrt [3]{x^2-3 x}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3 \sqrt [3]{\frac {3}{2}} \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2} \sqrt [3]{3 x-x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (\frac {2 x}{9}+\frac {2}{3}\right ) \sqrt {\frac {-\frac {2 x}{9}+\left (1-9 \left (\frac {1}{3}-\frac {2 x}{9}\right )^2\right )^{2/3}+\frac {4}{3}}{\left (\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {2 x}{9}+\sqrt {3}+\frac {2}{3}}{\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2} \sqrt {-\frac {\frac {2 x}{9}+\frac {2}{3}}{\left (\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}\right )^2}}}+\frac {\sqrt {2+\sqrt {3}} \left (\frac {2 x}{9}+\frac {2}{3}\right ) \sqrt {\frac {-\frac {2 x}{9}+\left (1-9 \left (\frac {1}{3}-\frac {2 x}{9}\right )^2\right )^{2/3}+\frac {4}{3}}{\left (\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}\right )^2}} E\left (\arcsin \left (\frac {\frac {2 x}{9}+\sqrt {3}+\frac {2}{3}}{\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2} \sqrt {-\frac {\frac {2 x}{9}+\frac {2}{3}}{\left (\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}\right )^2}}}-\frac {6 \sqrt {-\left (\frac {1}{3}-\frac {2 x}{9}\right )^2}}{\frac {2 x}{9}-\sqrt {3}+\frac {2}{3}}\right )}{2 \left (\frac {1}{3}-\frac {2 x}{9}\right ) \sqrt [3]{x^2-3 x}}\)

Input:

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

Output:

(3*(3/2)^(1/3)*Sqrt[-(1/3 - (2*x)/9)^2]*(3*x - x^2)^(1/3)*((-6*Sqrt[-(1/3 
- (2*x)/9)^2])/(2/3 - Sqrt[3] + (2*x)/9) + (Sqrt[2 + Sqrt[3]]*(2/3 + (2*x) 
/9)*Sqrt[(4/3 + (1 - 9*(1/3 - (2*x)/9)^2)^(2/3) - (2*x)/9)/(2/3 - Sqrt[3] 
+ (2*x)/9)^2]*EllipticE[ArcSin[(2/3 + Sqrt[3] + (2*x)/9)/(2/3 - Sqrt[3] + 
(2*x)/9)], -7 + 4*Sqrt[3]])/(3^(3/4)*Sqrt[-(1/3 - (2*x)/9)^2]*Sqrt[-((2/3 
+ (2*x)/9)/(2/3 - Sqrt[3] + (2*x)/9)^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt 
[3])*(2/3 + (2*x)/9)*Sqrt[(4/3 + (1 - 9*(1/3 - (2*x)/9)^2)^(2/3) - (2*x)/9 
)/(2/3 - Sqrt[3] + (2*x)/9)^2]*EllipticF[ArcSin[(2/3 + Sqrt[3] + (2*x)/9)/ 
(2/3 - Sqrt[3] + (2*x)/9)], -7 + 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[-(1/3 - (2*x) 
/9)^2]*Sqrt[-((2/3 + (2*x)/9)/(2/3 - Sqrt[3] + (2*x)/9)^2)])))/(2*(1/3 - ( 
2*x)/9)*(-3*x + x^2)^(1/3))
 

Defintions of rubi rules used

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, 
b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
 

rule 1093

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b*x + c*x^2)^p/((- 
c)*((b*x + c*x^2)/b^2))^p   Int[((-c)*(x/b) - c^2*(x^2/b^2))^p, x], x] /; F 
reeQ[{b, c}, x] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05

method	result	size
meijerg	\(\frac {3^{\frac {2}{3}} \left (-\operatorname {signum}\left (-3+x \right )\right )^{\frac {1}{3}} x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], \frac {x}{3}\right )}{2 \operatorname {signum}\left (-3+x \right )^{\frac {1}{3}}}\)	\(32\)

Input:

int(1/(x^2-3*x)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

1/2*3^(2/3)/signum(-3+x)^(1/3)*(-signum(-3+x))^(1/3)*x^(2/3)*hypergeom([1/ 
3,2/3],[5/3],1/3*x)
 

Fricas [F]

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

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

Output:

integral((x^2 - 3*x)^(-1/3), x)
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{-3 x+x^2}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} - 3 x}}\, dx \] Input:

integrate(1/(x**2-3*x)**(1/3),x)
 

Output:

Integral((x**2 - 3*x)**(-1/3), x)
 

Maxima [F]

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

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

Output:

integrate((x^2 - 3*x)^(-1/3), x)
 

Giac [F]

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

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

Output:

integrate((x^2 - 3*x)^(-1/3), x)
 

Mupad [B] (verification not implemented)

Time = 23.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.04 \[ \int \frac {1}{\sqrt [3]{-3 x+x^2}} \, dx=\frac {3^{2/3}\,x\,{\left (3-x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ \frac {x}{3}\right )}{2\,{\left (x^2-3\,x\right )}^{1/3}} \] Input:

int(1/(x^2 - 3*x)^(1/3),x)
 

Output:

(3^(2/3)*x*(3 - x)^(1/3)*hypergeom([1/3, 2/3], 5/3, x/3))/(2*(x^2 - 3*x)^( 
1/3))
 

Reduce [F]

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

int(1/(x^2-3*x)^(1/3),x)
 

Output:

int(1/(x**(1/3)*(x - 3)**(1/3)),x)