\(\int \frac {\sqrt [3]{-x^2+x^3}}{x} \, dx\) [1529]

Optimal result

Integrand size = 17, antiderivative size = 106 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{x} \, dx=\sqrt [3]{-x^2+x^3}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-x^2+x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \] Output:

(x^3-x^2)^(1/3)+1/3*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^3-x^2)^(1/3)))+1/3*ln 
(-x+(x^3-x^2)^(1/3))-1/6*ln(x^2+x*(x^3-x^2)^(1/3)+(x^3-x^2)^(2/3))
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{x} \, dx=\frac {(-1+x)^{2/3} x^{4/3} \left (6 \sqrt [3]{-1+x} x^{2/3}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+2 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )-\log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )\right )}{6 \left ((-1+x) x^2\right )^{2/3}} \] Input:

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

Output:

((-1 + x)^(2/3)*x^(4/3)*(6*(-1 + x)^(1/3)*x^(2/3) + 2*Sqrt[3]*ArcTan[(Sqrt 
[3]*x^(1/3))/(2*(-1 + x)^(1/3) + x^(1/3))] + 2*Log[(-1 + x)^(1/3) - x^(1/3 
)] - Log[(-1 + x)^(2/3) + (-1 + x)^(1/3)*x^(1/3) + x^(2/3)]))/(6*((-1 + x) 
*x^2)^(2/3))
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1927, 1938, 71}

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.

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

\(\Big \downarrow \) 1927

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

\(\Big \downarrow \) 1938

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

\(\Big \downarrow \) 71

\(\displaystyle \sqrt [3]{x^3-x^2}-\frac {(x-1)^{2/3} x^{4/3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )-\frac {1}{2} \log (x-1)\right )}{3 \left (x^3-x^2\right )^{2/3}}\)

Input:

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

Output:

(-x^2 + x^3)^(1/3) - ((-1 + x)^(2/3)*x^(4/3)*(-(Sqrt[3]*ArcTan[1/Sqrt[3] + 
 (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))]) - (3*Log[-1 + x^(1/3)/(-1 + x)^(1/ 
3)])/2 - Log[-1 + x]/2))/(3*(-x^2 + x^3)^(2/3))
 

Defintions of rubi rules used

rule 71

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> 
 With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( 
Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + 
 b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
 

rule 1927

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a* 
(n - j)*(p/(c^j*(m + n*p + 1)))   Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1) 
, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (Int 
egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.69 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.25

method	result	size
meijerg	\(\frac {3 \operatorname {signum}\left (-1+x \right )^{\frac {1}{3}} x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x\right )}{2 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}}}\)	\(27\)
pseudoelliptic	\(\frac {x^{2} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-6 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-2 \ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right )\right )}{6 \left (\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}\right ) \left (\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x \right )}\)	\(135\)
trager	\(\left (x^{3}-x^{2}\right )^{\frac {1}{3}}+\frac {\ln \left (-\frac {36 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{2}-72 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x -144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+90 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +60 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-78 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x -15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+25 x^{2}-15 x}{x}\right )}{3}+2 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \ln \left (-\frac {144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{2}-288 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x +144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-54 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -66 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-36 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x +9 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-20 x^{2}+5 x}{x}\right )\)	\(338\)
risch	\(\text {Expression too large to display}\)	\(638\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} + \frac {1}{3} \, \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \] Input:

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

Output:

-1/3*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - x^2)^(1/3))/x) + (x^ 
3 - x^2)^(1/3) + 1/3*log(-(x - (x^3 - x^2)^(1/3))/x) - 1/6*log((x^2 + (x^3 
 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + x {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \] Input:

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

Output:

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x + 1)^(1/3) + 1)) + x*(-1/x + 1)^( 
1/3) - 1/6*log((-1/x + 1)^(2/3) + (-1/x + 1)^(1/3) + 1) + 1/3*log(abs((-1/ 
x + 1)^(1/3) - 1))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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