\(\int x^7 \sqrt [3]{-1+x^3} \, dx\) [1585]

Optimal result

Integrand size = 13, antiderivative size = 109 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\frac {1}{162} \sqrt [3]{-1+x^3} \left (-5 x^2-3 x^5+18 x^8\right )+\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {5}{486} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Output:

1/162*(x^3-1)^(1/3)*(18*x^8-3*x^5-5*x^2)+5/243*arctan(3^(1/2)*x/(x+2*(x^3- 
1)^(1/3)))*3^(1/2)+5/243*ln(-x+(x^3-1)^(1/3))-5/486*ln(x^2+x*(x^3-1)^(1/3) 
+(x^3-1)^(2/3))
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\frac {1}{486} \left (3 x^2 \sqrt [3]{-1+x^3} \left (-5-3 x^3+18 x^6\right )+10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+10 \log \left (-x+\sqrt [3]{-1+x^3}\right )-5 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \] Input:

Integrate[x^7*(-1 + x^3)^(1/3),x]
 

Output:

(3*x^2*(-1 + x^3)^(1/3)*(-5 - 3*x^3 + 18*x^6) + 10*Sqrt[3]*ArcTan[(Sqrt[3] 
*x)/(x + 2*(-1 + x^3)^(1/3))] + 10*Log[-x + (-1 + x^3)^(1/3)] - 5*Log[x^2 
+ x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/486
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {811, 843, 843, 853}

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.

\(\displaystyle \int x^7 \sqrt [3]{x^3-1} \, dx\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {1}{9} x^8 \sqrt [3]{x^3-1}-\frac {1}{9} \int \frac {x^7}{\left (x^3-1\right )^{2/3}}dx\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{9} \left (-\frac {5}{6} \int \frac {x^4}{\left (x^3-1\right )^{2/3}}dx-\frac {1}{6} \sqrt [3]{x^3-1} x^5\right )+\frac {1}{9} \sqrt [3]{x^3-1} x^8\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{9} \left (-\frac {5}{6} \left (\frac {2}{3} \int \frac {x}{\left (x^3-1\right )^{2/3}}dx+\frac {1}{3} \sqrt [3]{x^3-1} x^2\right )-\frac {1}{6} \sqrt [3]{x^3-1} x^5\right )+\frac {1}{9} \sqrt [3]{x^3-1} x^8\)

\(\Big \downarrow \) 853

\(\displaystyle \frac {1}{9} \left (-\frac {5}{6} \left (\frac {2}{3} \left (-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )\right )+\frac {1}{3} \sqrt [3]{x^3-1} x^2\right )-\frac {1}{6} \sqrt [3]{x^3-1} x^5\right )+\frac {1}{9} \sqrt [3]{x^3-1} x^8\)

Input:

Int[x^7*(-1 + x^3)^(1/3),x]
 

Output:

(x^8*(-1 + x^3)^(1/3))/9 + (-1/6*(x^5*(-1 + x^3)^(1/3)) - (5*((x^2*(-1 + x 
^3)^(1/3))/3 + (2*(-(ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3]) 
 - Log[x - (-1 + x^3)^(1/3)]/2))/3))/6)/9
 

Defintions of rubi rules used

rule 811

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

rule 843

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

rule 853

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

Maple [C] (warning: unable to verify)

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

Time = 1.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30

method	result	size
meijerg	\(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{8} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], x^{3}\right )}{8 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\)	\(33\)
risch	\(\frac {x^{2} \left (18 x^{6}-3 x^{3}-5\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{162}-\frac {5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{162 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(58\)
pseudoelliptic	\(\frac {-5 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )+10 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\left (54 x^{8}-9 x^{5}-15 x^{2}\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{486 {\left (-\left (x^{3}-1\right )^{\frac {1}{3}}+x \right )}^{3} \left (x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )^{3}}\)	\(133\)
trager	\(\frac {x^{2} \left (18 x^{6}-3 x^{3}-5\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{162}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{243}-\frac {5 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{243}-\frac {5 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{243}\)	\(320\)

Input:

int(x^7*(x^3-1)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

1/8*signum(x^3-1)^(1/3)/(-signum(x^3-1))^(1/3)*x^8*hypergeom([-1/3,8/3],[1 
1/3],x^3)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=-\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{162} \, {\left (18 \, x^{8} - 3 \, x^{5} - 5 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \frac {5}{243} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{486} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \] Input:

integrate(x^7*(x^3-1)^(1/3),x, algorithm="fricas")
 

Output:

-5/243*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - 1)^(1/3))/x) + 1/1 
62*(18*x^8 - 3*x^5 - 5*x^2)*(x^3 - 1)^(1/3) + 5/243*log(-(x - (x^3 - 1)^(1 
/3))/x) - 5/486*log((x^2 + (x^3 - 1)^(1/3)*x + (x^3 - 1)^(2/3))/x^2)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.65 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.33 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=- \frac {x^{8} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \] Input:

integrate(x**7*(x**3-1)**(1/3),x)
 

Output:

-x**8*exp(-2*I*pi/3)*gamma(8/3)*hyper((-1/3, 8/3), (11/3,), x**3)/(3*gamma 
(11/3))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.33 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=-\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}}}{x^{4}} - \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}}}{x^{7}}}{162 \, {\left (\frac {3 \, {\left (x^{3} - 1\right )}}{x^{3}} - \frac {3 \, {\left (x^{3} - 1\right )}^{2}}{x^{6}} + \frac {{\left (x^{3} - 1\right )}^{3}}{x^{9}} - 1\right )}} - \frac {5}{486} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {5}{243} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \] Input:

integrate(x^7*(x^3-1)^(1/3),x, algorithm="maxima")
 

Output:

-5/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3)/x + 1)) - 1/162*(10*( 
x^3 - 1)^(1/3)/x + 13*(x^3 - 1)^(4/3)/x^4 - 5*(x^3 - 1)^(7/3)/x^7)/(3*(x^3 
 - 1)/x^3 - 3*(x^3 - 1)^2/x^6 + (x^3 - 1)^3/x^9 - 1) - 5/486*log((x^3 - 1) 
^(1/3)/x + (x^3 - 1)^(2/3)/x^2 + 1) + 5/243*log((x^3 - 1)^(1/3)/x - 1)
 

Giac [F]

\[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\int { {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{7} \,d x } \] Input:

integrate(x^7*(x^3-1)^(1/3),x, algorithm="giac")
 

Output:

integrate((x^3 - 1)^(1/3)*x^7, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\int x^7\,{\left (x^3-1\right )}^{1/3} \,d x \] Input:

int(x^7*(x^3 - 1)^(1/3),x)
 

Output:

int(x^7*(x^3 - 1)^(1/3), x)
 

Reduce [F]

\[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\frac {\left (x^{3}-1\right )^{\frac {1}{3}} x^{8}}{9}-\frac {\left (x^{3}-1\right )^{\frac {1}{3}} x^{5}}{54}-\frac {5 \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}}{162}-\frac {5 \left (\int \frac {x}{\left (x^{3}-1\right )^{\frac {2}{3}}}d x \right )}{81} \] Input:

int(x^7*(x^3-1)^(1/3),x)
 

Output:

(18*(x**3 - 1)**(1/3)*x**8 - 3*(x**3 - 1)**(1/3)*x**5 - 5*(x**3 - 1)**(1/3 
)*x**2 - 10*int(((x**3 - 1)**(1/3)*x)/(x**3 - 1),x))/162