\(\int x^4 \sqrt [3]{1+x^3} \, dx\) [1438]

Optimal result

Integrand size = 13, antiderivative size = 102 \[ \int x^4 \sqrt [3]{1+x^3} \, dx=\frac {1}{18} \sqrt [3]{1+x^3} \left (x^2+3 x^5\right )+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{9 \sqrt {3}}+\frac {1}{27} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{54} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]

[Out]

1/18*(x^3+1)^(1/3)*(3*x^5+x^2)+1/27*arctan(3^(1/2)*x/(x+2*(x^3+1)^(1/3)))*3^(1/2)+1/27*ln(-x+(x^3+1)^(1/3))-1/
54*ln(x^2+x*(x^3+1)^(1/3)+(x^3+1)^(2/3))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {285, 327, 337} \[ \int x^4 \sqrt [3]{1+x^3} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{18} \log \left (x-\sqrt [3]{x^3+1}\right )+\frac {1}{6} \sqrt [3]{x^3+1} x^5+\frac {1}{18} \sqrt [3]{x^3+1} x^2 \]

[In]

Int[x^4*(1 + x^3)^(1/3),x]

[Out]

(x^2*(1 + x^3)^(1/3))/18 + (x^5*(1 + x^3)^(1/3))/6 + ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) +
Log[x - (1 + x^3)^(1/3)]/18

Rule 285

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] + Dist[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]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

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] - Dist[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 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^5 \sqrt [3]{1+x^3}+\frac {1}{6} \int \frac {x^4}{\left (1+x^3\right )^{2/3}} \, dx \\ & = \frac {1}{18} x^2 \sqrt [3]{1+x^3}+\frac {1}{6} x^5 \sqrt [3]{1+x^3}-\frac {1}{9} \int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx \\ & = \frac {1}{18} x^2 \sqrt [3]{1+x^3}+\frac {1}{6} x^5 \sqrt [3]{1+x^3}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{18} \log \left (x-\sqrt [3]{1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.96 \[ \int x^4 \sqrt [3]{1+x^3} \, dx=\frac {1}{54} \left (3 \sqrt [3]{1+x^3} \left (x^2+3 x^5\right )+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )+2 \log \left (-x+\sqrt [3]{1+x^3}\right )-\log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[x^4*(1 + x^3)^(1/3),x]

[Out]

(3*(1 + x^3)^(1/3)*(x^2 + 3*x^5) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))] + 2*Log[-x + (1 + x^3
)^(1/3)] - Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/54

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 1.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.17

method	result	size
meijerg	\(\frac {x^{5} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x^{3}\right )}{5}\)	\(17\)
risch	\(\frac {x^{2} \left (3 x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{18}-\frac {x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{18}\)	\(37\)
pseudoelliptic	\(\frac {9 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )-\ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{54 \left (x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right )^{2} {\left (-x +\left (x^{3}+1\right )^{\frac {1}{3}}\right )}^{2}}\)	\(133\)
trager	\(\frac {x^{2} \left (3 x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{18}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{27}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+4 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{27}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+4 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2\right )}{27}\)	\(313\)

[In]

int(x^4*(x^3+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/5*x^5*hypergeom([-1/3,5/3],[8/3],-x^3)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92 \[ \int x^4 \sqrt [3]{1+x^3} \, dx=-\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{18} \, {\left (3 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{27} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{54} \, \log \left (\frac {x^{2} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

integrate(x^4*(x^3+1)^(1/3),x, algorithm="fricas")

[Out]

-1/27*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + 1)^(1/3))/x) + 1/18*(3*x^5 + x^2)*(x^3 + 1)^(1/3) + 1/2
7*log(-(x - (x^3 + 1)^(1/3))/x) - 1/54*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 = 1.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.30 \[ \int x^4 \sqrt [3]{1+x^3} \, dx=\frac {x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} \]

[In]

integrate(x**4*(x**3+1)**(1/3),x)

[Out]

x**5*gamma(5/3)*hyper((-1/3, 5/3), (8/3,), x**3*exp_polar(I*pi))/(3*gamma(8/3))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.19 \[ \int x^4 \sqrt [3]{1+x^3} \, dx=-\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{x^{4}}}{18 \, {\left (\frac {2 \, {\left (x^{3} + 1\right )}}{x^{3}} - \frac {{\left (x^{3} + 1\right )}^{2}}{x^{6}} - 1\right )}} - \frac {1}{54} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {1}{27} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]

[In]

integrate(x^4*(x^3+1)^(1/3),x, algorithm="maxima")

[Out]

-1/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3)/x + 1)) - 1/18*(2*(x^3 + 1)^(1/3)/x + (x^3 + 1)^(4/3)/x^4)
/(2*(x^3 + 1)/x^3 - (x^3 + 1)^2/x^6 - 1) - 1/54*log((x^3 + 1)^(1/3)/x + (x^3 + 1)^(2/3)/x^2 + 1) + 1/27*log((x
^3 + 1)^(1/3)/x - 1)

Giac [F]

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

[In]

integrate(x^4*(x^3+1)^(1/3),x, algorithm="giac")

[Out]

integrate((x^3 + 1)^(1/3)*x^4, x)

Mupad [F(-1)]

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

[In]

int(x^4*(x^3 + 1)^(1/3),x)

[Out]

int(x^4*(x^3 + 1)^(1/3), x)