3.15.39 \(\int x^3 (1+x^3)^{2/3} \, dx\)

Optimal. Leaf size=102 \[ \frac {1}{27} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{9 \sqrt {3}}+\frac {1}{18} \left (x^3+1\right )^{2/3} \left (3 x^4+2 x\right )-\frac {1}{54} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 79, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {279, 321, 239} \begin {gather*} \frac {1}{9} \left (x^3+1\right )^{2/3} x+\frac {1}{18} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{6} \left (x^3+1\right )^{2/3} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(1 + x^3)^(2/3),x]

[Out]

(x*(1 + x^3)^(2/3))/9 + (x^4*(1 + x^3)^(2/3))/6 - ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) + Lo
g[-x + (1 + x^3)^(1/3)]/18

Rule 239

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

Rule 279

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 321

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]

Rubi steps

\begin {align*} \int x^3 \left (1+x^3\right )^{2/3} \, dx &=\frac {1}{6} x^4 \left (1+x^3\right )^{2/3}+\frac {1}{3} \int \frac {x^3}{\sqrt [3]{1+x^3}} \, dx\\ &=\frac {1}{9} x \left (1+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (1+x^3\right )^{2/3}-\frac {1}{9} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=\frac {1}{9} x \left (1+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (1+x^3\right )^{2/3}-\frac {\tan ^{-1}\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*}

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Mathematica [C]  time = 0.01, size = 32, normalized size = 0.31 \begin {gather*} \frac {1}{6} x \left (\left (x^3+1\right )^{5/3}-\, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};-x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(1 + x^3)^(2/3),x]

[Out]

(x*((1 + x^3)^(5/3) - Hypergeometric2F1[-2/3, 1/3, 4/3, -x^3]))/6

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IntegrateAlgebraic [A]  time = 0.21, size = 102, normalized size = 1.00 \begin {gather*} \frac {1}{18} \left (1+x^3\right )^{2/3} \left (2 x+3 x^4\right )-\frac {\tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x^3)^(2/3)*(2*x + 3*x^4))/18 - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/(9*Sqrt[3]) + Log[-x + (1 + x
^3)^(1/3)]/27 - Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/54

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fricas [A]  time = 0.50, size = 94, normalized size = 0.92 \begin {gather*} \frac {1}{18} \, {\left (3 \, x^{4} + 2 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + \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}{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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(3*x^4 + 2*x)*(x^3 + 1)^(2/3) + 1/27*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + 1)^(1/3))/x) + 1/27
*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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} + 1\right )}^{\frac {2}{3}} x^{3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^3 + 1)^(2/3)*x^3, x)

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maple [C]  time = 1.92, size = 17, normalized size = 0.17

method	result	size
meijerg	\(\frac {x^{4} \hypergeom \left (\left [-\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -x^{3}\right )}{4}\)	\(17\)
risch	\(\frac {x \left (3 x^{3}+2\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{18}-\frac {x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )}{9}\)	\(33\)
trager	\(\frac {x \left (3 x^{3}+2\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{18}+\frac {\ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-2 x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{27}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}-x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{27}\)	\(184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*hypergeom([-2/3,4/3],[7/3],-x^3)

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maxima [A]  time = 0.41, size = 121, normalized size = 1.19 \begin {gather*} \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 {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + \frac {2 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}}}{x^{5}}}{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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\left (x^3+1\right )}^{2/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^3 + 1)^(2/3),x)

[Out]

int(x^3*(x^3 + 1)^(2/3), x)

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sympy [C]  time = 1.00, size = 31, normalized size = 0.30 \begin {gather*} \frac {x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4*gamma(4/3)*hyper((-2/3, 4/3), (7/3,), x**3*exp_polar(I*pi))/(3*gamma(7/3))

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