3.17.9 \(\int \frac {(1-x^3)^{2/3} (-1+x^3)}{x^6 (-1+2 x^3)} \, dx\)

Optimal. Leaf size=110 \[ -\frac {1}{3} \log \left (\sqrt [3]{1-x^3}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{1-x^3}+x}\right )}{\sqrt {3}}+\frac {\left (1-x^3\right )^{2/3} \left (-3 x^3-2\right )}{10 x^5}+\frac {1}{6} \log \left (\sqrt [3]{1-x^3} x+\left (1-x^3\right )^{2/3}+x^2\right ) \]

________________________________________________________________________________________

Rubi [C]  time = 0.02, antiderivative size = 42, normalized size of antiderivative = 0.38, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {21, 510} \begin {gather*} -\frac {\left (1-2 x^3\right )^{5/3} \, _2F_1\left (-\frac {5}{3},-\frac {5}{3};-\frac {2}{3};-\frac {x^3}{1-2 x^3}\right )}{5 x^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((1 - x^3)^(2/3)*(-1 + x^3))/(x^6*(-1 + 2*x^3)),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx &=-\int \frac {\left (1-x^3\right )^{5/3}}{x^6 \left (-1+2 x^3\right )} \, dx\\ &=-\frac {\left (1-2 x^3\right )^{5/3} \, _2F_1\left (-\frac {5}{3},-\frac {5}{3};-\frac {2}{3};-\frac {x^3}{1-2 x^3}\right )}{5 x^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.02, size = 42, normalized size = 0.38 \begin {gather*} -\frac {\left (1-2 x^3\right )^{5/3} \, _2F_1\left (-\frac {5}{3},-\frac {5}{3};-\frac {2}{3};-\frac {x^3}{1-2 x^3}\right )}{5 x^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((1 - x^3)^(2/3)*(-1 + x^3))/(x^6*(-1 + 2*x^3)),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.20, size = 110, normalized size = 1.00 \begin {gather*} \frac {\left (-2-3 x^3\right ) \left (1-x^3\right )^{2/3}}{10 x^5}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1-x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1-x^3}\right )+\frac {1}{6} \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[((1 - x^3)^(2/3)*(-1 + x^3))/(x^6*(-1 + 2*x^3)),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.98, size = 136, normalized size = 1.24 \begin {gather*} \frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {4 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (x^{3} - 1\right )}}{7 \, x^{3} + 1}\right ) - 5 \, x^{5} \log \left (\frac {2 \, x^{3} - 3 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - 1}{2 \, x^{3} - 1}\right ) - 3 \, {\left (3 \, x^{3} + 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3+1)^(2/3)*(x^3-1)/x^6/(2*x^3-1),x, algorithm="fricas")

[Out]

1/30*(10*sqrt(3)*x^5*arctan(-(4*sqrt(3)*(-x^3 + 1)^(1/3)*x^2 - 2*sqrt(3)*(-x^3 + 1)^(2/3)*x - sqrt(3)*(x^3 - 1
))/(7*x^3 + 1)) - 5*x^5*log((2*x^3 - 3*(-x^3 + 1)^(1/3)*x^2 + 3*(-x^3 + 1)^(2/3)*x - 1)/(2*x^3 - 1)) - 3*(3*x^
3 + 2)*(-x^3 + 1)^(2/3))/x^5

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3+1)^(2/3)*(x^3-1)/x^6/(2*x^3-1),x, algorithm="giac")

[Out]

integrate((x^3 - 1)*(-x^3 + 1)^(2/3)/((2*x^3 - 1)*x^6), x)

________________________________________________________________________________________

maple [C]  time = 5.28, size = 278, normalized size = 2.53

method	result	size
risch	\(\frac {3 x^{6}-x^{3}-2}{10 x^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}}-\frac {\ln \left (-\frac {3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+\left (-x^{3}+1\right )^{\frac {2}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-1}{2 x^{3}-1}\right )}{3}+\RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+2 \left (-x^{3}+1\right )^{\frac {2}{3}} x -\left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{2 x^{3}-1}\right )\)	\(278\)
trager	\(-\frac {\left (3 x^{3}+2\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{10 x^{5}}+32 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \ln \left (\frac {374952960 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x [phone] \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{2}-34273056 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+421089 \left (-x^{3}+1\right )^{\frac {2}{3}} x +421089 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+252248 x^{3}-2999623680 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )-220717}{2 x^{3}-1}\right )+\frac {\ln \left (-\frac {-374952960 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x [phone] \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{2}-26461536 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}-527889 \left (-x^{3}+1\right )^{\frac {2}{3}} x -527889 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+64078 x^{3}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-10936032 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+9154}{2 x^{3}-1}\right )}{3}-32 \ln \left (-\frac {-374952960 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x [phone] \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{2}-26461536 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}-527889 \left (-x^{3}+1\right )^{\frac {2}{3}} x -527889 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+64078 x^{3}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-10936032 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+9154}{2 x^{3}-1}\right ) \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )\)	\(515\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^3+1)^(2/3)*(x^3-1)/x^6/(2*x^3-1),x,method=_RETURNVERBOSE)

[Out]

1/10*(3*x^6-x^3-2)/x^5/(-x^3+1)^(1/3)-1/3*ln(-(3*RootOf(9*_Z^2-3*_Z+1)*(-x^3+1)^(2/3)*x-6*RootOf(9*_Z^2-3*_Z+1
)*(-x^3+1)^(1/3)*x^2+3*RootOf(9*_Z^2-3*_Z+1)*x^3+(-x^3+1)^(2/3)*x+(-x^3+1)^(1/3)*x^2-1)/(2*x^3-1))+RootOf(9*_Z
^2-3*_Z+1)*ln(-(9*RootOf(9*_Z^2-3*_Z+1)^2*x^3-3*RootOf(9*_Z^2-3*_Z+1)*(-x^3+1)^(2/3)*x-3*RootOf(9*_Z^2-3*_Z+1)
*(-x^3+1)^(1/3)*x^2-3*RootOf(9*_Z^2-3*_Z+1)*x^3+2*(-x^3+1)^(2/3)*x-(-x^3+1)^(1/3)*x^2+3*RootOf(9*_Z^2-3*_Z+1))
/(2*x^3-1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3+1)^(2/3)*(x^3-1)/x^6/(2*x^3-1),x, algorithm="maxima")

[Out]

integrate((x^3 - 1)*(-x^3 + 1)^(2/3)/((2*x^3 - 1)*x^6), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (1-x^3\right )}^{5/3}}{x^6\,\left (2\,x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1 - x^3)^(5/3)/(x^6*(2*x^3 - 1)),x)

[Out]

-int((1 - x^3)^(5/3)/(x^6*(2*x^3 - 1)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{6} \left (2 x^{3} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**3+1)**(2/3)*(x**3-1)/x**6/(2*x**3-1),x)

[Out]

Integral((-(x - 1)*(x**2 + x + 1))**(2/3)*(x - 1)*(x**2 + x + 1)/(x**6*(2*x**3 - 1)), x)

________________________________________________________________________________________