3.26.69 \(\int \frac {x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx\)

Optimal. Leaf size=219 \[ -\log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )}{\sqrt [3]{2}}+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{2 \sqrt [3]{2}}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right )}{\sqrt [3]{2}} \]

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Rubi [A]  time = 0.10, antiderivative size = 295, normalized size of antiderivative = 1.35, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2042, 105, 59, 91} \begin {gather*} -\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x+1)}{2 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}}+\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{x^3-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((1 + x)*(-x^2 + x^3)^(1/3)),x]

[Out]

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

Rule 59

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt
[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x
)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[
b*c - a*d, 0] && PosQ[d/b]

Rule 91

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

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 2042

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol]
:> Dist[(e^IntPart[m]*(e*x)^FracPart[m]*(a*x^j + b*x^(j + n))^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a
+ b*x^n)^FracPart[p]), Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n,
p, q}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] &&  !(EqQ[n, 1] && EqQ[j, 1])

Rubi steps

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

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Mathematica [C]  time = 0.04, size = 61, normalized size = 0.28 \begin {gather*} \frac {3 \left ((x-1) x^2\right )^{2/3} \left (2 x^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};1-x\right )-\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x-1}{2 x}\right )\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.46, size = 219, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2}}-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{\sqrt [3]{2}}+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.42, size = 241, normalized size = 1.10 \begin {gather*} \frac {1}{4} \, {\left (2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}}} - 2^{\frac {2}{3}}\right )} \log \left (\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} x \sqrt {-2^{\frac {1}{3}}} + 2^{\frac {1}{3}} x + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{2 \, x}\right ) - \frac {1}{4} \, {\left (2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}}} + 2^{\frac {2}{3}}\right )} \log \left (-\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} x \sqrt {-2^{\frac {1}{3}}} - 2^{\frac {1}{3}} x - 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{2 \, x}\right ) + \frac {1}{2} \cdot 2^{\frac {2}{3}} \log \left (-\frac {3 \, {\left (2^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{x}\right ) - \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \log \left (-\frac {3 \, {\left (x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{x}\right ) + \frac {1}{2} \, \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*sqrt(3/2)*sqrt(-2^(1/3)) - 2^(2/3))*log(3/2*(2^(2/3)*sqrt(3/2)*x*sqrt(-2^(1/3)) + 2^(1/3)*x + 2*(x^3 -
x^2)^(1/3))/x) - 1/4*(2*sqrt(3/2)*sqrt(-2^(1/3)) + 2^(2/3))*log(-3/2*(2^(2/3)*sqrt(3/2)*x*sqrt(-2^(1/3)) - 2^(
1/3)*x - 2*(x^3 - x^2)^(1/3))/x) + 1/2*2^(2/3)*log(-3*(2^(1/3)*x - (x^3 - x^2)^(1/3))/x) - sqrt(3)*arctan(1/3*
(sqrt(3)*x + 2*sqrt(3)*(x^3 - x^2)^(1/3))/x) - log(-3*(x - (x^3 - x^2)^(1/3))/x) + 1/2*log((x^2 + (x^3 - x^2)^
(1/3)*x + (x^3 - x^2)^(2/3))/x^2)

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giac [A]  time = 0.32, size = 149, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{2} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.60, size = 1000, normalized size = 4.57

method	result	size
trager	\(-\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+48 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-30 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -16 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-36 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+96 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+18 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -64 x^{2}+16 x}{x}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+24 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -19 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-30 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+48 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+10 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -10 x^{2}+6 x}{x}\right )}{2}+\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \ln \left (\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{2}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +24 \left (x^{3}-x^{2}\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-48 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +26 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-52 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}-4\right ) x +20 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) x +36 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{x \left (1+x \right )}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{2}-\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -24 \left (x^{3}-x^{2}\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+48 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -44 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}+22 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{3}-4\right ) x -2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\textit {\_Z}^{2}\right ) x -60 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{x \left (1+x \right )}\right )}{2}\)	\(1000\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln((-RootOf(_Z^2-2*_Z+4)^2*x^2+48*RootOf(_Z^2-2*_Z+4)*(x^3-x^2)^(2/3)-30*RootOf(_Z^2-2*_Z+4)*(x^3-x^2)^(1/3)*
x+2*RootOf(_Z^2-2*_Z+4)^2*x-16*RootOf(_Z^2-2*_Z+4)*x^2-36*(x^3-x^2)^(2/3)+96*x*(x^3-x^2)^(1/3)+18*RootOf(_Z^2-
2*_Z+4)*x-64*x^2+16*x)/x)+1/2*RootOf(_Z^2-2*_Z+4)*ln(-(2*RootOf(_Z^2-2*_Z+4)^2*x^2+24*RootOf(_Z^2-2*_Z+4)*(x^3
-x^2)^(2/3)-9*RootOf(_Z^2-2*_Z+4)*(x^3-x^2)^(1/3)*x-4*RootOf(_Z^2-2*_Z+4)^2*x-19*RootOf(_Z^2-2*_Z+4)*x^2-30*(x
^3-x^2)^(2/3)+48*x*(x^3-x^2)^(1/3)+10*RootOf(_Z^2-2*_Z+4)*x-10*x^2+6*x)/x)+1/2*RootOf(RootOf(_Z^3-4)^2+_Z*Root
Of(_Z^3-4)+_Z^2)*ln((RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^2-2*RootOf(RootOf(_Z^3
-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^2-2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(
_Z^3-4)^3*x+4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x+24*(x^3-x^2)^(2/3)*RootOf(R
ootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2-48*RootOf(_Z^3-4)^2*(x^3-x^2)^(1/3)*x-30*RootOf(RootO
f(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*(x^3-x^2)^(1/3)*x+26*RootOf(_Z^3-4)*x^2-52*RootOf(RootOf(_Z
^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^2-10*RootOf(_Z^3-4)*x+20*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x+3
6*(x^3-x^2)^(2/3))/x/(1+x))+1/2*RootOf(_Z^3-4)*ln((2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z
^3-4)^3*x^2-RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^2-4*RootOf(RootOf(_Z^3-4)^2+_
Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x+2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2
*x-24*(x^3-x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2+48*RootOf(_Z^3-4)^2*(x^
3-x^2)^(1/3)*x+18*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*(x^3-x^2)^(1/3)*x-44*RootOf(_
Z^3-4)*x^2+22*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^2+4*RootOf(_Z^3-4)*x-2*RootOf(RootOf(_Z^3-4)^2
+_Z*RootOf(_Z^3-4)+_Z^2)*x-60*(x^3-x^2)^(2/3))/x/(1+x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((x^3 - x^2)^(1/3)*(x + 1)),x)

[Out]

int(x/((x^3 - x^2)^(1/3)*(x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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