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

Optimal. Leaf size=150 \[ -\frac {\sqrt [3]{(x-1)^2 (x+1)}}{x}+\frac {\log (x)}{6}-\frac {2}{3} \log (x+1)-\frac {3}{2} \log \left (1-\frac {x-1}{\sqrt [3]{(x-1)^2 (x+1)}}\right )-\frac {1}{2} \log \left (\frac {x-1}{\sqrt [3]{(x-1)^2 (x+1)}}+1\right )-\frac {\tan ^{-1}\left (\frac {1-\frac {2 (x-1)}{\sqrt [3]{(x-1)^2 (x+1)}}}{\sqrt {3}}\right )}{\sqrt {3}}-\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 (x-1)}{\sqrt [3]{(x-1)^2 (x+1)}}+1}{\sqrt {3}}\right ) \]

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Rubi [B]  time = 0.33, antiderivative size = 404, normalized size of antiderivative = 2.69, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2081, 2077, 97, 157, 60, 91} \[ -\frac {\sqrt [3]{x^3-x^2-x+1}}{x}+\frac {\sqrt [3]{x^3-x^2-x+1} \log (x)}{2 \sqrt [3]{3} (3-3 x)^{2/3} \sqrt [3]{x+1}}-\frac {3^{2/3} \sqrt [3]{x^3-x^2-x+1} \log \left (\frac {4 (x+1)}{3}\right )}{2 (3-3 x)^{2/3} \sqrt [3]{x+1}}-\frac {3\ 3^{2/3} \sqrt [3]{x^3-x^2-x+1} \log \left (\frac {\sqrt [3]{3-3 x}}{\sqrt [3]{3} \sqrt [3]{x+1}}+1\right )}{2 (3-3 x)^{2/3} \sqrt [3]{x+1}}-\frac {3^{2/3} \sqrt [3]{x^3-x^2-x+1} \log \left (\left (\frac {2}{3}\right )^{2/3} \sqrt [3]{3-3 x}-\frac {2^{2/3} \sqrt [3]{x+1}}{\sqrt [3]{3}}\right )}{2 (3-3 x)^{2/3} \sqrt [3]{x+1}}-\frac {3 \sqrt [6]{3} \sqrt [3]{x^3-x^2-x+1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{3-3 x}}{3^{5/6} \sqrt [3]{x+1}}\right )}{(3-3 x)^{2/3} \sqrt [3]{x+1}}-\frac {\sqrt [6]{3} \sqrt [3]{x^3-x^2-x+1} \tan ^{-1}\left (\frac {2 \sqrt [3]{3-3 x}}{3^{5/6} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{(3-3 x)^{2/3} \sqrt [3]{x+1}} \]

Antiderivative was successfully verified.

[In]

Int[((-1 + x)^2*(1 + x))^(1/3)/x^2,x]

[Out]

-((1 - x - x^2 + x^3)^(1/3)/x) - (3*3^(1/6)*(1 - x - x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] - (2*(3 - 3*x)^(1/3))/(
3^(5/6)*(1 + x)^(1/3))])/((3 - 3*x)^(2/3)*(1 + x)^(1/3)) - (3^(1/6)*(1 - x - x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3]
+ (2*(3 - 3*x)^(1/3))/(3^(5/6)*(1 + x)^(1/3))])/((3 - 3*x)^(2/3)*(1 + x)^(1/3)) + ((1 - x - x^2 + x^3)^(1/3)*
Log[x])/(2*3^(1/3)*(3 - 3*x)^(2/3)*(1 + x)^(1/3)) - (3^(2/3)*(1 - x - x^2 + x^3)^(1/3)*Log[(4*(1 + x))/3])/(2*
(3 - 3*x)^(2/3)*(1 + x)^(1/3)) - (3*3^(2/3)*(1 - x - x^2 + x^3)^(1/3)*Log[1 + (3 - 3*x)^(1/3)/(3^(1/3)*(1 + x)
^(1/3))])/(2*(3 - 3*x)^(2/3)*(1 + x)^(1/3)) - (3^(2/3)*(1 - x - x^2 + x^3)^(1/3)*Log[(2/3)^(2/3)*(3 - 3*x)^(1/
3) - (2^(2/3)*(1 + x)^(1/3))/3^(1/3)])/(2*(3 - 3*x)^(2/3)*(1 + x)^(1/3))

Rule 60

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-(d/b), 3]}, Simp[(Sq
rt[3]*q*ArcTan[1/Sqrt[3] - (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3))])/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] && NegQ[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 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 2077

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[(a + b*x + d*x^3)^p/
((3*a - b*x)^p*(3*a + 2*b*x)^(2*p)), Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b,
d, e, f, m, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {16}{27}-\frac {4 x}{3}+x^3}}{\left (\frac {1}{3}+x\right )^2} \, dx,x,-\frac {1}{3}+x\right )\\ &=\frac {\left (3 \sqrt [3]{1-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {16}{9}-\frac {8 x}{3}\right )^{2/3} \sqrt [3]{\frac {16}{9}+\frac {4 x}{3}}}{\left (\frac {1}{3}+x\right )^2} \, dx,x,-\frac {1}{3}+x\right )}{4\ 2^{2/3} (1-x)^{2/3} \sqrt [3]{1+x}}\\ &=-\frac {\sqrt [3]{1-x-x^2+x^3}}{x}+\frac {\left (3 \sqrt [3]{1-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {-\frac {64}{27}-\frac {32 x}{9}}{\sqrt [3]{\frac {16}{9}-\frac {8 x}{3}} \left (\frac {1}{3}+x\right ) \left (\frac {16}{9}+\frac {4 x}{3}\right )^{2/3}} \, dx,x,-\frac {1}{3}+x\right )}{4\ 2^{2/3} (1-x)^{2/3} \sqrt [3]{1+x}}\\ &=-\frac {\sqrt [3]{1-x-x^2+x^3}}{x}-\frac {\left (4 \sqrt [3]{2} \sqrt [3]{1-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {16}{9}-\frac {8 x}{3}} \left (\frac {1}{3}+x\right ) \left (\frac {16}{9}+\frac {4 x}{3}\right )^{2/3}} \, dx,x,-\frac {1}{3}+x\right )}{9 (1-x)^{2/3} \sqrt [3]{1+x}}-\frac {\left (4 \sqrt [3]{2} \sqrt [3]{1-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {16}{9}-\frac {8 x}{3}} \left (\frac {16}{9}+\frac {4 x}{3}\right )^{2/3}} \, dx,x,-\frac {1}{3}+x\right )}{3 (1-x)^{2/3} \sqrt [3]{1+x}}\\ &=-\frac {\sqrt [3]{1-x-x^2+x^3}}{x}-\frac {\sqrt {3} \sqrt [3]{1-x-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{(1-x)^{2/3} \sqrt [3]{1+x}}-\frac {\sqrt [3]{1-x-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{\sqrt {3} (1-x)^{2/3} \sqrt [3]{1+x}}+\frac {\sqrt [3]{1-x-x^2+x^3} \log (x)}{6 (1-x)^{2/3} \sqrt [3]{1+x}}-\frac {\sqrt [3]{1-x-x^2+x^3} \log (1+x)}{2 (1-x)^{2/3} \sqrt [3]{1+x}}-\frac {\sqrt [3]{1-x-x^2+x^3} \log \left (\sqrt [3]{1-x}-\sqrt [3]{1+x}\right )}{2 (1-x)^{2/3} \sqrt [3]{1+x}}-\frac {3 \sqrt [3]{1-x-x^2+x^3} \log \left (\frac {3 \left (\sqrt [3]{1-x}+\sqrt [3]{1+x}\right )}{\sqrt [3]{1+x}}\right )}{2 (1-x)^{2/3} \sqrt [3]{1+x}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 112, normalized size = 0.75 \[ \frac {\sqrt [3]{(x-1)^2 (x+1)} \left (3 (x+1) \left (3\ 2^{2/3} \sqrt [3]{1-x} x \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {x+1}{2}\right )-2 x+2\right )-2 \left (\frac {1}{x}+1\right )^{2/3} \sqrt [3]{\frac {x-1}{x}} x F_1\left (1;\frac {1}{3},\frac {2}{3};2;\frac {1}{x},-\frac {1}{x}\right )\right )}{6 x \left (x^2-1\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((-1 + x)^2*(1 + x))^(1/3)*(-2*(1 + x^(-1))^(2/3)*((-1 + x)/x)^(1/3)*x*AppellF1[1, 1/3, 2/3, 2, x^(-1), -x^(-
1)] + 3*(1 + x)*(2 - 2*x + 3*2^(2/3)*(1 - x)^(1/3)*x*Hypergeometric2F1[1/3, 1/3, 4/3, (1 + x)/2])))/(6*x*(-1 +
x^2))

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IntegrateAlgebraic [A]  time = 8.04, size = 247, normalized size = 1.65 \[ \frac {\sqrt [3]{x-1} (x+1)^{2/3} \sqrt [3]{(x-1)^2 (x+1)} \left (-\frac {(x-1)^{2/3} \sqrt [3]{x+1}}{x}-\log \left (\sqrt [3]{x-1}-\sqrt [3]{x+1}\right )-\frac {1}{3} \log \left (\sqrt [3]{x-1}+\sqrt [3]{x+1}\right )+\frac {1}{6} \log \left ((x-1)^{2/3}-\sqrt [3]{x+1} \sqrt [3]{x-1}+(x+1)^{2/3}\right )+\frac {1}{2} \log \left ((x-1)^{2/3}+\sqrt [3]{x+1} \sqrt [3]{x-1}+(x+1)^{2/3}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x+1}}{2 \sqrt [3]{x-1}-\sqrt [3]{x+1}}\right )}{\sqrt {3}}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x+1}}{2 \sqrt [3]{x-1}+\sqrt [3]{x+1}}\right )\right )}{x^2-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.88, size = 280, normalized size = 1.87 \[ \frac {6 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{3 \, {\left (x - 1\right )}}\right ) - 2 \, \sqrt {3} x \arctan \left (-\frac {\sqrt {3} {\left (x - 1\right )} - 2 \, \sqrt {3} {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{3 \, {\left (x - 1\right )}}\right ) + 3 \, x \log \left (\frac {x^{2} + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 2 \, x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}} + 1}{x^{2} - 2 \, x + 1}\right ) + x \log \left (\frac {x^{2} - {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 2 \, x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}} + 1}{x^{2} - 2 \, x + 1}\right ) - 2 \, x \log \left (\frac {x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} - 1}{x - 1}\right ) - 6 \, x \log \left (-\frac {x - {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} - 1}{x - 1}\right ) - 6 \, {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{6 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*sqrt(3)*x*arctan(1/3*(sqrt(3)*(x - 1) + 2*sqrt(3)*(x^3 - x^2 - x + 1)^(1/3))/(x - 1)) - 2*sqrt(3)*x*arc
tan(-1/3*(sqrt(3)*(x - 1) - 2*sqrt(3)*(x^3 - x^2 - x + 1)^(1/3))/(x - 1)) + 3*x*log((x^2 + (x^3 - x^2 - x + 1)
^(1/3)*(x - 1) - 2*x + (x^3 - x^2 - x + 1)^(2/3) + 1)/(x^2 - 2*x + 1)) + x*log((x^2 - (x^3 - x^2 - x + 1)^(1/3
)*(x - 1) - 2*x + (x^3 - x^2 - x + 1)^(2/3) + 1)/(x^2 - 2*x + 1)) - 2*x*log((x + (x^3 - x^2 - x + 1)^(1/3) - 1
)/(x - 1)) - 6*x*log(-(x - (x^3 - x^2 - x + 1)^(1/3) - 1)/(x - 1)) - 6*(x^3 - x^2 - x + 1)^(1/3))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 4.80, size = 1247, normalized size = 8.31

method	result	size
risch	\(-\frac {\left (\left (-1+x \right )^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}+\frac {\left (-\frac {\ln \left (-\frac {-157880368143+288529720857 x [phone] x^{5}-2841846626574 x^{3}[phone] x^{4}-2395436537574 x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{5}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{5}-223110876816 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{3}-477460395840 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}-266744567736 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x [phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{4}-1955796480 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{3}-21459433600 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}-19612292480 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x [phone] \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x^{3}-5266768885533 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{4}[phone] \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x^{2}-3511179257022 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{3}-114489133128 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x [phone] \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{2}[phone] \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -12395048712 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )-108655360 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2}-195065514279 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-38163044376 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-1755589628511 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x^{3}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{4}-585196542837 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{3}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x -627609879612 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{2}-104601646602 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x}{x \left (1+x \right )}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \ln \left (-\frac {33401336760+117256110840 x -901836092520 x^{5}[phone] x^{3}-901836092520 x^{4}[phone] x^{2}-1454977597671 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{5}-65021838093 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-12721014792 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{5}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{3}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -1454977597671 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{4}-9523415232 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{3}-104493028240 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}-95498691632 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x -4236366687381 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x^{3}[phone] \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{4}-1412122229127 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x^{2}[phone] \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{3}[phone] \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x -2340786171348 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{2}-390131028558 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x [phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )-529078624 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2}[phone] \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}[phone] \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x^{3}-343467399384 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x^{2}-228978266256 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{3}-195065514279 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}} x [phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x}{x \left (1+x \right )}\right )}{9}\right ) \left (\left (-1+x \right )^{2} \left (1+x \right )\right )^{\frac {1}{3}} \left (\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}{\left (-1+x \right ) \left (1+x \right )}\)	\(1247\)
trager	\(\text {Expression too large to display}\)	\(2057\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((-1+x)^2*(1+x))^(1/3)/x+(-1/3*ln(-(-157880368143+288529720857*x+4262769939861*x^5-2841846626574*x^3+42627699
39861*x^4-2395436537574*x^2-108655360*RootOf(_Z^2-3*_Z+9)^2-12395048712*RootOf(_Z^2-3*_Z+9)[phone]*RootOf(
_Z^2-3*_Z+9)^2*x^4-1955796480*RootOf(_Z^2-3*_Z+9)^2*x^3-21459433600*RootOf(_Z^2-3*_Z+9)^2*x^2-19612292480*Root
Of(_Z^2-3*_Z+9)^2*x+334666315224*RootOf(_Z^2-3*_Z+9)*x^5+1030402198152*(x^3+x^2-x-1)^(2/3)*x^3-5266768885533*(
x^3+x^2-x-1)^(1/3)*x^4+343467399384*(x^3+x^2-x-1)^(2/3)*x^2-3511179257022*(x^3+x^2-x-1)^(1/3)*x^3+65021838093*
RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(2/3)-114489133128*(x^3+x^2-x-1)^(2/3)*x+2340786171348*(x^3+x^2-x-1)^(1/3)*x
^2+52300823301*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)[phone]*(x^3+x^2-x-1)^(1/3)*x+2933694720*RootOf(_Z
^2-3*_Z+9)^2*x^5-195065514279*(x^3+x^2-x-1)^(1/3)-38163044376*(x^3+x^2-x-1)^(2/3)-1755589628511*RootOf(_Z^2-3*
_Z+9)*(x^3+x^2-x-1)^(2/3)*x^3+1412122229127*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)*x^4-585196542837*RootOf(_Z
^2-3*_Z+9)*(x^3+x^2-x-1)^(2/3)*x^2+941414819418*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)*x^3+195065514279*RootO
f(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(2/3)*x-627609879612*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)*x^2-104601646602*Roo
tOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)*x-223110876816*RootOf(_Z^2-3*_Z+9)*x^3-477460395840*RootOf(_Z^2-3*_Z+9)*x
^2-266744567736*RootOf(_Z^2-3*_Z+9)*x+334666315224*RootOf(_Z^2-3*_Z+9)*x^4)/x/(1+x))+1/9*RootOf(_Z^2-3*_Z+9)*l
n(-(33401336760+117256110840*x-901836092520*x^5+601224061680*x^3-901836092520*x^4+685078835760*x^2-529078624*R
ootOf(_Z^2-3*_Z+9)^2+53888059173*RootOf(_Z^2-3*_Z+9)[phone]*RootOf(_Z^2-3*_Z+9)^2*x^4-9523415232*RootOf(_
Z^2-3*_Z+9)^2*x^3-104493028240*RootOf(_Z^2-3*_Z+9)^2*x^2-95498691632*RootOf(_Z^2-3*_Z+9)^2*x-1454977597671*Roo
tOf(_Z^2-3*_Z+9)*x^5-4236366687381*(x^3+x^2-x-1)^(2/3)*x^3+5266768885533*(x^3+x^2-x-1)^(1/3)*x^4-1412122229127
*(x^3+x^2-x-1)^(2/3)*x^2+3511179257022*(x^3+x^2-x-1)^(1/3)*x^3-65021838093*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(
2/3)[phone]*(x^3+x^2-x-1)^(2/3)*x-2340786171348*(x^3+x^2-x-1)^(1/3)*x^2-12721014792*RootOf(_Z^2-3*_Z+9)*
(x^3+x^2-x-1)^(1/3)-390131028558*(x^3+x^2-x-1)^(1/3)*x+14285122848*RootOf(_Z^2-3*_Z+9)^2*x^5+195065514279*(x^3
+x^2-x-1)^(1/3)[phone]*(x^3+x^2-x-1)^(2/3)[phone]*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(2/3)*x^3-343
467399384*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)*x^4+585196542837*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(2/3)*x^2
-228978266256*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)*x^3-195065514279*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(2/3)
*x+152652177504*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)*x^2+25442029584*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3
)*x+969985065114*RootOf(_Z^2-3*_Z+9)*x^3+1047579629778*RootOf(_Z^2-3*_Z+9)*x^2+131482623837*RootOf(_Z^2-3*_Z+9
)*x-1454977597671*RootOf(_Z^2-3*_Z+9)*x^4)/x/(1+x)))*((-1+x)^2*(1+x))^(1/3)*((-1+x)*(1+x)^2)^(1/3)/(-1+x)/(1+x
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x - 1)^2*(x + 1))^(1/3)/x^2,x)

[Out]

int(((x - 1)^2*(x + 1))^(1/3)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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