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

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

[Out]

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

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Rubi [B]  time = 0.327181, 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 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]

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 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 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]

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*}

Mathematica [C]  time = 0.0668508, 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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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt [3]{ \left ( -1+x \right ) ^{2} \left ( 1+x \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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Fricas [B]  time = 1.88514, size = 730, normalized size = 4.87 \begin{align*} \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} \end{align*}

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

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

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)