3.3.28 \(\int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx\) [228]

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

[Out]

-((-1+x)^2*(1+x))^(1/3)/x+1/6*ln(x)-2/3*ln(1+x)-3/2*ln(1+(1-x)/((-1+x)^2*(1+x))^(1/3))-1/2*ln(1+(-1+x)/((-1+x)
^2*(1+x))^(1/3))-1/3*arctan(1/3*(1-2*(-1+x)/((-1+x)^2*(1+x))^(1/3))*3^(1/2))*3^(1/2)-arctan(1/3*(1+2*(-1+x)/((
-1+x)^2*(1+x))^(1/3))*3^(1/2))*3^(1/2)

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(404\) vs. \(2(150)=300\).
time = 0.22, 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 = {2106, 2102, 99, 163, 62, 93} \begin {gather*} -\frac {3 \sqrt [6]{3} \sqrt [3]{x^3-x^2-x+1} \text {ArcTan}\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} \text {ArcTan}\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}}-\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}} \end {gather*}

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 62

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

Rule 93

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 99

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 163

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 2102

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 2106

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 &=\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 0.36, size = 229, normalized size = 1.53 \begin {gather*} -\frac {(-1+x)^{4/3} (1+x)^{2/3} \left (18 (-1+x)^{2/3} \sqrt [3]{1+x}-6 \sqrt {3} x \tan ^{-1}\left (\frac {1-\frac {2}{\sqrt [3]{\frac {-1+x}{1+x}}}}{\sqrt {3}}\right )-18 \sqrt {3} x \tan ^{-1}\left (\frac {1+\frac {2}{\sqrt [3]{\frac {-1+x}{1+x}}}}{\sqrt {3}}\right )-10 x \log \left (\frac {2}{-1+x}\right )-3 x \log \left (1+\frac {1}{\left (\frac {-1+x}{1+x}\right )^{2/3}}-\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )+28 x \log \left (-1+\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )+6 x \log \left (1+\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )+x \log \left (1+\frac {1}{\left (\frac {-1+x}{1+x}\right )^{2/3}}+\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )\right )}{18 x \left ((-1+x)^2 (1+x)\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18*((-1 + x)^(4/3)*(1 + x)^(2/3)*(18*(-1 + x)^(2/3)*(1 + x)^(1/3) - 6*Sqrt[3]*x*ArcTan[(1 - 2/((-1 + x)/(1
+ x))^(1/3))/Sqrt[3]] - 18*Sqrt[3]*x*ArcTan[(1 + 2/((-1 + x)/(1 + x))^(1/3))/Sqrt[3]] - 10*x*Log[2/(-1 + x)] -
3*x*Log[1 + ((-1 + x)/(1 + x))^(-2/3) - ((-1 + x)/(1 + x))^(-1/3)] + 28*x*Log[-1 + ((-1 + x)/(1 + x))^(-1/3)]
+ 6*x*Log[1 + ((-1 + x)/(1 + x))^(-1/3)] + x*Log[1 + ((-1 + x)/(1 + x))^(-2/3) + ((-1 + x)/(1 + x))^(-1/3)]))
/(x*((-1 + x)^2*(1 + x))^(2/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.97, size = 1247, normalized size = 8.31

method	result	size
risch	\(\text {Expression too large to display}\)	\(1247\)
trager	\(\text {Expression too large to display}\)	\(2055\)

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^5-1955796480*RootOf(_Z^2-3*_
Z+9)^2*x^3-21459433600*RootOf(_Z^2-3*_Z+9)^2*x^2-108655360*RootOf(_Z^2-3*_Z+9)^2+2933694720*RootOf(_Z^2-3*_Z+9
)^2*x^5+4262769939861*x^4-2395436537574*x^2-2841846626574*x^3+2933694720*RootOf(_Z^2-3*_Z+9)^2*x^4-17555896285
11*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^4-58519
6542837*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+1
95065514279*RootOf(_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*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)*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
[phone]*(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-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^4-12395048712*RootOf(_Z^2-3*_Z+9)-195065514279*(x^3+x^2-x-1)
^(1/3)-38163044376*(x^3+x^2-x-1)^(2/3)-19612292480*RootOf(_Z^2-3*_Z+9)^2*x)/x/(1+x))+1/9*RootOf(_Z^2-3*_Z+9)*l
n(-(33401336760+117256110840*x-901836092520*x^5-9523415232*RootOf(_Z^2-3*_Z+9)^2*x^3-104493028240*RootOf(_Z^2-
3*_Z+9)^2*x^2-529078624*RootOf(_Z^2-3*_Z+9)^2+14285122848*RootOf(_Z^2-3*_Z+9)^2*x^5-901836092520*x^4+685078835
760*x^2+601224061680*x^3+14285122848*RootOf(_Z^2-3*_Z+9)^2*x^4+1755589628511*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)
^(2/3)*x^3-343467399384*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-1454977597671*RootOf(_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+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+53888059173*RootOf(_Z^2-3*_Z+9)[phone]*(x^3+x^2-x-1)^(1/3)[phone]*(x^3+x^2-x-1)^
(2/3)-95498691632*RootOf(_Z^2-3*_Z+9)^2*x)/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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (126) = 252\).
time = 0.42, size = 280, normalized size = 1.87 \begin {gather*} \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 {gather*}

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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