3.2.35 \(\int \frac {(2+x^3) \sqrt [3]{-x+x^4}}{(-1+x^3)^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac {3 x^2}{2 \left (x^4-x\right )^{2/3}} \]

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Rubi [A]  time = 0.07, antiderivative size = 25, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2056, 449} \begin {gather*} \frac {3 x \sqrt [3]{x^4-x}}{2 \left (1-x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(-x + x^4)^(1/3))/(2*(1 - x^3))

Rule 449

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

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps

\begin {align*} \int \frac {\left (2+x^3\right ) \sqrt [3]{-x+x^4}}{\left (-1+x^3\right )^2} \, dx &=\frac {\sqrt [3]{-x+x^4} \int \frac {\sqrt [3]{x} \left (2+x^3\right )}{\left (-1+x^3\right )^{5/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^3}}\\ &=\frac {3 x \sqrt [3]{-x+x^4}}{2 \left (1-x^3\right )}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 61, normalized size = 3.39 \begin {gather*} \frac {3 \sqrt [3]{x \left (x^3-1\right )} \left (13 x \, _2F_1\left (\frac {4}{9},\frac {5}{3};\frac {13}{9};x^3\right )+2 x^4 \, _2F_1\left (\frac {13}{9},\frac {5}{3};\frac {22}{9};x^3\right )\right )}{26 \sqrt [3]{1-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(x*(-1 + x^3))^(1/3)*(13*x*Hypergeometric2F1[4/9, 5/3, 13/9, x^3] + 2*x^4*Hypergeometric2F1[13/9, 5/3, 22/9
, x^3]))/(26*(1 - x^3)^(1/3))

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IntegrateAlgebraic [A]  time = 0.19, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 x^2}{2 \left (-x+x^4\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x^2)/(2*(-x + x^4)^(2/3))

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fricas [A]  time = 0.47, size = 19, normalized size = 1.06 \begin {gather*} -\frac {3 \, {\left (x^{4} - x\right )}^{\frac {1}{3}} x}{2 \, {\left (x^{3} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^4-x)^(1/3)/(x^3-1)^2,x, algorithm="fricas")

[Out]

-3/2*(x^4 - x)^(1/3)*x/(x^3 - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{{\left (x^{3} - 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^4-x)^(1/3)/(x^3-1)^2,x, algorithm="giac")

[Out]

integrate((x^4 - x)^(1/3)*(x^3 + 2)/(x^3 - 1)^2, x)

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maple [A]  time = 0.12, size = 20, normalized size = 1.11

method	result	size
gosper	\(-\frac {3 x \left (x^{4}-x \right )^{\frac {1}{3}}}{2 \left (x^{3}-1\right )}\)	\(20\)
trager	\(-\frac {3 x \left (x^{4}-x \right )^{\frac {1}{3}}}{2 \left (x^{3}-1\right )}\)	\(20\)
risch	\(-\frac {3 \left (x \left (x^{3}-1\right )\right )^{\frac {1}{3}} x}{2 \left (x^{3}-1\right )}\)	\(20\)
meijerg	\(\frac {3 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [\frac {4}{9}, \frac {5}{3}\right ], \left [\frac {13}{9}\right ], x^{3}\right ) x^{\frac {4}{3}}}{2 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}+\frac {3 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [\frac {13}{9}, \frac {5}{3}\right ], \left [\frac {22}{9}\right ], x^{3}\right ) x^{\frac {13}{3}}}{13 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+2)*(x^4-x)^(1/3)/(x^3-1)^2,x,method=_RETURNVERBOSE)

[Out]

-3/2*x*(x^4-x)^(1/3)/(x^3-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{{\left (x^{3} - 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^4-x)^(1/3)/(x^3-1)^2,x, algorithm="maxima")

[Out]

integrate((x^4 - x)^(1/3)*(x^3 + 2)/(x^3 - 1)^2, x)

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mupad [B]  time = 0.15, size = 21, normalized size = 1.17 \begin {gather*} -\frac {3\,x\,{\left (x^4-x\right )}^{1/3}}{2\,\left (x^3-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 - x)^(1/3)*(x^3 + 2))/(x^3 - 1)^2,x)

[Out]

-(3*x*(x^4 - x)^(1/3))/(2*(x^3 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+2)*(x**4-x)**(1/3)/(x**3-1)**2,x)

[Out]

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

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