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

Optimal. Leaf size=18 \[ -\frac {3 \left (x^4+x\right )^{2/3}}{x^3+1} \]

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Rubi [A]  time = 0.08, antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2056, 449} \begin {gather*} -\frac {3 x}{\sqrt [3]{x^4+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x)/(x + x^4)^(1/3)

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

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Mathematica [C]  time = 0.04, size = 60, normalized size = 3.33 \begin {gather*} \frac {3 \sqrt [3]{x^3+1} \left (x^4 \, _2F_1\left (\frac {11}{9},\frac {4}{3};\frac {20}{9};-x^3\right )-11 x \, _2F_1\left (\frac {2}{9},\frac {4}{3};\frac {11}{9};-x^3\right )\right )}{11 \sqrt [3]{x^4+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 + x^3)^(1/3)*(-11*x*Hypergeometric2F1[2/9, 4/3, 11/9, -x^3] + x^4*Hypergeometric2F1[11/9, 4/3, 20/9, -x^
3]))/(11*(x + x^4)^(1/3))

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IntegrateAlgebraic [A]  time = 0.20, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x+x^4\right )^{2/3}}{1+x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(x + x^4)^(2/3))/(1 + x^3)

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fricas [A]  time = 0.44, size = 16, normalized size = 0.89 \begin {gather*} -\frac {3 \, {\left (x^{4} + x\right )}^{\frac {2}{3}}}{x^{3} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(x^4 + x)^(2/3)/(x^3 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 11, normalized size = 0.61

method	result	size
gosper	\(-\frac {3 x}{\left (x^{4}+x \right )^{\frac {1}{3}}}\)	\(11\)
risch	\(-\frac {3 x}{\left (x \left (x^{3}+1\right )\right )^{\frac {1}{3}}}\)	\(13\)
trager	\(-\frac {3 \left (x^{4}+x \right )^{\frac {2}{3}}}{x^{3}+1}\)	\(17\)
meijerg	\(-3 \hypergeom \left (\left [\frac {2}{9}, \frac {4}{3}\right ], \left [\frac {11}{9}\right ], -x^{3}\right ) x^{\frac {2}{3}}+\frac {3 \hypergeom \left (\left [\frac {11}{9}, \frac {4}{3}\right ], \left [\frac {20}{9}\right ], -x^{3}\right ) x^{\frac {11}{3}}}{11}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.15, size = 16, normalized size = 0.89 \begin {gather*} -\frac {3\,{\left (x^4+x\right )}^{2/3}}{x^3+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*(x + x^4)^(2/3))/(x^3 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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