\(\int \frac {-2+x^3}{(1+x^3) \sqrt [3]{x+x^4}} \, dx\) [138]

Optimal result

Integrand size = 22, antiderivative size = 18 \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3 \left (x+x^4\right )^{2/3}}{1+x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 460} \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3 x}{\sqrt [3]{x^4+x}} \]

[In]

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

[Out]

(-3*x)/(x + x^4)^(1/3)

Rule 460

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 2081

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 5.86 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3 x}{\sqrt [3]{x+x^4}} \]

[In]

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

[Out]

(-3*x)/(x + x^4)^(1/3)

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3 \, {\left (x^{4} + x\right )}^{\frac {2}{3}}}{x^{3} + 1} \]

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [B] (verification not implemented)

Time = 4.98 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3\,{\left (x^4+x\right )}^{2/3}}{x^3+1} \]

[In]

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

[Out]

-(3*(x + x^4)^(2/3))/(x^3 + 1)