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

Optimal. Leaf size=16 \[ -\frac {3 x^2}{2 \left (x^4+x\right )^{2/3}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 21, normalized size of antiderivative = 1.31, 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}}{2 \left (x^3+1\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*}

________________________________________________________________________________________

Mathematica [C]  time = 0.04, size = 61, normalized size = 3.81 \begin {gather*} \frac {3 \sqrt [3]{x^4+x} \left (2 x^4 \, _2F_1\left (\frac {13}{9},\frac {5}{3};\frac {22}{9};-x^3\right )-13 x \, _2F_1\left (\frac {4}{9},\frac {5}{3};\frac {13}{9};-x^3\right )\right )}{26 \sqrt [3]{x^3+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(x + x^4)^(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))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.19, size = 16, 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))

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 17, 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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 18, normalized size = 1.12

method	result	size
gosper	\(-\frac {3 x \left (x^{4}+x \right )^{\frac {1}{3}}}{2 \left (x^{3}+1\right )}\)	\(18\)
trager	\(-\frac {3 x \left (x^{4}+x \right )^{\frac {1}{3}}}{2 \left (x^{3}+1\right )}\)	\(18\)
risch	\(-\frac {3 \left (x \left (x^{3}+1\right )\right )^{\frac {1}{3}} x}{2 \left (x^{3}+1\right )}\)	\(20\)
meijerg	\(-\frac {3 \hypergeom \left (\left [\frac {4}{9}, \frac {5}{3}\right ], \left [\frac {13}{9}\right ], -x^{3}\right ) x^{\frac {4}{3}}}{2}+\frac {3 \hypergeom \left (\left [\frac {13}{9}, \frac {5}{3}\right ], \left [\frac {22}{9}\right ], -x^{3}\right ) x^{\frac {13}{3}}}{13}\)	\(34\)

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 19, normalized size = 1.19 \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^3 - 2)*(x + x^4)^(1/3))/(x^3 + 1)^2,x)

[Out]

-(3*x*(x + x^4)^(1/3))/(2*(x^3 + 1))

________________________________________________________________________________________

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)

________________________________________________________________________________________