Optimal. Leaf size=18 \[ -\frac {3 \left (x^4+x\right )^{2/3}}{x^3+1} \]
________________________________________________________________________________________
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]
[Out]
Rule 449
Rule 2056
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*}
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________