Optimal. Leaf size=14 \[ \frac {4 x}{\sqrt [4]{x^2 (x+1)}} \]
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Rubi [A] time = 0.06, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2056, 74} \begin {gather*} \frac {4 x}{\sqrt [4]{x^3+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 2056
Rubi steps
\begin {align*} \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x}\right ) \int \frac {2+x}{\sqrt {x} (1+x)^{5/4}} \, dx}{\sqrt [4]{x^2+x^3}}\\ &=\frac {4 x}{\sqrt [4]{x^2+x^3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {4 x}{\sqrt [4]{x^2 (x+1)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 6.44, size = 14, normalized size = 1.00 \begin {gather*} \frac {4 x}{\sqrt [4]{x^2 (1+x)}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 18, normalized size = 1.29 \begin {gather*} \frac {4 \, {\left (x^{3} + x^{2}\right )}^{\frac {3}{4}}}{x^{2} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 11, normalized size = 0.79 \begin {gather*} \frac {4}{{\left (\frac {1}{x} + \frac {1}{x^{2}}\right )}^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 13, normalized size = 0.93
method | result | size |
gosper | \(\frac {4 x}{\left (x^{3}+x^{2}\right )^{\frac {1}{4}}}\) | \(13\) |
risch | \(\frac {4 x}{\left (x^{2} \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(13\) |
trager | \(\frac {4 \left (x^{3}+x^{2}\right )^{\frac {3}{4}}}{x \left (1+x \right )}\) | \(20\) |
meijerg | \(4 \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -x \right ) \sqrt {x}+\frac {2 \hypergeom \left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], -x \right ) x^{\frac {3}{2}}}{3}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 2}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 19, normalized size = 1.36 \begin {gather*} \frac {4\,{\left (x^3+x^2\right )}^{3/4}}{x\,\left (x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 2}{\sqrt [4]{x^{2} \left (x + 1\right )} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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