Optimal. Leaf size=23 \[ \frac {2 (2 x-1) \sqrt {x^4+x^3}}{3 x^3} \]
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Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2016, 2000} \begin {gather*} \frac {4 \sqrt {x^4+x^3}}{3 x^2}-\frac {2 \sqrt {x^4+x^3}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2000
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {x^3+x^4}} \, dx &=-\frac {2 \sqrt {x^3+x^4}}{3 x^3}-\frac {2}{3} \int \frac {1}{\sqrt {x^3+x^4}} \, dx\\ &=-\frac {2 \sqrt {x^3+x^4}}{3 x^3}+\frac {4 \sqrt {x^3+x^4}}{3 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \left (2 x^2+x-1\right )}{3 \sqrt {x^3 (x+1)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 (-1+2 x) \sqrt {x^3+x^4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 26, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left (2 \, x^{3} + \sqrt {x^{4} + x^{3}} {\left (2 \, x - 1\right )}\right )}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {3}{2}} + 2 \, \sqrt {\frac {1}{x} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 16, normalized size = 0.70
method | result | size |
meijerg | \(-\frac {2 \left (1-2 x \right ) \sqrt {1+x}}{3 x^{\frac {3}{2}}}\) | \(16\) |
gosper | \(\frac {2 \left (1+x \right ) \left (-1+2 x \right )}{3 \sqrt {x^{4}+x^{3}}}\) | \(20\) |
trager | \(\frac {2 \left (-1+2 x \right ) \sqrt {x^{4}+x^{3}}}{3 x^{3}}\) | \(20\) |
risch | \(\frac {-\frac {2}{3}+\frac {2}{3} x +\frac {4}{3} x^{2}}{\sqrt {x^{3} \left (1+x \right )}}\) | \(20\) |
default | \(\frac {2 \sqrt {x \left (1+x \right )}\, \sqrt {x^{2}+x}\, \left (-1+2 x \right )}{3 x \sqrt {x^{4}+x^{3}}}\) | \(34\) |
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 {1}{\sqrt {x^{4} + x^{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 29, normalized size = 1.26 \begin {gather*} \frac {4\,x\,\sqrt {x^4+x^3}-2\,\sqrt {x^4+x^3}}{3\,x^3} \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 {1}{x \sqrt {x^{3} \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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