Optimal. Leaf size=38 \[ \log \left (x^3-x^2+\sqrt {x^6-2 x^5+x^4+4 x^3-4 x^2+5}+2\right ) \]
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Rubi [F] time = 0.40, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx &=\int \frac {x (-2+3 x)}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx\\ &=\int \left (-\frac {2 x}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}}+\frac {3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}}\right ) \, dx\\ &=-\left (2 \int \frac {x}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx\right )+3 \int \frac {x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.06, size = 11, normalized size = 0.29 \begin {gather*} \sinh ^{-1}\left (x^3-x^2+2\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 38, normalized size = 1.00 \begin {gather*} \log \left (2-x^2+x^3+\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 36, normalized size = 0.95 \begin {gather*} \log \left (x^{3} - x^{2} + \sqrt {x^{6} - 2 \, x^{5} + x^{4} + 4 \, x^{3} - 4 \, x^{2} + 5} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 38, normalized size = 1.00 \begin {gather*} -\log \left (-x^{3} + x^{2} + \sqrt {4 \, x^{3} + {\left (x^{3} - x^{2}\right )}^{2} - 4 \, x^{2} + 5} - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 37, normalized size = 0.97
method | result | size |
trager | \(\ln \left (2-x^{2}+x^{3}+\sqrt {x^{6}-2 x^{5}+x^{4}+4 x^{3}-4 x^{2}+5}\right )\) | \(37\) |
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 {3 \, x^{2} - 2 \, x}{\sqrt {x^{6} - 2 \, x^{5} + x^{4} + 4 \, x^{3} - 4 \, x^{2} + 5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2\,x-3\,x^2}{\sqrt {x^6-2\,x^5+x^4+4\,x^3-4\,x^2+5}} \,d x \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 \left (3 x - 2\right )}{\sqrt {x^{6} - 2 x^{5} + x^{4} + 4 x^{3} - 4 x^{2} + 5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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