Optimal. Leaf size=40 \[ \log \left (x^3+2 x^2+\sqrt {x^6+4 x^5+4 x^4+2 x^3+4 x^2-5}+1\right ) \]
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Rubi [F] time = 0.42, 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 {4 x+3 x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {4 x+3 x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx &=\int \frac {x (4+3 x)}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx\\ &=\int \left (\frac {4 x}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}}+\frac {3 x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}}\right ) \, dx\\ &=3 \int \frac {x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx+4 \int \frac {x}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 x+3 x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.17, size = 40, normalized size = 1.00 \begin {gather*} \log \left (1+2 x^2+x^3+\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 38, normalized size = 0.95 \begin {gather*} \log \left (x^{3} + 2 \, x^{2} + \sqrt {x^{6} + 4 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} + 4 \, x^{2} - 5} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 41, normalized size = 1.02 \begin {gather*} -\log \left ({\left | -x^{3} - 2 \, x^{2} + \sqrt {2 \, x^{3} + {\left (x^{3} + 2 \, x^{2}\right )}^{2} + 4 \, x^{2} - 5} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 39, normalized size = 0.98
method | result | size |
trager | \(\ln \left (1+2 x^{2}+x^{3}+\sqrt {x^{6}+4 x^{5}+4 x^{4}+2 x^{3}+4 x^{2}-5}\right )\) | \(39\) |
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} + 4 \, x}{\sqrt {x^{6} + 4 \, x^{5} + 4 \, x^{4} + 2 \, 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 = -0.02 \begin {gather*} \int \frac {3\,x^2+4\,x}{\sqrt {x^6+4\,x^5+4\,x^4+2\,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 + 4\right )}{\sqrt {x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} - 5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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