Optimal. Leaf size=20 \[ x^4 (-1+\log (x)) \left (-4-e+x+4 \log ^2(x)\right ) \]
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Rubi [B] time = 0.16, antiderivative size = 75, normalized size of antiderivative = 3.75, number of steps used = 12, number of rules used = 6, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {6, 1593, 43, 2334, 2305, 2304} \begin {gather*} -x^5+\frac {1}{4} (6+e) x^4+\frac {3}{4} (4+e) x^4-\frac {x^4}{2}+4 x^4 \log ^3(x)-4 x^4 \log ^2(x)+2 x^4 \log (x)-\left ((6+e) x^4-x^5\right ) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 43
Rule 1593
Rule 2304
Rule 2305
Rule 2334
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left ((12+3 e) x^3-4 x^4+\left (-24 x^3-4 e x^3+5 x^4\right ) \log (x)-4 x^3 \log ^2(x)+16 x^3 \log ^3(x)\right ) \, dx\\ &=\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}-4 \int x^3 \log ^2(x) \, dx+16 \int x^3 \log ^3(x) \, dx+\int \left (-24 x^3-4 e x^3+5 x^4\right ) \log (x) \, dx\\ &=\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}-x^4 \log ^2(x)+4 x^4 \log ^3(x)+2 \int x^3 \log (x) \, dx-12 \int x^3 \log ^2(x) \, dx+\int \left ((-24-4 e) x^3+5 x^4\right ) \log (x) \, dx\\ &=-\frac {x^4}{8}+\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}+\frac {1}{2} x^4 \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x)+6 \int x^3 \log (x) \, dx+\int x^3 (-24-4 e+5 x) \log (x) \, dx\\ &=-\frac {x^4}{2}+\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}+2 x^4 \log (x)-\left ((6+e) x^4-x^5\right ) \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x)-\int x^3 (-6-e+x) \, dx\\ &=-\frac {x^4}{2}+\frac {3}{4} (4+e) x^4-\frac {4 x^5}{5}+2 x^4 \log (x)-\left ((6+e) x^4-x^5\right ) \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x)-\int \left (-\left ((6+e) x^3\right )+x^4\right ) \, dx\\ &=-\frac {x^4}{2}+\frac {3}{4} (4+e) x^4+\frac {1}{4} (6+e) x^4-x^5+2 x^4 \log (x)-\left ((6+e) x^4-x^5\right ) \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x)\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.01, size = 55, normalized size = 2.75 \begin {gather*} 4 x^4+e x^4-x^5-4 x^4 \log (x)-e x^4 \log (x)+x^5 \log (x)-4 x^4 \log ^2(x)+4 x^4 \log ^3(x) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 54, normalized size = 2.70 \begin {gather*} 4 \, x^{4} \log \relax (x)^{3} - 4 \, x^{4} \log \relax (x)^{2} - x^{5} + x^{4} e + 4 \, x^{4} + {\left (x^{5} - x^{4} e - 4 \, x^{4}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 57, normalized size = 2.85 \begin {gather*} 4 \, x^{4} \log \relax (x)^{3} + x^{5} \log \relax (x) - x^{4} e \log \relax (x) - 4 \, x^{4} \log \relax (x)^{2} - x^{5} + x^{4} e - 4 \, x^{4} \log \relax (x) + 4 \, x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 51, normalized size = 2.55
method | result | size |
norman | \(x^{5} \ln \relax (x )+\left ({\mathrm e}+4\right ) x^{4}+\left (-{\mathrm e}-4\right ) x^{4} \ln \relax (x )-x^{5}-4 x^{4} \ln \relax (x )^{2}+4 x^{4} \ln \relax (x )^{3}\) | \(51\) |
default | \(-{\mathrm e} \ln \relax (x ) x^{4}+x^{4} {\mathrm e}+x^{5} \ln \relax (x )-x^{5}-4 x^{4} \ln \relax (x )+4 x^{4}-4 x^{4} \ln \relax (x )^{2}+4 x^{4} \ln \relax (x )^{3}\) | \(58\) |
risch | \(-{\mathrm e} \ln \relax (x ) x^{4}+x^{4} {\mathrm e}+x^{5} \ln \relax (x )-x^{5}-4 x^{4} \ln \relax (x )+4 x^{4}-4 x^{4} \ln \relax (x )^{2}+4 x^{4} \ln \relax (x )^{3}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 86, normalized size = 4.30 \begin {gather*} \frac {1}{8} \, {\left (32 \, \log \relax (x)^{3} - 24 \, \log \relax (x)^{2} + 12 \, \log \relax (x) - 3\right )} x^{4} - \frac {1}{8} \, {\left (8 \, \log \relax (x)^{2} - 4 \, \log \relax (x) + 1\right )} x^{4} - x^{5} + \frac {1}{4} \, x^{4} {\left (e + 6\right )} + \frac {3}{4} \, x^{4} e + 3 \, x^{4} + {\left (x^{5} - x^{4} e - 6 \, x^{4}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 21, normalized size = 1.05 \begin {gather*} x^4\,\left (\ln \relax (x)-1\right )\,\left (4\,{\ln \relax (x)}^2+x-\mathrm {e}-4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 49, normalized size = 2.45 \begin {gather*} - x^{5} + 4 x^{4} \log {\relax (x )}^{3} - 4 x^{4} \log {\relax (x )}^{2} + x^{4} \left (e + 4\right ) + \left (x^{5} - 4 x^{4} - e x^{4}\right ) \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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