Optimal. Leaf size=40 \[ \frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}-\frac {16}{567} \log (2+3 x)+\frac {\log (1+5 x)}{4375} \]
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Rubi [A]
time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1599, 715, 646,
31} \begin {gather*} \frac {x^3}{45}-\frac {13 x^2}{450}+\frac {139 x}{3375}-\frac {16}{567} \log (3 x+2)+\frac {\log (5 x+1)}{4375} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 715
Rule 1599
Rubi steps
\begin {align*} \int \frac {x^5}{2 x+13 x^2+15 x^3} \, dx &=\int \frac {x^4}{2+13 x+15 x^2} \, dx\\ &=\int \left (\frac {139}{3375}-\frac {13 x}{225}+\frac {x^2}{15}-\frac {278+1417 x}{3375 \left (2+13 x+15 x^2\right )}\right ) \, dx\\ &=\frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}-\frac {\int \frac {278+1417 x}{2+13 x+15 x^2} \, dx}{3375}\\ &=\frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}+\frac {3}{875} \int \frac {1}{3+15 x} \, dx-\frac {80}{189} \int \frac {1}{10+15 x} \, dx\\ &=\frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}-\frac {16}{567} \log (2+3 x)+\frac {\log (1+5 x)}{4375}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 40, normalized size = 1.00 \begin {gather*} \frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}-\frac {16}{567} \log (2+3 x)+\frac {\log (1+5 x)}{4375} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 31, normalized size = 0.78
method | result | size |
default | \(\frac {139 x}{3375}-\frac {13 x^{2}}{450}+\frac {x^{3}}{45}-\frac {16 \ln \left (2+3 x \right )}{567}+\frac {\ln \left (5 x +1\right )}{4375}\) | \(31\) |
norman | \(\frac {139 x}{3375}-\frac {13 x^{2}}{450}+\frac {x^{3}}{45}-\frac {16 \ln \left (2+3 x \right )}{567}+\frac {\ln \left (5 x +1\right )}{4375}\) | \(31\) |
risch | \(\frac {139 x}{3375}-\frac {13 x^{2}}{450}+\frac {x^{3}}{45}-\frac {16 \ln \left (2+3 x \right )}{567}+\frac {\ln \left (5 x +1\right )}{4375}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 30, normalized size = 0.75 \begin {gather*} \frac {1}{45} \, x^{3} - \frac {13}{450} \, x^{2} + \frac {139}{3375} \, x + \frac {1}{4375} \, \log \left (5 \, x + 1\right ) - \frac {16}{567} \, \log \left (3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.36, size = 30, normalized size = 0.75 \begin {gather*} \frac {1}{45} \, x^{3} - \frac {13}{450} \, x^{2} + \frac {139}{3375} \, x + \frac {1}{4375} \, \log \left (5 \, x + 1\right ) - \frac {16}{567} \, \log \left (3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 34, normalized size = 0.85 \begin {gather*} \frac {x^{3}}{45} - \frac {13 x^{2}}{450} + \frac {139 x}{3375} + \frac {\log {\left (x + \frac {1}{5} \right )}}{4375} - \frac {16 \log {\left (x + \frac {2}{3} \right )}}{567} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 32, normalized size = 0.80 \begin {gather*} \frac {1}{45} \, x^{3} - \frac {13}{450} \, x^{2} + \frac {139}{3375} \, x + \frac {1}{4375} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac {16}{567} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 26, normalized size = 0.65 \begin {gather*} \frac {139\,x}{3375}-\frac {16\,\ln \left (x+\frac {2}{3}\right )}{567}+\frac {\ln \left (x+\frac {1}{5}\right )}{4375}-\frac {13\,x^2}{450}+\frac {x^3}{45} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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