Optimal. Leaf size=40 \[ \frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375} \]
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Rubi [A] time = 0.020967, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1386, 701, 632, 31} \[ \frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375} \]
Antiderivative was successfully verified.
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Rule 1386
Rule 701
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x^3}{13+\frac{2}{x}+15 x} \, 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*}
Mathematica [A] time = 0.0051427, size = 40, normalized size = 1. \[ \frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 31, normalized size = 0.8 \begin{align*}{\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 ( 1+5\,x \right ) }{4375}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2908, size = 41, normalized size = 1.02 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18517, size = 108, normalized size = 2.7 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.10749, size = 34, normalized size = 0.85 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13038, size = 43, normalized size = 1.08 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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