Optimal. Leaf size=33 \[ \frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1) \]
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Rubi [A] time = 0.0203888, antiderivative size = 33, 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} \[ \frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1) \]
Antiderivative was successfully verified.
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Rule 1386
Rule 701
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2}{13+\frac{2}{x}+15 x} \, dx &=\int \frac{x^3}{2+13 x+15 x^2} \, dx\\ &=\int \left (-\frac{13}{225}+\frac{x}{15}+\frac{26+139 x}{225 \left (2+13 x+15 x^2\right )}\right ) \, dx\\ &=-\frac{13 x}{225}+\frac{x^2}{30}+\frac{1}{225} \int \frac{26+139 x}{2+13 x+15 x^2} \, dx\\ &=-\frac{13 x}{225}+\frac{x^2}{30}-\frac{3}{175} \int \frac{1}{3+15 x} \, dx+\frac{40}{63} \int \frac{1}{10+15 x} \, dx\\ &=-\frac{13 x}{225}+\frac{x^2}{30}+\frac{8}{189} \log (2+3 x)-\frac{1}{875} \log (1+5 x)\\ \end{align*}
Mathematica [A] time = 0.0040892, size = 33, normalized size = 1. \[ \frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 26, normalized size = 0.8 \begin{align*} -{\frac{13\,x}{225}}+{\frac{{x}^{2}}{30}}+{\frac{8\,\ln \left ( 2+3\,x \right ) }{189}}-{\frac{\ln \left ( 1+5\,x \right ) }{875}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02046, size = 34, normalized size = 1.03 \begin{align*} \frac{1}{30} \, x^{2} - \frac{13}{225} \, x - \frac{1}{875} \, \log \left (5 \, x + 1\right ) + \frac{8}{189} \, \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 = 1.24822, size = 85, normalized size = 2.58 \begin{align*} \frac{1}{30} \, x^{2} - \frac{13}{225} \, x - \frac{1}{875} \, \log \left (5 \, x + 1\right ) + \frac{8}{189} \, \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 = 0.106369, size = 27, normalized size = 0.82 \begin{align*} \frac{x^{2}}{30} - \frac{13 x}{225} - \frac{\log{\left (x + \frac{1}{5} \right )}}{875} + \frac{8 \log{\left (x + \frac{2}{3} \right )}}{189} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24597, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{30} \, x^{2} - \frac{13}{225} \, x - \frac{1}{875} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) + \frac{8}{189} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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