Optimal. Leaf size=48 \[ \frac{13}{8 x^2}-\frac{1}{6 x^3}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1) \]
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Rubi [A] time = 0.0423328, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1354, 709, 800} \[ \frac{13}{8 x^2}-\frac{1}{6 x^3}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1) \]
Antiderivative was successfully verified.
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Rule 1354
Rule 709
Rule 800
Rubi steps
\begin{align*} \int \frac{1}{\left (15+\frac{2}{x^2}+\frac{13}{x}\right ) x^6} \, dx &=\int \frac{1}{x^4 \left (2+13 x+15 x^2\right )} \, dx\\ &=-\frac{1}{6 x^3}+\frac{1}{2} \int \frac{-13-15 x}{x^3 \left (2+13 x+15 x^2\right )} \, dx\\ &=-\frac{1}{6 x^3}+\frac{1}{2} \int \left (-\frac{13}{2 x^3}+\frac{139}{4 x^2}-\frac{1417}{8 x}-\frac{243}{56 (2+3 x)}+\frac{6250}{7 (1+5 x)}\right ) \, dx\\ &=-\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (2+3 x)+\frac{625}{7} \log (1+5 x)\\ \end{align*}
Mathematica [A] time = 0.0047954, size = 48, normalized size = 1. \[ \frac{13}{8 x^2}-\frac{1}{6 x^3}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 37, normalized size = 0.8 \begin{align*} -{\frac{1}{6\,{x}^{3}}}+{\frac{13}{8\,{x}^{2}}}-{\frac{139}{8\,x}}-{\frac{1417\,\ln \left ( x \right ) }{16}}-{\frac{81\,\ln \left ( 2+3\,x \right ) }{112}}+{\frac{625\,\ln \left ( 1+5\,x \right ) }{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08354, size = 49, normalized size = 1.02 \begin{align*} -\frac{417 \, x^{2} - 39 \, x + 4}{24 \, x^{3}} + \frac{625}{7} \, \log \left (5 \, x + 1\right ) - \frac{81}{112} \, \log \left (3 \, x + 2\right ) - \frac{1417}{16} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75177, size = 138, normalized size = 2.88 \begin{align*} \frac{30000 \, x^{3} \log \left (5 \, x + 1\right ) - 243 \, x^{3} \log \left (3 \, x + 2\right ) - 29757 \, x^{3} \log \left (x\right ) - 5838 \, x^{2} + 546 \, x - 56}{336 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.245562, size = 41, normalized size = 0.85 \begin{align*} - \frac{1417 \log{\left (x \right )}}{16} + \frac{625 \log{\left (x + \frac{1}{5} \right )}}{7} - \frac{81 \log{\left (x + \frac{2}{3} \right )}}{112} - \frac{417 x^{2} - 39 x + 4}{24 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13612, size = 53, normalized size = 1.1 \begin{align*} -\frac{417 \, x^{2} - 39 \, x + 4}{24 \, x^{3}} + \frac{625}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac{81}{112} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{1417}{16} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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