Optimal. Leaf size=29 \[ \frac{x^2}{2}+6 x+\frac{169}{4} \log (7-x)-\frac{1}{4} \log (x+1) \]
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Rubi [A] time = 0.0165118, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1657, 632, 31} \[ \frac{x^2}{2}+6 x+\frac{169}{4} \log (7-x)-\frac{1}{4} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 1657
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{2-x+x^3}{-7-6 x+x^2} \, dx &=\int \left (6+x+\frac{2 (22+21 x)}{-7-6 x+x^2}\right ) \, dx\\ &=6 x+\frac{x^2}{2}+2 \int \frac{22+21 x}{-7-6 x+x^2} \, dx\\ &=6 x+\frac{x^2}{2}-\frac{1}{4} \int \frac{1}{1+x} \, dx+\frac{169}{4} \int \frac{1}{-7+x} \, dx\\ &=6 x+\frac{x^2}{2}+\frac{169}{4} \log (7-x)-\frac{1}{4} \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0058195, size = 29, normalized size = 1. \[ \frac{x^2}{2}+6 x+\frac{169}{4} \log (7-x)-\frac{1}{4} \log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 22, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+6\,x+{\frac{169\,\ln \left ( x-7 \right ) }{4}}-{\frac{\ln \left ( 1+x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07431, size = 28, normalized size = 0.97 \begin{align*} \frac{1}{2} \, x^{2} + 6 \, x - \frac{1}{4} \, \log \left (x + 1\right ) + \frac{169}{4} \, \log \left (x - 7\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49636, size = 69, normalized size = 2.38 \begin{align*} \frac{1}{2} \, x^{2} + 6 \, x - \frac{1}{4} \, \log \left (x + 1\right ) + \frac{169}{4} \, \log \left (x - 7\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.101816, size = 22, normalized size = 0.76 \begin{align*} \frac{x^{2}}{2} + 6 x + \frac{169 \log{\left (x - 7 \right )}}{4} - \frac{\log{\left (x + 1 \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13385, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{2} \, x^{2} + 6 \, x - \frac{1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{169}{4} \, \log \left ({\left | x - 7 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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