Optimal. Leaf size=36 \[ \frac{7 x+13}{9 \left (x^2+5 x+4\right )}+\frac{7}{27} \log (x+1)-\frac{7}{27} \log (x+4) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0079996, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {638, 616, 31} \[ \frac{7 x+13}{9 \left (x^2+5 x+4\right )}+\frac{7}{27} \log (x+1)-\frac{7}{27} \log (x+4) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 638
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{-1+x}{\left (4+5 x+x^2\right )^2} \, dx &=\frac{13+7 x}{9 \left (4+5 x+x^2\right )}+\frac{7}{9} \int \frac{1}{4+5 x+x^2} \, dx\\ &=\frac{13+7 x}{9 \left (4+5 x+x^2\right )}+\frac{7}{27} \int \frac{1}{1+x} \, dx-\frac{7}{27} \int \frac{1}{4+x} \, dx\\ &=\frac{13+7 x}{9 \left (4+5 x+x^2\right )}+\frac{7}{27} \log (1+x)-\frac{7}{27} \log (4+x)\\ \end{align*}
Mathematica [A] time = 0.0172021, size = 33, normalized size = 0.92 \[ \frac{1}{27} \left (\frac{21 x+39}{x^2+5 x+4}+7 \log (x+1)-7 \log (x+4)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 28, normalized size = 0.8 \begin{align*}{\frac{2}{9+9\,x}}+{\frac{7\,\ln \left ( 1+x \right ) }{27}}+{\frac{5}{36+9\,x}}-{\frac{7\,\ln \left ( 4+x \right ) }{27}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.923678, size = 41, normalized size = 1.14 \begin{align*} \frac{7 \, x + 13}{9 \,{\left (x^{2} + 5 \, x + 4\right )}} - \frac{7}{27} \, \log \left (x + 4\right ) + \frac{7}{27} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.72885, size = 131, normalized size = 3.64 \begin{align*} -\frac{7 \,{\left (x^{2} + 5 \, x + 4\right )} \log \left (x + 4\right ) - 7 \,{\left (x^{2} + 5 \, x + 4\right )} \log \left (x + 1\right ) - 21 \, x - 39}{27 \,{\left (x^{2} + 5 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.113766, size = 31, normalized size = 0.86 \begin{align*} \frac{7 x + 13}{9 x^{2} + 45 x + 36} + \frac{7 \log{\left (x + 1 \right )}}{27} - \frac{7 \log{\left (x + 4 \right )}}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0542, size = 43, normalized size = 1.19 \begin{align*} \frac{7 \, x + 13}{9 \,{\left (x^{2} + 5 \, x + 4\right )}} - \frac{7}{27} \, \log \left ({\left | x + 4 \right |}\right ) + \frac{7}{27} \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]