Optimal. Leaf size=51 \[ \log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac{1}{2} (x+2)^2 (f-6 g)+\frac{1}{3} g (x+2)^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0847926, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {1586, 1850} \[ \log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac{1}{2} (x+2)^2 (f-6 g)+\frac{1}{3} g (x+2)^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1586
Rule 1850
Rubi steps
\begin{align*} \int \frac{\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx &=\int \frac{d+e x+f x^2+g x^3}{2+x} \, dx\\ &=\int \left (e-4 f+12 g+\frac{d-2 e+4 f-8 g}{2+x}+(f-6 g) (2+x)+g (2+x)^2\right ) \, dx\\ &=(e-4 f+12 g) x+\frac{1}{2} (f-6 g) (2+x)^2+\frac{1}{3} g (2+x)^3+(d-2 e+4 f-8 g) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0265888, size = 45, normalized size = 0.88 \[ \log (x+2) (d-2 e+4 f-8 g)+\frac{1}{6} (x+2) \left (6 e+3 f (x-6)+2 g \left (x^2-5 x+22\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 58, normalized size = 1.1 \begin{align*}{\frac{g{x}^{3}}{3}}+{\frac{f{x}^{2}}{2}}-g{x}^{2}+ex-2\,fx+4\,gx+\ln \left ( 2+x \right ) d-2\,\ln \left ( 2+x \right ) e+4\,\ln \left ( 2+x \right ) f-8\,\ln \left ( 2+x \right ) g \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15627, size = 58, normalized size = 1.14 \begin{align*} \frac{1}{3} \, g x^{3} + \frac{1}{2} \,{\left (f - 2 \, g\right )} x^{2} +{\left (e - 2 \, f + 4 \, g\right )} x +{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.46849, size = 116, normalized size = 2.27 \begin{align*} \frac{1}{3} \, g x^{3} + \frac{1}{2} \,{\left (f - 2 \, g\right )} x^{2} +{\left (e - 2 \, f + 4 \, g\right )} x +{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.317787, size = 41, normalized size = 0.8 \begin{align*} \frac{g x^{3}}{3} + x^{2} \left (\frac{f}{2} - g\right ) + x \left (e - 2 f + 4 g\right ) + \left (d - 2 e + 4 f - 8 g\right ) \log{\left (x + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.08436, size = 66, normalized size = 1.29 \begin{align*} \frac{1}{3} \, g x^{3} + \frac{1}{2} \, f x^{2} - g x^{2} - 2 \, f x + 4 \, g x + x e +{\left (d + 4 \, f - 8 \, g - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]