Optimal. Leaf size=66 \[ \log (x+1) (d-e+f-g+h)-\log (x+2) (d-2 e+4 f-8 g+16 h)+x (f-3 g+7 h)+\frac{1}{2} x^2 (g-3 h)+\frac{h x^3}{3} \]
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Rubi [A] time = 0.0853165, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {1586, 1657, 632, 31} \[ \log (x+1) (d-e+f-g+h)-\log (x+2) (d-2 e+4 f-8 g+16 h)+x (f-3 g+7 h)+\frac{1}{2} x^2 (g-3 h)+\frac{h x^3}{3} \]
Antiderivative was successfully verified.
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Rule 1586
Rule 1657
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{4-5 x^2+x^4} \, dx &=\int \frac{d+e x+f x^2+g x^3+h x^4}{2+3 x+x^2} \, dx\\ &=\int \left (f-3 g+7 h+(g-3 h) x+h x^2+\frac{d-2 f+6 g-14 h+(e-3 f+7 g-15 h) x}{2+3 x+x^2}\right ) \, dx\\ &=(f-3 g+7 h) x+\frac{1}{2} (g-3 h) x^2+\frac{h x^3}{3}+\int \frac{d-2 f+6 g-14 h+(e-3 f+7 g-15 h) x}{2+3 x+x^2} \, dx\\ &=(f-3 g+7 h) x+\frac{1}{2} (g-3 h) x^2+\frac{h x^3}{3}+(d-e+f-g+h) \int \frac{1}{1+x} \, dx-(d-2 e+4 f-8 g+16 h) \int \frac{1}{2+x} \, dx\\ &=(f-3 g+7 h) x+\frac{1}{2} (g-3 h) x^2+\frac{h x^3}{3}+(d-e+f-g+h) \log (1+x)-(d-2 e+4 f-8 g+16 h) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.024024, size = 67, normalized size = 1.02 \[ \log (x+1) (d-e+f-g+h)+\log (x+2) (-d+2 e-4 f+8 g-16 h)+x (f-3 g+7 h)+\frac{1}{2} x^2 (g-3 h)+\frac{h x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 98, normalized size = 1.5 \begin{align*}{\frac{h{x}^{3}}{3}}+{\frac{g{x}^{2}}{2}}-{\frac{3\,h{x}^{2}}{2}}+fx-3\,gx+7\,hx-\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-16\,\ln \left ( 2+x \right ) h+\ln \left ( 1+x \right ) d-\ln \left ( 1+x \right ) e+\ln \left ( 1+x \right ) f-\ln \left ( 1+x \right ) g+\ln \left ( 1+x \right ) h \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973296, size = 84, normalized size = 1.27 \begin{align*} \frac{1}{3} \, h x^{3} + \frac{1}{2} \,{\left (g - 3 \, h\right )} x^{2} +{\left (f - 3 \, g + 7 \, h\right )} x -{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h\right )} \log \left (x + 2\right ) +{\left (d - e + f - g + h\right )} \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51943, size = 170, normalized size = 2.58 \begin{align*} \frac{1}{3} \, h x^{3} + \frac{1}{2} \,{\left (g - 3 \, h\right )} x^{2} +{\left (f - 3 \, g + 7 \, h\right )} x -{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h\right )} \log \left (x + 2\right ) +{\left (d - e + f - g + h\right )} \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.50584, size = 94, normalized size = 1.42 \begin{align*} \frac{h x^{3}}{3} + x^{2} \left (\frac{g}{2} - \frac{3 h}{2}\right ) + x \left (f - 3 g + 7 h\right ) + \left (- d + 2 e - 4 f + 8 g - 16 h\right ) \log{\left (x + \frac{4 d - 6 e + 10 f - 18 g + 34 h}{2 d - 3 e + 5 f - 9 g + 17 h} \right )} + \left (d - e + f - g + h\right ) \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08079, size = 93, normalized size = 1.41 \begin{align*} \frac{1}{3} \, h x^{3} + \frac{1}{2} \, g x^{2} - \frac{3}{2} \, h x^{2} + f x - 3 \, g x + 7 \, h x -{\left (d + 4 \, f - 8 \, g + 16 \, h - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) +{\left (d + f - g + h - e\right )} \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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