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 = 0.09, antiderivative size = 66, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 1586
Rule 1657
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*}
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Mathematica [A] time = 0.02, size = 67, normalized size = 1.02 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.92, size = 62, normalized size = 0.94 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 69, normalized size = 1.05 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 98, normalized size = 1.48 \begin {gather*} \frac {h \,x^{3}}{3}+\frac {g \,x^{2}}{2}-\frac {3 h \,x^{2}}{2}-d \ln \left (x +2\right )+d \ln \left (x +1\right )+2 e \ln \left (x +2\right )-e \ln \left (x +1\right )+f x -4 f \ln \left (x +2\right )+f \ln \left (x +1\right )-3 g x +8 g \ln \left (x +2\right )-g \ln \left (x +1\right )+7 h x -16 h \ln \left (x +2\right )+h \ln \left (x +1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 62, normalized size = 0.94 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 63, normalized size = 0.95 \begin {gather*} x^2\,\left (\frac {g}{2}-\frac {3\,h}{2}\right )+x\,\left (f-3\,g+7\,h\right )-\ln \left (x+2\right )\,\left (d-2\,e+4\,f-8\,g+16\,h\right )+\frac {h\,x^3}{3}+\ln \left (x+1\right )\,\left (d-e+f-g+h\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.53, size = 94, normalized size = 1.42 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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