Optimal. Leaf size=105 \[ -\frac {d-e}{36 (x+1)}+\frac {d+e}{12 (1-x)}+\frac {d+2 e}{36 (2-x)}+\frac {1}{36} (2 d+5 e) \log (1-x)-\frac {1}{432} (35 d+58 e) \log (2-x)+\frac {1}{108} (2 d+e) \log (x+1)+\frac {1}{144} (d-2 e) \log (x+2) \]
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Rubi [A] time = 0.20, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1586, 6742} \begin {gather*} -\frac {d-e}{36 (x+1)}+\frac {d+e}{12 (1-x)}+\frac {d+2 e}{36 (2-x)}+\frac {1}{36} (2 d+5 e) \log (1-x)-\frac {1}{432} (35 d+58 e) \log (2-x)+\frac {1}{108} (2 d+e) \log (x+1)+\frac {1}{144} (d-2 e) \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 1586
Rule 6742
Rubi steps
\begin {align*} \int \frac {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {d+e x}{(2+x) \left (2-x-2 x^2+x^3\right )^2} \, dx\\ &=\int \left (\frac {d+2 e}{36 (-2+x)^2}+\frac {-35 d-58 e}{432 (-2+x)}+\frac {d+e}{12 (-1+x)^2}+\frac {2 d+5 e}{36 (-1+x)}+\frac {d-e}{36 (1+x)^2}+\frac {2 d+e}{108 (1+x)}+\frac {d-2 e}{144 (2+x)}\right ) \, dx\\ &=\frac {d+e}{12 (1-x)}+\frac {d+2 e}{36 (2-x)}-\frac {d-e}{36 (1+x)}+\frac {1}{36} (2 d+5 e) \log (1-x)-\frac {1}{432} (35 d+58 e) \log (2-x)+\frac {1}{108} (2 d+e) \log (1+x)+\frac {1}{144} (d-2 e) \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.09, size = 97, normalized size = 0.92 \begin {gather*} \frac {1}{432} \left (\frac {12 \left (d \left (-5 x^2+6 x+5\right )+2 e \left (5-2 x^2\right )\right )}{x^3-2 x^2-x+2}+12 (2 d+5 e) \log (1-x)-(35 d+58 e) \log (2-x)+4 (2 d+e) \log (x+1)+3 (d-2 e) \log (x+2)\right ) \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 {(2+x) (d+e x)}{\left (4-5 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.29, size = 211, normalized size = 2.01 \begin {gather*} -\frac {12 \, {\left (5 \, d + 4 \, e\right )} x^{2} - 72 \, d x - 3 \, {\left ({\left (d - 2 \, e\right )} x^{3} - 2 \, {\left (d - 2 \, e\right )} x^{2} - {\left (d - 2 \, e\right )} x + 2 \, d - 4 \, e\right )} \log \left (x + 2\right ) - 4 \, {\left ({\left (2 \, d + e\right )} x^{3} - 2 \, {\left (2 \, d + e\right )} x^{2} - {\left (2 \, d + e\right )} x + 4 \, d + 2 \, e\right )} \log \left (x + 1\right ) - 12 \, {\left ({\left (2 \, d + 5 \, e\right )} x^{3} - 2 \, {\left (2 \, d + 5 \, e\right )} x^{2} - {\left (2 \, d + 5 \, e\right )} x + 4 \, d + 10 \, e\right )} \log \left (x - 1\right ) + {\left ({\left (35 \, d + 58 \, e\right )} x^{3} - 2 \, {\left (35 \, d + 58 \, e\right )} x^{2} - {\left (35 \, d + 58 \, e\right )} x + 70 \, d + 116 \, e\right )} \log \left (x - 2\right ) - 60 \, d - 120 \, e}{432 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 98, normalized size = 0.93 \begin {gather*} \frac {1}{144} \, {\left (d - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{108} \, {\left (2 \, d + e\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{36} \, {\left (2 \, d + 5 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{432} \, {\left (35 \, d + 58 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {{\left (5 \, d + 4 \, e\right )} x^{2} - 6 \, d x - 5 \, d - 10 \, e}{36 \, {\left (x + 1\right )} {\left (x - 1\right )} {\left (x - 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 106, normalized size = 1.01 \begin {gather*} \frac {d \ln \left (x +2\right )}{144}-\frac {35 d \ln \left (x -2\right )}{432}+\frac {d \ln \left (x -1\right )}{18}+\frac {d \ln \left (x +1\right )}{54}-\frac {e \ln \left (x +2\right )}{72}-\frac {29 e \ln \left (x -2\right )}{216}+\frac {5 e \ln \left (x -1\right )}{36}+\frac {e \ln \left (x +1\right )}{108}-\frac {d}{36 \left (x -2\right )}-\frac {d}{36 \left (x +1\right )}-\frac {d}{12 \left (x -1\right )}-\frac {e}{18 \left (x -2\right )}+\frac {e}{36 x +36}-\frac {e}{12 \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 = 88, normalized size = 0.84 \begin {gather*} \frac {1}{144} \, {\left (d - 2 \, e\right )} \log \left (x + 2\right ) + \frac {1}{108} \, {\left (2 \, d + e\right )} \log \left (x + 1\right ) + \frac {1}{36} \, {\left (2 \, d + 5 \, e\right )} \log \left (x - 1\right ) - \frac {1}{432} \, {\left (35 \, d + 58 \, e\right )} \log \left (x - 2\right ) - \frac {{\left (5 \, d + 4 \, e\right )} x^{2} - 6 \, d x - 5 \, d - 10 \, e}{36 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 90, normalized size = 0.86 \begin {gather*} \ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {5\,e}{36}\right )-\frac {\left (-\frac {5\,d}{36}-\frac {e}{9}\right )\,x^2+\frac {d\,x}{6}+\frac {5\,d}{36}+\frac {5\,e}{18}}{-x^3+2\,x^2+x-2}+\ln \left (x+1\right )\,\left (\frac {d}{54}+\frac {e}{108}\right )+\ln \left (x+2\right )\,\left (\frac {d}{144}-\frac {e}{72}\right )-\ln \left (x-2\right )\,\left (\frac {35\,d}{432}+\frac {29\,e}{216}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.79, size = 1034, normalized size = 9.85
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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