Optimal. Leaf size=16 \[ \frac {9 x (7+x)}{2 e (2+x)^4} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(35\) vs. \(2(16)=32\).
time = 0.05, antiderivative size = 35, normalized size of antiderivative = 2.19, 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 = {12, 2099}
\begin {gather*} \frac {9}{2 e (x+2)^2}+\frac {27}{2 e (x+2)^3}-\frac {45}{e (x+2)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {126-153 x-18 x^2}{64+160 x+160 x^2+80 x^3+20 x^4+2 x^5} \, dx}{e}\\ &=\frac {\int \left (\frac {180}{(2+x)^5}-\frac {81}{2 (2+x)^4}-\frac {9}{(2+x)^3}\right ) \, dx}{e}\\ &=-\frac {45}{e (2+x)^4}+\frac {27}{2 e (2+x)^3}+\frac {9}{2 e (2+x)^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {9 x (7+x)}{2 e (2+x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 27, normalized size = 1.69
method | result | size |
norman | \(\frac {\frac {9 x^{2} {\mathrm e}^{-1}}{2}+\frac {63 \,{\mathrm e}^{-1} x}{2}}{\left (2+x \right )^{4}}\) | \(24\) |
default | \(\frac {9 \,{\mathrm e}^{-1} \left (-\frac {10}{\left (2+x \right )^{4}}+\frac {1}{\left (2+x \right )^{2}}+\frac {3}{\left (2+x \right )^{3}}\right )}{2}\) | \(27\) |
gosper | \(\frac {9 \left (x +7\right ) x \,{\mathrm e}^{-1}}{2 \left (x^{4}+8 x^{3}+24 x^{2}+32 x +16\right )}\) | \(31\) |
risch | \(\frac {{\mathrm e}^{-1} \left (\frac {9}{2} x^{2}+\frac {63}{2} x \right )}{x^{4}+8 x^{3}+24 x^{2}+32 x +16}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (13) = 26\).
time = 0.26, size = 31, normalized size = 1.94 \begin {gather*} \frac {9 \, {\left (x^{2} + 7 \, x\right )} e^{\left (-1\right )}}{2 \, {\left (x^{4} + 8 \, x^{3} + 24 \, x^{2} + 32 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (13) = 26\).
time = 0.36, size = 31, normalized size = 1.94 \begin {gather*} \frac {9 \, {\left (x^{2} + 7 \, x\right )} e^{\left (-1\right )}}{2 \, {\left (x^{4} + 8 \, x^{3} + 24 \, x^{2} + 32 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (15) = 30\).
time = 0.12, size = 48, normalized size = 3.00 \begin {gather*} - \frac {- 9 x^{2} - 63 x}{2 e x^{4} + 16 e x^{3} + 48 e x^{2} + 64 e x + 32 e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 16, normalized size = 1.00 \begin {gather*} \frac {9 \, {\left (x^{2} + 7 \, x\right )} e^{\left (-1\right )}}{2 \, {\left (x + 2\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 28, normalized size = 1.75 \begin {gather*} \frac {9\,{\mathrm {e}}^{-1}}{2\,{\left (x+2\right )}^2}+\frac {27\,{\mathrm {e}}^{-1}}{2\,{\left (x+2\right )}^3}-\frac {45\,{\mathrm {e}}^{-1}}{{\left (x+2\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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