Optimal. Leaf size=27 \[ 1-e^x+\frac {2}{25} x^2 \left (5+\frac {x^2}{4+x}\right )^2 \]
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Rubi [A] time = 0.16, antiderivative size = 50, normalized size of antiderivative = 1.85, number of steps used = 5, number of rules used = 3, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6688, 2194, 1620} \begin {gather*} \frac {2 x^4}{25}+\frac {4 x^3}{25}+\frac {66 x^2}{25}-\frac {192 x}{25}-e^x-\frac {7168}{25 (x+4)}+\frac {8192}{25 (x+4)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1620
Rule 2194
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^x+\frac {4 x \left (1600+1200 x+620 x^2+165 x^3+27 x^4+2 x^5\right )}{25 (4+x)^3}\right ) \, dx\\ &=\frac {4}{25} \int \frac {x \left (1600+1200 x+620 x^2+165 x^3+27 x^4+2 x^5\right )}{(4+x)^3} \, dx-\int e^x \, dx\\ &=-e^x+\frac {4}{25} \int \left (-48+33 x+3 x^2+2 x^3-\frac {4096}{(4+x)^3}+\frac {1792}{(4+x)^2}\right ) \, dx\\ &=-e^x-\frac {192 x}{25}+\frac {66 x^2}{25}+\frac {4 x^3}{25}+\frac {2 x^4}{25}+\frac {8192}{25 (4+x)^2}-\frac {7168}{25 (4+x)}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.03, size = 58, normalized size = 2.15 \begin {gather*} -e^x+\frac {8192}{25 (4+x)^2}-\frac {7168}{25 (4+x)}-\frac {208 (4+x)}{5}+\frac {42}{5} (4+x)^2-\frac {28}{25} (4+x)^3+\frac {2}{25} (4+x)^4 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 54, normalized size = 2.00 \begin {gather*} \frac {2 \, x^{6} + 20 \, x^{5} + 130 \, x^{4} + 400 \, x^{3} - 480 \, x^{2} - 25 \, {\left (x^{2} + 8 \, x + 16\right )} e^{x} - 10240 \, x - 20480}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 58, normalized size = 2.15 \begin {gather*} \frac {2 \, x^{6} + 20 \, x^{5} + 130 \, x^{4} + 400 \, x^{3} - 25 \, x^{2} e^{x} - 480 \, x^{2} - 200 \, x e^{x} - 10240 \, x - 400 \, e^{x} - 20480}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 38, normalized size = 1.41
method | result | size |
default | \(-\frac {7168}{25 \left (4+x \right )}+\frac {8192}{25 \left (4+x \right )^{2}}-\frac {192 x}{25}+\frac {66 x^{2}}{25}+\frac {4 x^{3}}{25}+\frac {2 x^{4}}{25}-{\mathrm e}^{x}\) | \(38\) |
risch | \(\frac {2 x^{4}}{25}+\frac {4 x^{3}}{25}+\frac {66 x^{2}}{25}-\frac {192 x}{25}+\frac {-\frac {7168 x}{25}-\frac {4096}{5}}{x^{2}+8 x +16}-{\mathrm e}^{x}\) | \(40\) |
norman | \(\frac {-256 x +16 x^{3}+\frac {26 x^{4}}{5}+\frac {4 x^{5}}{5}+\frac {2 x^{6}}{25}-8 \,{\mathrm e}^{x} x -{\mathrm e}^{x} x^{2}-16 \,{\mathrm e}^{x}-512}{\left (4+x \right )^{2}}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2}{25} \, x^{4} + \frac {4}{25} \, x^{3} + \frac {66}{25} \, x^{2} - \frac {192}{25} \, x - \frac {{\left (x^{3} + 12 \, x^{2} + 48 \, x\right )} e^{x}}{x^{3} + 12 \, x^{2} + 48 \, x + 64} - \frac {27648 \, {\left (5 \, x + 18\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} + \frac {16384 \, {\left (3 \, x + 11\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} - \frac {7936 \, {\left (3 \, x + 10\right )}}{5 \, {\left (x^{2} + 8 \, x + 16\right )}} + \frac {16896 \, {\left (2 \, x + 7\right )}}{5 \, {\left (x^{2} + 8 \, x + 16\right )}} + \frac {1536 \, {\left (x + 3\right )}}{x^{2} + 8 \, x + 16} - \frac {256 \, {\left (x + 2\right )}}{x^{2} + 8 \, x + 16} + \frac {64 \, e^{\left (-4\right )} E_{3}\left (-x - 4\right )}{{\left (x + 4\right )}^{2}} + 192 \, \int \frac {e^{x}}{x^{4} + 16 \, x^{3} + 96 \, x^{2} + 256 \, x + 256}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 35, normalized size = 1.30 \begin {gather*} \frac {66\,x^2}{25}-{\mathrm {e}}^x-\frac {\frac {7168\,x}{25}+\frac {4096}{5}}{{\left (x+4\right )}^2}-\frac {192\,x}{25}+\frac {4\,x^3}{25}+\frac {2\,x^4}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 44, normalized size = 1.63 \begin {gather*} \frac {2 x^{4}}{25} + \frac {4 x^{3}}{25} + \frac {66 x^{2}}{25} - \frac {192 x}{25} + \frac {- 7168 x - 20480}{25 x^{2} + 200 x + 400} - e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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