Optimal. Leaf size=24 \[ -1+e^5-x-\frac {2 e^{-2 x}}{\left (1+\frac {5 x}{2}\right )^2} \]
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Rubi [A]
time = 0.22, antiderivative size = 18, normalized size of antiderivative = 0.75, number of steps
used = 3, number of rules used = 2, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6820, 2228}
\begin {gather*} -x-\frac {8 e^{-2 x}}{(5 x+2)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2228
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {16 e^{-2 x} (7+5 x)}{(2+5 x)^3}\right ) \, dx\\ &=-x+16 \int \frac {e^{-2 x} (7+5 x)}{(2+5 x)^3} \, dx\\ &=-x-\frac {8 e^{-2 x}}{(2+5 x)^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.32, size = 28, normalized size = 1.17 \begin {gather*} -x-\frac {8 e^{\frac {4}{5}-\frac {2}{5} (2+5 x)}}{(2+5 x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs.
\(2(20)=40\).
time = 0.18, size = 51, normalized size = 2.12
method | result | size |
risch | \(-x -\frac {8 \,{\mathrm e}^{-2 x}}{\left (5 x +2\right )^{2}}\) | \(18\) |
default | \(-x +\frac {56 \,{\mathrm e}^{-2 x} \left (10 x -1\right )}{25 \left (25 x^{2}+20 x +4\right )}-\frac {16 \,{\mathrm e}^{-2 x} \left (35 x +9\right )}{25 \left (25 x^{2}+20 x +4\right )}\) | \(51\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 39, normalized size = 1.62 \begin {gather*} -\frac {{\left ({\left (25 \, x^{3} + 20 \, x^{2} + 4 \, x\right )} e^{\left (2 \, x\right )} + 8\right )} e^{\left (-2 \, x\right )}}{25 \, x^{2} + 20 \, x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 0.79 \begin {gather*} - x - \frac {8 e^{- 2 x}}{25 x^{2} + 20 x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 34, normalized size = 1.42 \begin {gather*} -\frac {25 \, x^{3} + 20 \, x^{2} + 4 \, x + 8 \, e^{\left (-2 \, x\right )}}{25 \, x^{2} + 20 \, x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 17, normalized size = 0.71 \begin {gather*} -x-\frac {8\,{\mathrm {e}}^{-2\,x}}{{\left (5\,x+2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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