Optimal. Leaf size=16 \[ \frac {e^{a x}}{a^2 (1+a x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2228}
\begin {gather*} \frac {e^{a x}}{a^2 (a x+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2228
Rubi steps
\begin {align*} \int \frac {e^{a x} x}{(1+a x)^2} \, dx &=\frac {e^{a x}}{a^2 (1+a x)}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 16, normalized size = 1.00 \begin {gather*} \frac {e^{a x}}{a^2 (1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 16, normalized size = 1.00
method | result | size |
gosper | \(\frac {{\mathrm e}^{a x}}{a^{2} \left (a x +1\right )}\) | \(16\) |
derivativedivides | \(\frac {{\mathrm e}^{a x}}{a^{2} \left (a x +1\right )}\) | \(16\) |
default | \(\frac {{\mathrm e}^{a x}}{a^{2} \left (a x +1\right )}\) | \(16\) |
norman | \(\frac {{\mathrm e}^{a x}}{a^{2} \left (a x +1\right )}\) | \(16\) |
risch | \(\frac {{\mathrm e}^{a x}}{a^{2} \left (a x +1\right )}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.78, size = 16, normalized size = 1.00 \begin {gather*} \frac {e^{\left (a x\right )}}{a^{3} x + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.71, size = 16, normalized size = 1.00 \begin {gather*} \frac {e^{\left (a x\right )}}{a^{3} x + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 12, normalized size = 0.75 \begin {gather*} \frac {e^{a x}}{a^{3} x + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (15) = 30\).
time = 0.65, size = 45, normalized size = 2.81 \begin {gather*} -\frac {e^{\left (-{\left (a x + 1\right )} {\left (\frac {1}{a x + 1} - 1\right )}\right )}}{{\left (a x + 1\right )} a^{2} {\left (\frac {1}{a x + 1} - 1\right )} - a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 15, normalized size = 0.94 \begin {gather*} \frac {{\mathrm {e}}^{a\,x}}{a^2\,\left (a\,x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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