Optimal. Leaf size=17 \[ 5+e^x-\frac {3}{8 x^2}+e^x x \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 14, 2176, 2194} \begin {gather*} -\frac {3}{8 x^2}+e^x (x+2)-e^x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2194
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {3+e^x \left (8 x^3+4 x^4\right )}{x^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {3}{x^3}+4 e^x (2+x)\right ) \, dx\\ &=-\frac {3}{8 x^2}+\int e^x (2+x) \, dx\\ &=-\frac {3}{8 x^2}+e^x (2+x)-\int e^x \, dx\\ &=-e^x-\frac {3}{8 x^2}+e^x (2+x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 0.88 \begin {gather*} -\frac {3}{8 x^2}+e^x (1+x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 18, normalized size = 1.06 \begin {gather*} \frac {8 \, {\left (x^{3} + x^{2}\right )} e^{x} - 3}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 21, normalized size = 1.24 \begin {gather*} \frac {8 \, x^{3} e^{x} + 8 \, x^{2} e^{x} - 3}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 13, normalized size = 0.76
method | result | size |
default | \(-\frac {3}{8 x^{2}}+{\mathrm e}^{x} x +{\mathrm e}^{x}\) | \(13\) |
risch | \(-\frac {3}{8 x^{2}}+\frac {\left (4 x +4\right ) {\mathrm e}^{x}}{4}\) | \(16\) |
norman | \(\frac {-\frac {3}{8}+{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} x^{3}}{x^{2}}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 16, normalized size = 0.94 \begin {gather*} {\left (x - 1\right )} e^{x} - \frac {3}{8 \, x^{2}} + 2 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 12, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^x+x\,{\mathrm {e}}^x-\frac {3}{8\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 12, normalized size = 0.71 \begin {gather*} \left (x + 1\right ) e^{x} - \frac {3}{8 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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