Optimal. Leaf size=28 \[ 3+2 x+\frac {x^2}{1+x}-\frac {5 e^5}{3+4 x^2} \]
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Rubi [A] time = 0.14, antiderivative size = 23, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 2, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2074, 261} \begin {gather*} -\frac {5 e^5}{4 x^2+3}+3 x+\frac {1}{x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 261
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (3-\frac {1}{(1+x)^2}+\frac {40 e^5 x}{\left (3+4 x^2\right )^2}\right ) \, dx\\ &=3 x+\frac {1}{1+x}+\left (40 e^5\right ) \int \frac {x}{\left (3+4 x^2\right )^2} \, dx\\ &=3 x+\frac {1}{1+x}-\frac {5 e^5}{3+4 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 1.32 \begin {gather*} 3 x+\frac {3+4 x^2-5 e^5 (1+x)}{3+3 x+4 x^2+4 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 45, normalized size = 1.61 \begin {gather*} \frac {12 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} - 5 \, {\left (x + 1\right )} e^{5} + 9 \, x + 3}{4 \, x^{3} + 4 \, x^{2} + 3 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 38, normalized size = 1.36 \begin {gather*} 3 \, x + \frac {4 \, x^{2} - 5 \, x e^{5} - 5 \, e^{5} + 3}{4 \, x^{3} + 4 \, x^{2} + 3 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 23, normalized size = 0.82
method | result | size |
default | \(3 x -\frac {5 \,{\mathrm e}^{5}}{4 x^{2}+3}+\frac {1}{x +1}\) | \(23\) |
risch | \(3 x +\frac {x^{2}-\frac {5 x \,{\mathrm e}^{5}}{4}+\frac {3}{4}-\frac {5 \,{\mathrm e}^{5}}{4}}{x^{3}+x^{2}+\frac {3}{4} x +\frac {3}{4}}\) | \(33\) |
norman | \(\frac {12 x^{4}-6+x^{2}-5 x \,{\mathrm e}^{5}-5 \,{\mathrm e}^{5}}{4 x^{3}+4 x^{2}+3 x +3}\) | \(38\) |
gosper | \(-\frac {-12 x^{4}+5 x \,{\mathrm e}^{5}-x^{2}+5 \,{\mathrm e}^{5}+6}{4 x^{3}+4 x^{2}+3 x +3}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 38, normalized size = 1.36 \begin {gather*} 3 \, x + \frac {4 \, x^{2} - 5 \, x e^{5} - 5 \, e^{5} + 3}{4 \, x^{3} + 4 \, x^{2} + 3 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 44, normalized size = 1.57 \begin {gather*} \frac {9\,x-5\,{\mathrm {e}}^5-5\,x\,{\mathrm {e}}^5+13\,x^2+12\,x^3+12\,x^4+3}{\left (4\,x^2+3\right )\,\left (x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 36, normalized size = 1.29 \begin {gather*} 3 x + \frac {4 x^{2} - 5 x e^{5} - 5 e^{5} + 3}{4 x^{3} + 4 x^{2} + 3 x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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