Optimal. Leaf size=31 \[ e^x+x-\frac {\frac {4}{5}+(3-x)^2+x+2 x^2}{9+x^2} \]
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Rubi [A] time = 0.16, antiderivative size = 46, normalized size of antiderivative = 1.48, number of steps used = 12, number of rules used = 8, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {28, 6742, 2194, 199, 203, 261, 288, 321} \begin {gather*} \frac {x}{2 \left (x^2+9\right )}+\frac {86}{5 \left (x^2+9\right )}-\frac {x^3}{2 \left (x^2+9\right )}+\frac {3 x}{2}+e^x \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 199
Rule 203
Rule 261
Rule 288
Rule 321
Rule 2194
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=5 \int \frac {630-172 x+65 x^2+5 x^4+e^x \left (405+90 x^2+5 x^4\right )}{\left (45+5 x^2\right )^2} \, dx\\ &=5 \int \left (\frac {e^x}{5}+\frac {126}{5 \left (9+x^2\right )^2}-\frac {172 x}{25 \left (9+x^2\right )^2}+\frac {13 x^2}{5 \left (9+x^2\right )^2}+\frac {x^4}{5 \left (9+x^2\right )^2}\right ) \, dx\\ &=13 \int \frac {x^2}{\left (9+x^2\right )^2} \, dx-\frac {172}{5} \int \frac {x}{\left (9+x^2\right )^2} \, dx+126 \int \frac {1}{\left (9+x^2\right )^2} \, dx+\int e^x \, dx+\int \frac {x^4}{\left (9+x^2\right )^2} \, dx\\ &=e^x+\frac {86}{5 \left (9+x^2\right )}+\frac {x}{2 \left (9+x^2\right )}-\frac {x^3}{2 \left (9+x^2\right )}+\frac {3}{2} \int \frac {x^2}{9+x^2} \, dx+\frac {13}{2} \int \frac {1}{9+x^2} \, dx+7 \int \frac {1}{9+x^2} \, dx\\ &=e^x+\frac {3 x}{2}+\frac {86}{5 \left (9+x^2\right )}+\frac {x}{2 \left (9+x^2\right )}-\frac {x^3}{2 \left (9+x^2\right )}+\frac {9}{2} \tan ^{-1}\left (\frac {x}{3}\right )-\frac {27}{2} \int \frac {1}{9+x^2} \, dx\\ &=e^x+\frac {3 x}{2}+\frac {86}{5 \left (9+x^2\right )}+\frac {x}{2 \left (9+x^2\right )}-\frac {x^3}{2 \left (9+x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 31, normalized size = 1.00 \begin {gather*} \frac {86+70 x+5 x^3+5 e^x \left (9+x^2\right )}{5 \left (9+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 28, normalized size = 0.90 \begin {gather*} \frac {5 \, x^{3} + 5 \, {\left (x^{2} + 9\right )} e^{x} + 70 \, x + 86}{5 \, {\left (x^{2} + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 30, normalized size = 0.97 \begin {gather*} \frac {5 \, x^{3} + 5 \, x^{2} e^{x} + 70 \, x + 45 \, e^{x} + 86}{5 \, {\left (x^{2} + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 18, normalized size = 0.58
method | result | size |
risch | \(x +\frac {5 x +\frac {86}{5}}{x^{2}+9}+{\mathrm e}^{x}\) | \(18\) |
default | \(\frac {5 x}{x^{2}+9}+\frac {86}{5 \left (x^{2}+9\right )}+x +{\mathrm e}^{x}\) | \(24\) |
norman | \(\frac {x^{3}+{\mathrm e}^{x} x^{2}+14 x +9 \,{\mathrm e}^{x}+\frac {86}{5}}{x^{2}+9}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 23, normalized size = 0.74 \begin {gather*} x + \frac {5 \, x}{x^{2} + 9} + \frac {86}{5 \, {\left (x^{2} + 9\right )}} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 19, normalized size = 0.61 \begin {gather*} x+{\mathrm {e}}^x+\frac {25\,x+86}{5\,x^2+45} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 15, normalized size = 0.48 \begin {gather*} x + \frac {25 x + 86}{5 x^{2} + 45} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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