Optimal. Leaf size=26 \[ -3+x+e^x \left (4+e^5-x\right ) x \left (1-x+3 x^2\right ) \]
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Rubi [B] time = 0.60, antiderivative size = 59, normalized size of antiderivative = 2.27, number of steps used = 32, number of rules used = 5, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.063, Rules used = {1594, 6688, 2196, 2194, 2176} \begin {gather*} -3 e^x x^4+13 e^x x^3+3 e^{x+5} x^3-5 e^x x^2-e^{x+5} x^2+4 e^x x+e^{x+5} x+x \end {gather*}
Antiderivative was successfully verified.
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Rule 1594
Rule 2176
Rule 2194
Rule 2196
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x-x^2+3 x^3+e^x x \left (1-x+3 x^2\right ) \left (4-6 x+34 x^2+x^3-3 x^4+e^5 \left (1-x+8 x^2+3 x^3\right )\right )}{x \left (1-x+3 x^2\right )} \, dx\\ &=\int \left (1+e^{5+x} \left (1-x+8 x^2+3 x^3\right )+e^x \left (4-6 x+34 x^2+x^3-3 x^4\right )\right ) \, dx\\ &=x+\int e^{5+x} \left (1-x+8 x^2+3 x^3\right ) \, dx+\int e^x \left (4-6 x+34 x^2+x^3-3 x^4\right ) \, dx\\ &=x+\int \left (e^{5+x}-e^{5+x} x+8 e^{5+x} x^2+3 e^{5+x} x^3\right ) \, dx+\int \left (4 e^x-6 e^x x+34 e^x x^2+e^x x^3-3 e^x x^4\right ) \, dx\\ &=x+3 \int e^{5+x} x^3 \, dx-3 \int e^x x^4 \, dx+4 \int e^x \, dx-6 \int e^x x \, dx+8 \int e^{5+x} x^2 \, dx+34 \int e^x x^2 \, dx+\int e^{5+x} \, dx-\int e^{5+x} x \, dx+\int e^x x^3 \, dx\\ &=4 e^x+e^{5+x}+x-6 e^x x-e^{5+x} x+34 e^x x^2+8 e^{5+x} x^2+e^x x^3+3 e^{5+x} x^3-3 e^x x^4-3 \int e^x x^2 \, dx+6 \int e^x \, dx-9 \int e^{5+x} x^2 \, dx+12 \int e^x x^3 \, dx-16 \int e^{5+x} x \, dx-68 \int e^x x \, dx+\int e^{5+x} \, dx\\ &=10 e^x+2 e^{5+x}+x-74 e^x x-17 e^{5+x} x+31 e^x x^2-e^{5+x} x^2+13 e^x x^3+3 e^{5+x} x^3-3 e^x x^4+6 \int e^x x \, dx+16 \int e^{5+x} \, dx+18 \int e^{5+x} x \, dx-36 \int e^x x^2 \, dx+68 \int e^x \, dx\\ &=78 e^x+18 e^{5+x}+x-68 e^x x+e^{5+x} x-5 e^x x^2-e^{5+x} x^2+13 e^x x^3+3 e^{5+x} x^3-3 e^x x^4-6 \int e^x \, dx-18 \int e^{5+x} \, dx+72 \int e^x x \, dx\\ &=72 e^x+x+4 e^x x+e^{5+x} x-5 e^x x^2-e^{5+x} x^2+13 e^x x^3+3 e^{5+x} x^3-3 e^x x^4-72 \int e^x \, dx\\ &=x+4 e^x x+e^{5+x} x-5 e^x x^2-e^{5+x} x^2+13 e^x x^3+3 e^{5+x} x^3-3 e^x x^4\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 51, normalized size = 1.96 \begin {gather*} x+e^x \left (e^5 x-e^5 x^2+3 e^5 x^3\right )+e^x \left (4 x-5 x^2+13 x^3-3 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 27, normalized size = 1.04 \begin {gather*} -{\left (x - e^{5} - 4\right )} e^{\left (x + \log \left (3 \, x^{2} - x + 1\right ) + \log \relax (x)\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 52, normalized size = 2.00 \begin {gather*} -3 \, x^{4} e^{x} + 3 \, x^{3} e^{\left (x + 5\right )} + 13 \, x^{3} e^{x} - x^{2} e^{\left (x + 5\right )} - 5 \, x^{2} e^{x} + x e^{\left (x + 5\right )} + 4 \, x e^{x} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 1.65
method | result | size |
risch | \(x +\left (3 x^{3} {\mathrm e}^{5}-3 x^{4}-x^{2} {\mathrm e}^{5}+13 x^{3}+x \,{\mathrm e}^{5}-5 x^{2}+4 x \right ) {\mathrm e}^{x}\) | \(43\) |
default | \(x +13 \,{\mathrm e}^{x} x^{3}-5 \,{\mathrm e}^{x} x^{2}+{\mathrm e}^{5} {\mathrm e}^{x}+4 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{x} x^{4}+3 \,{\mathrm e}^{5} \left ({\mathrm e}^{x} x^{3}-3 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}\right )-{\mathrm e}^{5} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+8 \,{\mathrm e}^{5} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 38, normalized size = 1.46 \begin {gather*} -{\left (3 \, x^{4} - x^{3} {\left (3 \, e^{5} + 13\right )} + x^{2} {\left (e^{5} + 5\right )} - x {\left (e^{5} + 4\right )}\right )} e^{x} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 52, normalized size = 2.00 \begin {gather*} x+x\,{\mathrm {e}}^{x+5}-5\,x^2\,{\mathrm {e}}^x+13\,x^3\,{\mathrm {e}}^x-3\,x^4\,{\mathrm {e}}^x-x^2\,{\mathrm {e}}^{x+5}+3\,x^3\,{\mathrm {e}}^{x+5}+4\,x\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 42, normalized size = 1.62 \begin {gather*} x + \left (- 3 x^{4} + 13 x^{3} + 3 x^{3} e^{5} - x^{2} e^{5} - 5 x^{2} + 4 x + x e^{5}\right ) e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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