Optimal. Leaf size=18 \[ e^{9+\frac {4}{(8+e) x}} x^2 \]
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Rubi [A]
time = 0.13, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6, 12, 2259,
2258, 2237, 2241, 2245} \begin {gather*} e^{\frac {4}{(8+e) x}+9} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 2237
Rule 2241
Rule 2245
Rule 2258
Rule 2259
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{9+\frac {4}{8 x+e x}} (-4+(16+2 e) x)}{8+e} \, dx\\ &=\frac {\int e^{9+\frac {4}{8 x+e x}} (-4+(16+2 e) x) \, dx}{8+e}\\ &=\frac {\int e^{9+\frac {4}{(8+e) x}} (-4+(16+2 e) x) \, dx}{8+e}\\ &=\frac {\int \left (-4 e^{9+\frac {4}{(8+e) x}}+2 e^{9+\frac {4}{(8+e) x}} (8+e) x\right ) \, dx}{8+e}\\ &=2 \int e^{9+\frac {4}{(8+e) x}} x \, dx-\frac {4 \int e^{9+\frac {4}{(8+e) x}} \, dx}{8+e}\\ &=-\frac {4 e^{9+\frac {4}{(8+e) x}} x}{8+e}+e^{9+\frac {4}{(8+e) x}} x^2-\frac {16 \int \frac {e^{9+\frac {4}{(8+e) x}}}{x} \, dx}{(8+e)^2}+\frac {4 \int e^{9+\frac {4}{(8+e) x}} \, dx}{8+e}\\ &=e^{9+\frac {4}{(8+e) x}} x^2+\frac {16 e^9 \text {Ei}\left (\frac {4}{(8+e) x}\right )}{(8+e)^2}+\frac {16 \int \frac {e^{9+\frac {4}{(8+e) x}}}{x} \, dx}{(8+e)^2}\\ &=e^{9+\frac {4}{(8+e) x}} x^2\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} e^{9+\frac {4}{(8+e) x}} x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.93, size = 180, normalized size = 10.00
method | result | size |
gosper | \({\mathrm e}^{9+\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2}\) | \(19\) |
norman | \(x^{2} {\mathrm e}^{4} {\mathrm e}^{5} {\mathrm e}^{\frac {4}{x \,{\mathrm e}+8 x}}\) | \(22\) |
risch | \(x^{2} {\mathrm e}^{\frac {9 x \,{\mathrm e}+72 x +4}{x \left ({\mathrm e}+8\right )}}\) | \(26\) |
derivativedivides | \(-\frac {16 \,{\mathrm e}^{4} {\mathrm e}^{5} \left (\frac {x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}}{4}+\expIntegral \left (1, -\frac {4}{\left ({\mathrm e}+8\right ) x}\right )+\frac {-\frac {{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} \left ({\mathrm e}+8\right )^{2}}{2}-2 x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}-8 \expIntegral \left (1, -\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{{\mathrm e}+8}+\frac {2 \,{\mathrm e} \left (-\frac {{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} \left ({\mathrm e}+8\right )^{2}}{32}-\frac {x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}}{8}-\frac {\expIntegral \left (1, -\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{2}\right )}{{\mathrm e}+8}\right )}{\left ({\mathrm e}+8\right )^{2}}\) | \(180\) |
default | \(-\frac {16 \,{\mathrm e}^{4} {\mathrm e}^{5} \left (\frac {x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}}{4}+\expIntegral \left (1, -\frac {4}{\left ({\mathrm e}+8\right ) x}\right )+\frac {-\frac {{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} \left ({\mathrm e}+8\right )^{2}}{2}-2 x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}-8 \expIntegral \left (1, -\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{{\mathrm e}+8}+\frac {2 \,{\mathrm e} \left (-\frac {{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} \left ({\mathrm e}+8\right )^{2}}{32}-\frac {x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}}{8}-\frac {\expIntegral \left (1, -\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{2}\right )}{{\mathrm e}+8}\right )}{\left ({\mathrm e}+8\right )^{2}}\) | \(180\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.30, size = 77, normalized size = 4.28 \begin {gather*} \frac {16 \, {\left (\frac {e^{9} \Gamma \left (-1, -\frac {4}{x {\left (e + 8\right )}}\right )}{e + 8} + \frac {2 \, e^{10} \Gamma \left (-2, -\frac {4}{x {\left (e + 8\right )}}\right )}{{\left (e + 8\right )}^{2}} + \frac {16 \, e^{9} \Gamma \left (-2, -\frac {4}{x {\left (e + 8\right )}}\right )}{{\left (e + 8\right )}^{2}}\right )}}{e + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 26, normalized size = 1.44 \begin {gather*} x^{2} e^{\left (\frac {9 \, x e + 72 \, x + 4}{x e + 8 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 17, normalized size = 0.94 \begin {gather*} x^{2} e^{9} e^{\frac {4}{e x + 8 x}} \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. 181 vs.
\(2 (18) = 36\).
time = 0.45, size = 181, normalized size = 10.06 \begin {gather*} \frac {16 \, e^{\left (\frac {9 \, x e + 72 \, x + 4}{x e + 8 \, x}\right )}}{\frac {{\left (9 \, x e + 72 \, x + 4\right )}^{2} e^{2}}{{\left (x e + 8 \, x\right )}^{2}} - \frac {18 \, {\left (9 \, x e + 72 \, x + 4\right )} e^{2}}{x e + 8 \, x} + \frac {16 \, {\left (9 \, x e + 72 \, x + 4\right )}^{2} e}{{\left (x e + 8 \, x\right )}^{2}} - \frac {288 \, {\left (9 \, x e + 72 \, x + 4\right )} e}{x e + 8 \, x} + \frac {64 \, {\left (9 \, x e + 72 \, x + 4\right )}^{2}}{{\left (x e + 8 \, x\right )}^{2}} - \frac {1152 \, {\left (9 \, x e + 72 \, x + 4\right )}}{x e + 8 \, x} + 81 \, e^{2} + 1296 \, e + 5184} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.15, size = 19, normalized size = 1.06 \begin {gather*} x^2\,{\mathrm {e}}^9\,{\mathrm {e}}^{\frac {4}{8\,x+x\,\mathrm {e}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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