Optimal. Leaf size=25 \[ x+\left (1+\frac {e^x}{15 x}+x\right ) \left (-\frac {5}{x}+2 x\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps
used = 13, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {12, 14, 2230,
2225, 2208, 2209} \begin {gather*} 2 x^2-\frac {e^x}{3 x^2}+3 x+\frac {2 e^x}{15}-\frac {5}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{15} \int \frac {75 x+45 x^3+60 x^4+e^x \left (10-5 x+2 x^3\right )}{x^3} \, dx\\ &=\frac {1}{15} \int \left (\frac {e^x \left (10-5 x+2 x^3\right )}{x^3}+\frac {15 \left (5+3 x^2+4 x^3\right )}{x^2}\right ) \, dx\\ &=\frac {1}{15} \int \frac {e^x \left (10-5 x+2 x^3\right )}{x^3} \, dx+\int \frac {5+3 x^2+4 x^3}{x^2} \, dx\\ &=\frac {1}{15} \int \left (2 e^x+\frac {10 e^x}{x^3}-\frac {5 e^x}{x^2}\right ) \, dx+\int \left (3+\frac {5}{x^2}+4 x\right ) \, dx\\ &=-\frac {5}{x}+3 x+2 x^2+\frac {2 \int e^x \, dx}{15}-\frac {1}{3} \int \frac {e^x}{x^2} \, dx+\frac {2}{3} \int \frac {e^x}{x^3} \, dx\\ &=\frac {2 e^x}{15}-\frac {e^x}{3 x^2}-\frac {5}{x}+\frac {e^x}{3 x}+3 x+2 x^2+\frac {1}{3} \int \frac {e^x}{x^2} \, dx-\frac {1}{3} \int \frac {e^x}{x} \, dx\\ &=\frac {2 e^x}{15}-\frac {e^x}{3 x^2}-\frac {5}{x}+3 x+2 x^2-\frac {\text {Ei}(x)}{3}+\frac {1}{3} \int \frac {e^x}{x} \, dx\\ &=\frac {2 e^x}{15}-\frac {e^x}{3 x^2}-\frac {5}{x}+3 x+2 x^2\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 31, normalized size = 1.24 \begin {gather*} \frac {2 e^x}{15}-\frac {e^x}{3 x^2}-\frac {5}{x}+3 x+2 x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 26, normalized size = 1.04
method | result | size |
default | \(2 x^{2}+3 x -\frac {5}{x}-\frac {{\mathrm e}^{x}}{3 x^{2}}+\frac {2 \,{\mathrm e}^{x}}{15}\) | \(26\) |
risch | \(2 x^{2}+3 x -\frac {5}{x}+\frac {\left (2 x^{2}-5\right ) {\mathrm e}^{x}}{15 x^{2}}\) | \(29\) |
norman | \(\frac {-5 x +3 x^{3}+2 x^{4}+\frac {2 \,{\mathrm e}^{x} x^{2}}{15}-\frac {{\mathrm e}^{x}}{3}}{x^{2}}\) | \(30\) |
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 = 0.29, size = 32, normalized size = 1.28 \begin {gather*} 2 \, x^{2} + 3 \, x - \frac {5}{x} + \frac {2}{15} \, e^{x} - \frac {1}{3} \, \Gamma \left (-1, -x\right ) - \frac {2}{3} \, \Gamma \left (-2, -x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 29, normalized size = 1.16 \begin {gather*} \frac {30 \, x^{4} + 45 \, x^{3} + {\left (2 \, x^{2} - 5\right )} e^{x} - 75 \, x}{15 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 26, normalized size = 1.04 \begin {gather*} 2 x^{2} + 3 x - \frac {5}{x} + \frac {\left (2 x^{2} - 5\right ) e^{x}}{15 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.37, size = 30, normalized size = 1.20 \begin {gather*} \frac {30 \, x^{4} + 45 \, x^{3} + 2 \, x^{2} e^{x} - 75 \, x - 5 \, e^{x}}{15 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 26, normalized size = 1.04 \begin {gather*} 3\,x+\frac {2\,{\mathrm {e}}^x}{15}-\frac {5\,x+\frac {{\mathrm {e}}^x}{3}}{x^2}+2\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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