Optimal. Leaf size=24 \[ \frac {25 \left (1-x-e^x \left (2+\left (\frac {3}{2}+x\right )^2\right )\right )}{x^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 38, normalized size of antiderivative = 1.58, number of steps
used = 13, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {12, 14, 37,
2230, 2225, 2208, 2209} \begin {gather*} \frac {25 (2-x)^2}{4 x^2}-\frac {425 e^x}{4 x^2}-25 e^x-\frac {75 e^x}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 37
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {-200+100 x+e^x \left (850-125 x-300 x^2-100 x^3\right )}{x^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {100 (-2+x)}{x^3}-\frac {25 e^x \left (-34+5 x+12 x^2+4 x^3\right )}{x^3}\right ) \, dx\\ &=-\left (\frac {25}{4} \int \frac {e^x \left (-34+5 x+12 x^2+4 x^3\right )}{x^3} \, dx\right )+25 \int \frac {-2+x}{x^3} \, dx\\ &=\frac {25 (2-x)^2}{4 x^2}-\frac {25}{4} \int \left (4 e^x-\frac {34 e^x}{x^3}+\frac {5 e^x}{x^2}+\frac {12 e^x}{x}\right ) \, dx\\ &=\frac {25 (2-x)^2}{4 x^2}-25 \int e^x \, dx-\frac {125}{4} \int \frac {e^x}{x^2} \, dx-75 \int \frac {e^x}{x} \, dx+\frac {425}{2} \int \frac {e^x}{x^3} \, dx\\ &=-25 e^x-\frac {425 e^x}{4 x^2}+\frac {25 (2-x)^2}{4 x^2}+\frac {125 e^x}{4 x}-75 \text {Ei}(x)-\frac {125}{4} \int \frac {e^x}{x} \, dx+\frac {425}{4} \int \frac {e^x}{x^2} \, dx\\ &=-25 e^x-\frac {425 e^x}{4 x^2}+\frac {25 (2-x)^2}{4 x^2}-\frac {75 e^x}{x}-\frac {425 \text {Ei}(x)}{4}+\frac {425}{4} \int \frac {e^x}{x} \, dx\\ &=-25 e^x-\frac {425 e^x}{4 x^2}+\frac {25 (2-x)^2}{4 x^2}-\frac {75 e^x}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.16, size = 27, normalized size = 1.12 \begin {gather*} -\frac {25 \left (4 (-1+x)+e^x \left (17+12 x+4 x^2\right )\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 30, normalized size = 1.25
method | result | size |
norman | \(\frac {25-25 x -75 \,{\mathrm e}^{x} x -25 \,{\mathrm e}^{x} x^{2}-\frac {425 \,{\mathrm e}^{x}}{4}}{x^{2}}\) | \(26\) |
risch | \(\frac {-100 x +100}{4 x^{2}}-\frac {25 \left (4 x^{2}+12 x +17\right ) {\mathrm e}^{x}}{4 x^{2}}\) | \(29\) |
default | \(\frac {25}{x^{2}}-\frac {25}{x}-\frac {425 \,{\mathrm e}^{x}}{4 x^{2}}-\frac {75 \,{\mathrm e}^{x}}{x}-25 \,{\mathrm e}^{x}\) | \(30\) |
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 = 0.28, size = 33, normalized size = 1.38 \begin {gather*} -\frac {25}{x} + \frac {25}{x^{2}} - 75 \, {\rm Ei}\left (x\right ) - 25 \, e^{x} - \frac {125}{4} \, \Gamma \left (-1, -x\right ) - \frac {425}{2} \, \Gamma \left (-2, -x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 23, normalized size = 0.96 \begin {gather*} -\frac {25 \, {\left ({\left (4 \, x^{2} + 12 \, x + 17\right )} e^{x} + 4 \, x - 4\right )}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 27, normalized size = 1.12 \begin {gather*} \frac {25 - 25 x}{x^{2}} + \frac {\left (- 100 x^{2} - 300 x - 425\right ) e^{x}}{4 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 26, normalized size = 1.08 \begin {gather*} -\frac {25 \, {\left (4 \, x^{2} e^{x} + 12 \, x e^{x} + 4 \, x + 17 \, e^{x} - 4\right )}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 24, normalized size = 1.00 \begin {gather*} -25\,{\mathrm {e}}^x-\frac {\frac {425\,{\mathrm {e}}^x}{4}+x\,\left (75\,{\mathrm {e}}^x+25\right )-25}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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