Optimal. Leaf size=29 \[ e^2+x-\frac {x+\frac {e^2}{\left (5-x-\frac {\log (x)}{3}\right )^2}}{x} \]
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Rubi [F]
time = 0.98, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-3375 x^2+2025 x^3-405 x^4+27 x^5+e^2 (-117+81 x)+\left (9 e^2+675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)}{-3375 x^2+2025 x^3-405 x^4+27 x^5+\left (675 x^2-270 x^3+27 x^4\right ) \log (x)+\left (-45 x^2+9 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {27 (-5+x)^3 x^2+9 e^2 (-13+9 x)+9 \left (e^2+3 (-5+x)^2 x^2\right ) \log (x)+9 (-5+x) x^2 \log ^2(x)+x^2 \log ^3(x)}{x^2 (3 (-5+x)+\log (x))^3} \, dx\\ &=\int \left (1+\frac {18 e^2 (1+3 x)}{x^2 (-15+3 x+\log (x))^3}+\frac {9 e^2}{x^2 (-15+3 x+\log (x))^2}\right ) \, dx\\ &=x+\left (9 e^2\right ) \int \frac {1}{x^2 (-15+3 x+\log (x))^2} \, dx+\left (18 e^2\right ) \int \frac {1+3 x}{x^2 (-15+3 x+\log (x))^3} \, dx\\ &=x+\left (9 e^2\right ) \int \frac {1}{x^2 (-15+3 x+\log (x))^2} \, dx+\left (18 e^2\right ) \int \left (\frac {1}{x^2 (-15+3 x+\log (x))^3}+\frac {3}{x (-15+3 x+\log (x))^3}\right ) \, dx\\ &=x+\left (9 e^2\right ) \int \frac {1}{x^2 (-15+3 x+\log (x))^2} \, dx+\left (18 e^2\right ) \int \frac {1}{x^2 (-15+3 x+\log (x))^3} \, dx+\left (54 e^2\right ) \int \frac {1}{x (-15+3 x+\log (x))^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.12, size = 19, normalized size = 0.66 \begin {gather*} x-\frac {9 e^2}{x (-15+3 x+\log (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs.
\(2(27)=54\).
time = 1.61, size = 58, normalized size = 2.00
method | result | size |
risch | \(x -\frac {9 \,{\mathrm e}^{2}}{x \left (\ln \left (x \right )+3 x -15\right )^{2}}\) | \(19\) |
default | \(\frac {-90 x^{3}+225 x^{2}+x^{2} \ln \left (x \right )^{2}-30 x^{2} \ln \left (x \right )-9 \,{\mathrm e}^{2}+9 x^{4}+6 x^{3} \ln \left (x \right )}{x \left (\ln \left (x \right )+3 x -15\right )^{2}}\) | \(58\) |
norman | \(\frac {-675 x^{2}+2250 x +x^{2} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right )-300 x \ln \left (x \right )+10 x \ln \left (x \right )^{2}+30 x^{2} \ln \left (x \right )+9 x^{4}-9 \,{\mathrm e}^{2}}{x \left (\ln \left (x \right )+3 x -15\right )^{2}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (24) = 48\).
time = 0.33, size = 75, normalized size = 2.59 \begin {gather*} \frac {9 \, x^{4} + x^{2} \log \left (x\right )^{2} - 90 \, x^{3} + 225 \, x^{2} + 6 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (x\right ) - 9 \, e^{2}}{9 \, x^{3} + x \log \left (x\right )^{2} - 90 \, x^{2} + 6 \, {\left (x^{2} - 5 \, x\right )} \log \left (x\right ) + 225 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (24) = 48\).
time = 0.38, size = 75, normalized size = 2.59 \begin {gather*} \frac {9 \, x^{4} + x^{2} \log \left (x\right )^{2} - 90 \, x^{3} + 225 \, x^{2} + 6 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (x\right ) - 9 \, e^{2}}{9 \, x^{3} + x \log \left (x\right )^{2} - 90 \, x^{2} + 6 \, {\left (x^{2} - 5 \, x\right )} \log \left (x\right ) + 225 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 37, normalized size = 1.28 \begin {gather*} x - \frac {9 e^{2}}{9 x^{3} - 90 x^{2} + x \log {\left (x \right )}^{2} + 225 x + \left (6 x^{2} - 30 x\right ) \log {\left (x \right )}} \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. 77 vs.
\(2 (24) = 48\).
time = 0.42, size = 77, normalized size = 2.66 \begin {gather*} \frac {9 \, x^{4} + 6 \, x^{3} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 90 \, x^{3} - 30 \, x^{2} \log \left (x\right ) + 225 \, x^{2} - 9 \, e^{2}}{9 \, x^{3} + 6 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} - 90 \, x^{2} - 30 \, x \log \left (x\right ) + 225 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (x\right )\,\left (27\,x^4-270\,x^3+675\,x^2+9\,{\mathrm {e}}^2\right )-{\ln \left (x\right )}^2\,\left (45\,x^2-9\,x^3\right )+x^2\,{\ln \left (x\right )}^3-3375\,x^2+2025\,x^3-405\,x^4+27\,x^5+{\mathrm {e}}^2\,\left (81\,x-117\right )}{\ln \left (x\right )\,\left (27\,x^4-270\,x^3+675\,x^2\right )-{\ln \left (x\right )}^2\,\left (45\,x^2-9\,x^3\right )+x^2\,{\ln \left (x\right )}^3-3375\,x^2+2025\,x^3-405\,x^4+27\,x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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