Optimal. Leaf size=27 \[ \frac {3 (5-x) (5+x)}{\left (-9+\frac {e^3}{3}-x\right ) x^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(27)=54\).
time = 0.07, antiderivative size = 62, normalized size of antiderivative = 2.30, number of steps
used = 3, number of rules used = 2, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6, 2099}
\begin {gather*} -\frac {225}{\left (27-e^3\right ) x^2}+\frac {9 \left (504-54 e^3+e^6\right )}{\left (27-e^3\right )^2 \left (3 x-e^3+27\right )}+\frac {675}{\left (27-e^3\right )^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12150-450 e^3+2025 x-27 x^3}{\left (729+e^6\right ) x^3+162 x^4+9 x^5+e^3 \left (-54 x^3-6 x^4\right )} \, dx\\ &=\int \left (-\frac {27 \left (504-54 e^3+e^6\right )}{(-3+e)^2 \left (9+3 e+e^2\right )^2 \left (-27+e^3-3 x\right )^2}-\frac {450}{\left (-27+e^3\right ) x^3}-\frac {675}{\left (-27+e^3\right )^2 x^2}\right ) \, dx\\ &=-\frac {225}{\left (27-e^3\right ) x^2}+\frac {675}{\left (27-e^3\right )^2 x}+\frac {9 \left (504-54 e^3+e^6\right )}{\left (27-e^3\right )^2 \left (27-e^3+3 x\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 21, normalized size = 0.78 \begin {gather*} -\frac {9 \left (-25+x^2\right )}{x^2 \left (e^3-3 (9+x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 20, normalized size = 0.74
method | result | size |
gosper | \(-\frac {9 \left (x^{2}-25\right )}{x^{2} \left (-27+{\mathrm e}^{3}-3 x \right )}\) | \(20\) |
norman | \(\frac {-9 x^{2}+225}{x^{2} \left (-27+{\mathrm e}^{3}-3 x \right )}\) | \(21\) |
risch | \(\frac {-9 x^{2}+225}{x^{2} \left (-27+{\mathrm e}^{3}-3 x \right )}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 24, normalized size = 0.89 \begin {gather*} \frac {9 \, {\left (x^{2} - 25\right )}}{3 \, x^{3} - x^{2} {\left (e^{3} - 27\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 27, normalized size = 1.00 \begin {gather*} \frac {9 \, {\left (x^{2} - 25\right )}}{3 \, x^{3} - x^{2} e^{3} + 27 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.46, size = 20, normalized size = 0.74 \begin {gather*} - \frac {225 - 9 x^{2}}{3 x^{3} + x^{2} \cdot \left (27 - e^{3}\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. 54 vs.
\(2 (22) = 44\).
time = 0.39, size = 54, normalized size = 2.00 \begin {gather*} \frac {9 \, {\left (e^{6} - 54 \, e^{3} + 504\right )}}{{\left (3 \, x - e^{3} + 27\right )} {\left (e^{6} - 54 \, e^{3} + 729\right )}} + \frac {225 \, {\left (3 \, x + e^{3} - 27\right )}}{x^{2} {\left (e^{6} - 54 \, e^{3} + 729\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.59, size = 25, normalized size = 0.93 \begin {gather*} \frac {9\,x^2-225}{3\,x^3-x^2\,\left ({\mathrm {e}}^3-27\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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