Optimal. Leaf size=24 \[ e^{\frac {11}{9}+\frac {\left (x+\log \left (2 x+25 x^2\right )\right )^4}{x^4}} \]
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Rubi [F]
time = 20.35, 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 {\exp \left (\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}\right ) \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}\right ) \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{x^5 (2+25 x)} \, dx\\ &=\int \frac {4 \exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} (x+\log (x (2+25 x)))^3 (2+50 x-(2+25 x) \log (x (2+25 x)))}{x^4} \, dx\\ &=4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} (x+\log (x (2+25 x)))^3 (2+50 x-(2+25 x) \log (x (2+25 x)))}{x^4} \, dx\\ &=4 \int \left (\frac {2 \exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (1+25 x) (x (2+25 x))^{-1+\frac {4}{x}}}{x}-\frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-6-148 x+25 x^2\right ) \log (x (2+25 x))}{x^2}-\frac {3 \exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-48 x+25 x^2\right ) \log ^2(x (2+25 x))}{x^3}-\frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-44 x+75 x^2\right ) \log ^3(x (2+25 x))}{x^4}-\frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (2+25 x) (x (2+25 x))^{-1+\frac {4}{x}} \log ^4(x (2+25 x))}{x^4}\right ) \, dx\\ &=-\left (4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-6-148 x+25 x^2\right ) \log (x (2+25 x))}{x^2} \, dx\right )-4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-44 x+75 x^2\right ) \log ^3(x (2+25 x))}{x^4} \, dx-4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (2+25 x) (x (2+25 x))^{-1+\frac {4}{x}} \log ^4(x (2+25 x))}{x^4} \, dx+8 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (1+25 x) (x (2+25 x))^{-1+\frac {4}{x}}}{x} \, dx-12 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-48 x+25 x^2\right ) \log ^2(x (2+25 x))}{x^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(24)=48\).
time = 0.09, size = 64, normalized size = 2.67 \begin {gather*} e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{4/x} \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. \(73\) vs.
\(2(21)=42\).
time = 0.63, size = 74, normalized size = 3.08
method | result | size |
risch | \({\mathrm e}^{\frac {9 \ln \left (25 x^{2}+2 x \right )^{4}+36 x \ln \left (25 x^{2}+2 x \right )^{3}+54 x^{2} \ln \left (25 x^{2}+2 x \right )^{2}+36 x^{3} \ln \left (25 x^{2}+2 x \right )+20 x^{4}}{9 x^{4}}}\) | \(74\) |
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. 175 vs.
\(2 (21) = 42\).
time = 0.68, size = 175, normalized size = 7.29 \begin {gather*} e^{\left (\frac {4 \, \log \left (25 \, x + 2\right )}{x} + \frac {6 \, \log \left (25 \, x + 2\right )^{2}}{x^{2}} + \frac {4 \, \log \left (25 \, x + 2\right )^{3}}{x^{3}} + \frac {\log \left (25 \, x + 2\right )^{4}}{x^{4}} + \frac {4 \, \log \left (x\right )}{x} + \frac {12 \, \log \left (25 \, x + 2\right ) \log \left (x\right )}{x^{2}} + \frac {12 \, \log \left (25 \, x + 2\right )^{2} \log \left (x\right )}{x^{3}} + \frac {4 \, \log \left (25 \, x + 2\right )^{3} \log \left (x\right )}{x^{4}} + \frac {6 \, \log \left (x\right )^{2}}{x^{2}} + \frac {12 \, \log \left (25 \, x + 2\right ) \log \left (x\right )^{2}}{x^{3}} + \frac {6 \, \log \left (25 \, x + 2\right )^{2} \log \left (x\right )^{2}}{x^{4}} + \frac {4 \, \log \left (x\right )^{3}}{x^{3}} + \frac {4 \, \log \left (25 \, x + 2\right ) \log \left (x\right )^{3}}{x^{4}} + \frac {\log \left (x\right )^{4}}{x^{4}} + \frac {20}{9}\right )} \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. 73 vs.
\(2 (21) = 42\).
time = 0.39, size = 73, normalized size = 3.04 \begin {gather*} e^{\left (\frac {20 \, x^{4} + 36 \, x^{3} \log \left (25 \, x^{2} + 2 \, x\right ) + 54 \, x^{2} \log \left (25 \, x^{2} + 2 \, x\right )^{2} + 36 \, x \log \left (25 \, x^{2} + 2 \, x\right )^{3} + 9 \, \log \left (25 \, x^{2} + 2 \, x\right )^{4}}{9 \, x^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (20) = 40\).
time = 0.36, size = 70, normalized size = 2.92 \begin {gather*} e^{\frac {\frac {20 x^{4}}{9} + 4 x^{3} \log {\left (25 x^{2} + 2 x \right )} + 6 x^{2} \log {\left (25 x^{2} + 2 x \right )}^{2} + 4 x \log {\left (25 x^{2} + 2 x \right )}^{3} + \log {\left (25 x^{2} + 2 x \right )}^{4}}{x^{4}}} \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. 68 vs.
\(2 (21) = 42\).
time = 1.10, size = 68, normalized size = 2.83 \begin {gather*} e^{\left (\frac {4 \, \log \left (25 \, x^{2} + 2 \, x\right )}{x} + \frac {6 \, \log \left (25 \, x^{2} + 2 \, x\right )^{2}}{x^{2}} + \frac {4 \, \log \left (25 \, x^{2} + 2 \, x\right )^{3}}{x^{3}} + \frac {\log \left (25 \, x^{2} + 2 \, x\right )^{4}}{x^{4}} + \frac {20}{9}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.31, size = 71, normalized size = 2.96 \begin {gather*} {\mathrm {e}}^{20/9}\,{\mathrm {e}}^{\frac {{\ln \left (25\,x^2+2\,x\right )}^4}{x^4}}\,{\mathrm {e}}^{\frac {6\,{\ln \left (25\,x^2+2\,x\right )}^2}{x^2}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (25\,x^2+2\,x\right )}^3}{x^3}}\,{\left (25\,x^2+2\,x\right )}^{4/x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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