Optimal. Leaf size=19 \[ \frac {1}{32} x^3 \left (-1+\log \left (x+\frac {x^2}{25}\right )\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 24, normalized size of antiderivative = 1.26, number of steps
used = 10, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {6820, 12,
6874, 78, 2581, 30, 45} \begin {gather*} \frac {1}{32} x^3 \log \left (\frac {1}{25} x (x+25)\right )-\frac {x^3}{32} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 45
Rule 78
Rule 2581
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (-50-x+3 (25+x) \log \left (\frac {1}{25} x (25+x)\right )\right )}{32 (25+x)} \, dx\\ &=\frac {1}{32} \int \frac {x^2 \left (-50-x+3 (25+x) \log \left (\frac {1}{25} x (25+x)\right )\right )}{25+x} \, dx\\ &=\frac {1}{32} \int \left (-\frac {x^2 (50+x)}{25+x}+3 x^2 \log \left (\frac {1}{25} x (25+x)\right )\right ) \, dx\\ &=-\left (\frac {1}{32} \int \frac {x^2 (50+x)}{25+x} \, dx\right )+\frac {3}{32} \int x^2 \log \left (\frac {1}{25} x (25+x)\right ) \, dx\\ &=\frac {1}{32} x^3 \log \left (\frac {1}{25} x (25+x)\right )-\frac {\int x^2 \, dx}{32}-\frac {1}{32} \int \frac {x^3}{25+x} \, dx-\frac {1}{32} \int \left (-625+25 x+x^2+\frac {15625}{25+x}\right ) \, dx\\ &=\frac {625 x}{32}-\frac {25 x^2}{64}-\frac {x^3}{48}-\frac {15625}{32} \log (25+x)+\frac {1}{32} x^3 \log \left (\frac {1}{25} x (25+x)\right )-\frac {1}{32} \int \left (625-25 x+x^2-\frac {15625}{25+x}\right ) \, dx\\ &=-\frac {x^3}{32}+\frac {1}{32} x^3 \log \left (\frac {1}{25} x (25+x)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 1.21 \begin {gather*} \frac {1}{32} \left (-x^3+x^3 \log \left (\frac {1}{25} x (25+x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 27, normalized size = 1.42
method | result | size |
norman | \(-\frac {x^{3}}{32}+\frac {x^{3} \ln \left (\frac {1}{25} x^{2}+x \right )}{32}\) | \(20\) |
risch | \(-\frac {x^{3}}{32}+\frac {x^{3} \ln \left (\frac {1}{25} x^{2}+x \right )}{32}\) | \(20\) |
default | \(-\frac {x^{3} \ln \left (5\right )}{16}+\frac {x^{3} \ln \left (x^{2}+25 x \right )}{32}-\frac {x^{3}}{32}\) | \(27\) |
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. 41 vs.
\(2 (15) = 30\).
time = 0.50, size = 41, normalized size = 2.16 \begin {gather*} -\frac {1}{48} \, x^{3} {\left (3 \, \log \left (5\right ) + 1\right )} + \frac {1}{32} \, x^{3} \log \left (x\right ) - \frac {1}{96} \, x^{3} + \frac {1}{32} \, {\left (x^{3} + 15625\right )} \log \left (x + 25\right ) - \frac {15625}{32} \, \log \left (x + 25\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{32} \, x^{3} \log \left (\frac {1}{25} \, x^{2} + x\right ) - \frac {1}{32} \, x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 17, normalized size = 0.89 \begin {gather*} \frac {x^{3} \log {\left (\frac {x^{2}}{25} + x \right )}}{32} - \frac {x^{3}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{32} \, x^{3} \log \left (\frac {1}{25} \, x^{2} + x\right ) - \frac {1}{32} \, x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.52, size = 15, normalized size = 0.79 \begin {gather*} \frac {x^3\,\left (\ln \left (\frac {x^2}{25}+x\right )-1\right )}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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