Optimal. Leaf size=22 \[ \frac {16 e^{16} \left (1+\frac {x}{5}\right )^4}{\log \left (\frac {x}{25}\right )} \]
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Rubi [B] time = 0.61, antiderivative size = 84, normalized size of antiderivative = 3.82, number of steps used = 29, number of rules used = 12, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {12, 6688, 6742, 2353, 2297, 2298, 2302, 30, 2306, 2309, 2178, 2330} \begin {gather*} \frac {16 e^{16} x^4}{625 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x^3}{125 \log \left (\frac {x}{25}\right )}+\frac {96 e^{16} x^2}{25 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x}{5 \log \left (\frac {x}{25}\right )}+\frac {16 e^{16}}{\log \left (\frac {x}{25}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2178
Rule 2297
Rule 2298
Rule 2302
Rule 2306
Rule 2309
Rule 2330
Rule 2353
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{625} \int \frac {e^{16} \left (-10000-8000 x-2400 x^2-320 x^3-16 x^4\right )+e^{16} \left (8000 x+4800 x^2+960 x^3+64 x^4\right ) \log \left (\frac {x}{25}\right )}{x \log ^2\left (\frac {x}{25}\right )} \, dx\\ &=\frac {1}{625} \int \frac {16 e^{16} (5+x)^3 \left (-5-x+4 x \log \left (\frac {x}{25}\right )\right )}{x \log ^2\left (\frac {x}{25}\right )} \, dx\\ &=\frac {1}{625} \left (16 e^{16}\right ) \int \frac {(5+x)^3 \left (-5-x+4 x \log \left (\frac {x}{25}\right )\right )}{x \log ^2\left (\frac {x}{25}\right )} \, dx\\ &=\frac {1}{625} \left (16 e^{16}\right ) \int \left (-\frac {(5+x)^4}{x \log ^2\left (\frac {x}{25}\right )}+\frac {4 (5+x)^3}{\log \left (\frac {x}{25}\right )}\right ) \, dx\\ &=-\left (\frac {1}{625} \left (16 e^{16}\right ) \int \frac {(5+x)^4}{x \log ^2\left (\frac {x}{25}\right )} \, dx\right )+\frac {1}{625} \left (64 e^{16}\right ) \int \frac {(5+x)^3}{\log \left (\frac {x}{25}\right )} \, dx\\ &=-\left (\frac {1}{625} \left (16 e^{16}\right ) \int \left (\frac {500}{\log ^2\left (\frac {x}{25}\right )}+\frac {625}{x \log ^2\left (\frac {x}{25}\right )}+\frac {150 x}{\log ^2\left (\frac {x}{25}\right )}+\frac {20 x^2}{\log ^2\left (\frac {x}{25}\right )}+\frac {x^3}{\log ^2\left (\frac {x}{25}\right )}\right ) \, dx\right )+\frac {1}{625} \left (64 e^{16}\right ) \int \left (\frac {125}{\log \left (\frac {x}{25}\right )}+\frac {75 x}{\log \left (\frac {x}{25}\right )}+\frac {15 x^2}{\log \left (\frac {x}{25}\right )}+\frac {x^3}{\log \left (\frac {x}{25}\right )}\right ) \, dx\\ &=-\left (\frac {1}{625} \left (16 e^{16}\right ) \int \frac {x^3}{\log ^2\left (\frac {x}{25}\right )} \, dx\right )+\frac {1}{625} \left (64 e^{16}\right ) \int \frac {x^3}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{125} \left (64 e^{16}\right ) \int \frac {x^2}{\log ^2\left (\frac {x}{25}\right )} \, dx+\frac {1}{125} \left (192 e^{16}\right ) \int \frac {x^2}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{25} \left (96 e^{16}\right ) \int \frac {x}{\log ^2\left (\frac {x}{25}\right )} \, dx+\frac {1}{25} \left (192 e^{16}\right ) \int \frac {x}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{5} \left (64 e^{16}\right ) \int \frac {1}{\log ^2\left (\frac {x}{25}\right )} \, dx+\frac {1}{5} \left (64 e^{16}\right ) \int \frac {1}{\log \left (\frac {x}{25}\right )} \, dx-\left (16 e^{16}\right ) \int \frac {1}{x \log ^2\left (\frac {x}{25}\right )} \, dx\\ &=\frac {64 e^{16} x}{5 \log \left (\frac {x}{25}\right )}+\frac {96 e^{16} x^2}{25 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x^3}{125 \log \left (\frac {x}{25}\right )}+\frac {16 e^{16} x^4}{625 \log \left (\frac {x}{25}\right )}+320 e^{16} \text {li}\left (\frac {x}{25}\right )-\frac {1}{625} \left (64 e^{16}\right ) \int \frac {x^3}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{125} \left (192 e^{16}\right ) \int \frac {x^2}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{25} \left (192 e^{16}\right ) \int \frac {x}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{5} \left (64 e^{16}\right ) \int \frac {1}{\log \left (\frac {x}{25}\right )} \, dx-\left (16 e^{16}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {x}{25}\right )\right )+\left (4800 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )+\left (24000 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )+\left (40000 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )\\ &=4800 e^{16} \text {Ei}\left (2 \log \left (\frac {x}{25}\right )\right )+24000 e^{16} \text {Ei}\left (3 \log \left (\frac {x}{25}\right )\right )+40000 e^{16} \text {Ei}\left (4 \log \left (\frac {x}{25}\right )\right )+\frac {16 e^{16}}{\log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x}{5 \log \left (\frac {x}{25}\right )}+\frac {96 e^{16} x^2}{25 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x^3}{125 \log \left (\frac {x}{25}\right )}+\frac {16 e^{16} x^4}{625 \log \left (\frac {x}{25}\right )}-\left (4800 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )-\left (24000 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )-\left (40000 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )\\ &=\frac {16 e^{16}}{\log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x}{5 \log \left (\frac {x}{25}\right )}+\frac {96 e^{16} x^2}{25 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x^3}{125 \log \left (\frac {x}{25}\right )}+\frac {16 e^{16} x^4}{625 \log \left (\frac {x}{25}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 20, normalized size = 0.91 \begin {gather*} \frac {16 e^{16} (5+x)^4}{625 \log \left (\frac {x}{25}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 28, normalized size = 1.27 \begin {gather*} \frac {16 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )} e^{16}}{625 \, \log \left (\frac {1}{25} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 38, normalized size = 1.73 \begin {gather*} \frac {16 \, {\left (x^{4} e^{16} + 20 \, x^{3} e^{16} + 150 \, x^{2} e^{16} + 500 \, x e^{16} + 625 \, e^{16}\right )}}{625 \, \log \left (\frac {1}{25} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 29, normalized size = 1.32
method | result | size |
risch | \(\frac {16 \,{\mathrm e}^{16} \left (x^{4}+20 x^{3}+150 x^{2}+500 x +625\right )}{625 \ln \left (\frac {x}{25}\right )}\) | \(29\) |
norman | \(\frac {16 \,{\mathrm e}^{16}+\frac {64 x \,{\mathrm e}^{16}}{5}+\frac {96 x^{2} {\mathrm e}^{16}}{25}+\frac {64 x^{3} {\mathrm e}^{16}}{125}+\frac {16 x^{4} {\mathrm e}^{16}}{625}}{\ln \left (\frac {x}{25}\right )}\) | \(49\) |
derivativedivides | \(-40000 \,{\mathrm e}^{16} \expIntegralEi \left (1, -4 \ln \left (\frac {x}{25}\right )\right )-24000 \,{\mathrm e}^{16} \expIntegralEi \left (1, -3 \ln \left (\frac {x}{25}\right )\right )-10000 \,{\mathrm e}^{16} \left (-\frac {x^{4}}{390625 \ln \left (\frac {x}{25}\right )}-4 \expIntegralEi \left (1, -4 \ln \left (\frac {x}{25}\right )\right )\right )-4800 \,{\mathrm e}^{16} \expIntegralEi \left (1, -2 \ln \left (\frac {x}{25}\right )\right )-8000 \,{\mathrm e}^{16} \left (-\frac {x^{3}}{15625 \ln \left (\frac {x}{25}\right )}-3 \expIntegralEi \left (1, -3 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \expIntegralEi \left (1, -\ln \left (\frac {x}{25}\right )\right )-2400 \,{\mathrm e}^{16} \left (-\frac {x^{2}}{625 \ln \left (\frac {x}{25}\right )}-2 \expIntegralEi \left (1, -2 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \left (-\frac {x}{25 \ln \left (\frac {x}{25}\right )}-\expIntegralEi \left (1, -\ln \left (\frac {x}{25}\right )\right )\right )+\frac {16 \,{\mathrm e}^{16}}{\ln \left (\frac {x}{25}\right )}\) | \(180\) |
default | \(-40000 \,{\mathrm e}^{16} \expIntegralEi \left (1, -4 \ln \left (\frac {x}{25}\right )\right )-24000 \,{\mathrm e}^{16} \expIntegralEi \left (1, -3 \ln \left (\frac {x}{25}\right )\right )-10000 \,{\mathrm e}^{16} \left (-\frac {x^{4}}{390625 \ln \left (\frac {x}{25}\right )}-4 \expIntegralEi \left (1, -4 \ln \left (\frac {x}{25}\right )\right )\right )-4800 \,{\mathrm e}^{16} \expIntegralEi \left (1, -2 \ln \left (\frac {x}{25}\right )\right )-8000 \,{\mathrm e}^{16} \left (-\frac {x^{3}}{15625 \ln \left (\frac {x}{25}\right )}-3 \expIntegralEi \left (1, -3 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \expIntegralEi \left (1, -\ln \left (\frac {x}{25}\right )\right )-2400 \,{\mathrm e}^{16} \left (-\frac {x^{2}}{625 \ln \left (\frac {x}{25}\right )}-2 \expIntegralEi \left (1, -2 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \left (-\frac {x}{25 \ln \left (\frac {x}{25}\right )}-\expIntegralEi \left (1, -\ln \left (\frac {x}{25}\right )\right )\right )+\frac {16 \,{\mathrm e}^{16}}{\ln \left (\frac {x}{25}\right )}\) | \(180\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 101, normalized size = 4.59 \begin {gather*} 40000 \, {\rm Ei}\left (4 \, \log \left (\frac {1}{25} \, x\right )\right ) e^{16} + 24000 \, {\rm Ei}\left (3 \, \log \left (\frac {1}{25} \, x\right )\right ) e^{16} + 4800 \, {\rm Ei}\left (2 \, \log \left (\frac {1}{25} \, x\right )\right ) e^{16} + 320 \, {\rm Ei}\left (\log \left (\frac {1}{25} \, x\right )\right ) e^{16} - 320 \, e^{16} \Gamma \left (-1, -\log \left (\frac {1}{25} \, x\right )\right ) - 4800 \, e^{16} \Gamma \left (-1, -2 \, \log \left (\frac {1}{25} \, x\right )\right ) - 24000 \, e^{16} \Gamma \left (-1, -3 \, \log \left (\frac {1}{25} \, x\right )\right ) - 40000 \, e^{16} \Gamma \left (-1, -4 \, \log \left (\frac {1}{25} \, x\right )\right ) + \frac {16 \, e^{16}}{\log \left (\frac {1}{25} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.43, size = 47, normalized size = 2.14 \begin {gather*} \frac {16\,{\mathrm {e}}^{16}\,x^6+320\,{\mathrm {e}}^{16}\,x^5+2400\,{\mathrm {e}}^{16}\,x^4+8000\,{\mathrm {e}}^{16}\,x^3+10000\,{\mathrm {e}}^{16}\,x^2}{625\,x^2\,\ln \left (\frac {x}{25}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 42, normalized size = 1.91 \begin {gather*} \frac {16 x^{4} e^{16} + 320 x^{3} e^{16} + 2400 x^{2} e^{16} + 8000 x e^{16} + 10000 e^{16}}{625 \log {\left (\frac {x}{25} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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