Optimal. Leaf size=26 \[ i \pi +\frac {e \log ^2(3+x (25+x))}{x}+\log (3+\log (2)) \]
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Rubi [A]
time = 1.70, antiderivative size = 16, normalized size of antiderivative = 0.62, number of steps
used = 60, number of rules used = 19, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1608, 6820,
12, 14, 814, 646, 31, 2605, 6874, 2608, 2604, 2404, 2354, 2438, 2465, 2437, 2338, 2441, 2440}
\begin {gather*} \frac {e \log ^2\left (x^2+25 x+3\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 31
Rule 646
Rule 814
Rule 1608
Rule 2338
Rule 2354
Rule 2404
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2465
Rule 2604
Rule 2605
Rule 2608
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e \left (50 x+4 x^2\right ) \log \left (3+25 x+x^2\right )+e \left (-3-25 x-x^2\right ) \log ^2\left (3+25 x+x^2\right )}{x^2 \left (3+25 x+x^2\right )} \, dx\\ &=\int \frac {e \left (\frac {2 x (25+2 x)}{3+25 x+x^2}-\log \left (3+25 x+x^2\right )\right ) \log \left (3+25 x+x^2\right )}{x^2} \, dx\\ &=e \int \frac {\left (\frac {2 x (25+2 x)}{3+25 x+x^2}-\log \left (3+25 x+x^2\right )\right ) \log \left (3+25 x+x^2\right )}{x^2} \, dx\\ &=e \int \left (\frac {2 (25+2 x) \log \left (3+25 x+x^2\right )}{x \left (3+25 x+x^2\right )}-\frac {\log ^2\left (3+25 x+x^2\right )}{x^2}\right ) \, dx\\ &=-\left (e \int \frac {\log ^2\left (3+25 x+x^2\right )}{x^2} \, dx\right )+(2 e) \int \frac {(25+2 x) \log \left (3+25 x+x^2\right )}{x \left (3+25 x+x^2\right )} \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-(2 e) \int \frac {(25+2 x) \log \left (3+25 x+x^2\right )}{x \left (3+25 x+x^2\right )} \, dx+(2 e) \int \left (\frac {25 \log \left (3+25 x+x^2\right )}{3 x}+\frac {(-619-25 x) \log \left (3+25 x+x^2\right )}{3 \left (3+25 x+x^2\right )}\right ) \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} (2 e) \int \frac {(-619-25 x) \log \left (3+25 x+x^2\right )}{3+25 x+x^2} \, dx-(2 e) \int \left (\frac {25 \log \left (3+25 x+x^2\right )}{3 x}+\frac {(-619-25 x) \log \left (3+25 x+x^2\right )}{3 \left (3+25 x+x^2\right )}\right ) \, dx+\frac {1}{3} (50 e) \int \frac {\log \left (3+25 x+x^2\right )}{x} \, dx\\ &=\frac {50}{3} e \log (x) \log \left (3+25 x+x^2\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-\frac {1}{3} (2 e) \int \frac {(-619-25 x) \log \left (3+25 x+x^2\right )}{3+25 x+x^2} \, dx+\frac {1}{3} (2 e) \int \left (\frac {\left (-25-\sqrt {613}\right ) \log \left (3+25 x+x^2\right )}{25-\sqrt {613}+2 x}+\frac {\left (-25+\sqrt {613}\right ) \log \left (3+25 x+x^2\right )}{25+\sqrt {613}+2 x}\right ) \, dx-\frac {1}{3} (50 e) \int \frac {(25+2 x) \log (x)}{3+25 x+x^2} \, dx-\frac {1}{3} (50 e) \int \frac {\log \left (3+25 x+x^2\right )}{x} \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-\frac {1}{3} (2 e) \int \left (\frac {\left (-25-\sqrt {613}\right ) \log \left (3+25 x+x^2\right )}{25-\sqrt {613}+2 x}+\frac {\left (-25+\sqrt {613}\right ) \log \left (3+25 x+x^2\right )}{25+\sqrt {613}+2 x}\right ) \, dx+\frac {1}{3} (50 e) \int \frac {(25+2 x) \log (x)}{3+25 x+x^2} \, dx-\frac {1}{3} (50 e) \int \left (\frac {2 \log (x)}{25-\sqrt {613}+2 x}+\frac {2 \log (x)}{25+\sqrt {613}+2 x}\right ) \, dx-\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (3+25 x+x^2\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (3+25 x+x^2\right )}{25-\sqrt {613}+2 x} \, dx\\ &=-\frac {1}{3} \left (25+\sqrt {613}\right ) e \log \left (25-\sqrt {613}+2 x\right ) \log \left (3+25 x+x^2\right )-\frac {1}{3} \left (25-\sqrt {613}\right ) e \log \left (25+\sqrt {613}+2 x\right ) \log \left (3+25 x+x^2\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} (50 e) \int \left (\frac {2 \log (x)}{25-\sqrt {613}+2 x}+\frac {2 \log (x)}{25+\sqrt {613}+2 x}\right ) \, dx-\frac {1}{3} (100 e) \int \frac {\log (x)}{25-\sqrt {613}+2 x} \, dx-\frac {1}{3} (100 e) \int \frac {\log (x)}{25+\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \int \frac {(25+2 x) \log \left (25+\sqrt {613}+2 x\right )}{3+25 x+x^2} \, dx+\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (3+25 x+x^2\right )}{25+\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \int \frac {(25+2 x) \log \left (25-\sqrt {613}+2 x\right )}{3+25 x+x^2} \, dx+\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (3+25 x+x^2\right )}{25-\sqrt {613}+2 x} \, dx\\ &=-\frac {50}{3} e \log (x) \log \left (1+\frac {2 x}{25-\sqrt {613}}\right )-\frac {50}{3} e \log (x) \log \left (1+\frac {2 x}{25+\sqrt {613}}\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} (50 e) \int \frac {\log \left (1+\frac {2 x}{25-\sqrt {613}}\right )}{x} \, dx+\frac {1}{3} (50 e) \int \frac {\log \left (1+\frac {2 x}{25+\sqrt {613}}\right )}{x} \, dx+\frac {1}{3} (100 e) \int \frac {\log (x)}{25-\sqrt {613}+2 x} \, dx+\frac {1}{3} (100 e) \int \frac {\log (x)}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \int \frac {(25+2 x) \log \left (25+\sqrt {613}+2 x\right )}{3+25 x+x^2} \, dx+\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \int \left (\frac {2 \log \left (25+\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x}+\frac {2 \log \left (25+\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x}\right ) \, dx-\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \int \frac {(25+2 x) \log \left (25-\sqrt {613}+2 x\right )}{3+25 x+x^2} \, dx+\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \int \left (\frac {2 \log \left (25-\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x}+\frac {2 \log \left (25-\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x}\right ) \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-\frac {50}{3} e \text {Li}_2\left (-\frac {2 x}{25-\sqrt {613}}\right )-\frac {50}{3} e \text {Li}_2\left (-\frac {2 x}{25+\sqrt {613}}\right )-\frac {1}{3} (50 e) \int \frac {\log \left (1+\frac {2 x}{25-\sqrt {613}}\right )}{x} \, dx-\frac {1}{3} (50 e) \int \frac {\log \left (1+\frac {2 x}{25+\sqrt {613}}\right )}{x} \, dx-\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \int \left (\frac {2 \log \left (25+\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x}+\frac {2 \log \left (25+\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x}\right ) \, dx+\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (25+\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (25+\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \int \left (\frac {2 \log \left (25-\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x}+\frac {2 \log \left (25-\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x}\right ) \, dx+\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (25-\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (25-\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x} \, dx\\ &=\frac {1}{3} \left (25-\sqrt {613}\right ) e \log \left (-\frac {25-\sqrt {613}+2 x}{2 \sqrt {613}}\right ) \log \left (25+\sqrt {613}+2 x\right )+\frac {1}{3} \left (25+\sqrt {613}\right ) e \log \left (25-\sqrt {613}+2 x\right ) \log \left (\frac {25+\sqrt {613}+2 x}{2 \sqrt {613}}\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25+\sqrt {613}+2 x\right )-\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (\frac {2 \left (25-\sqrt {613}+2 x\right )}{2 \left (25-\sqrt {613}\right )-2 \left (25+\sqrt {613}\right )}\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (25+\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (25+\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x} \, dx+\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25-\sqrt {613}+2 x\right )-\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (25-\sqrt {613}+2 x\right )}{25-\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (25-\sqrt {613}+2 x\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (\frac {2 \left (25+\sqrt {613}+2 x\right )}{-2 \left (25-\sqrt {613}\right )+2 \left (25+\sqrt {613}\right )}\right )}{25-\sqrt {613}+2 x} \, dx\\ &=\frac {1}{6} \left (25+\sqrt {613}\right ) e \log ^2\left (25-\sqrt {613}+2 x\right )+\frac {1}{6} \left (25-\sqrt {613}\right ) e \log ^2\left (25+\sqrt {613}+2 x\right )+\frac {e \log ^2\left (3+25 x+x^2\right )}{x}-\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25+\sqrt {613}+2 x\right )-\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (25-\sqrt {613}\right )-2 \left (25+\sqrt {613}\right )}\right )}{x} \, dx,x,25+\sqrt {613}+2 x\right )+\frac {1}{3} \left (2 \left (25-\sqrt {613}\right ) e\right ) \int \frac {\log \left (\frac {2 \left (25-\sqrt {613}+2 x\right )}{2 \left (25-\sqrt {613}\right )-2 \left (25+\sqrt {613}\right )}\right )}{25+\sqrt {613}+2 x} \, dx-\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25-\sqrt {613}+2 x\right )-\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (25-\sqrt {613}\right )+2 \left (25+\sqrt {613}\right )}\right )}{x} \, dx,x,25-\sqrt {613}+2 x\right )+\frac {1}{3} \left (2 \left (25+\sqrt {613}\right ) e\right ) \int \frac {\log \left (\frac {2 \left (25+\sqrt {613}+2 x\right )}{-2 \left (25-\sqrt {613}\right )+2 \left (25+\sqrt {613}\right )}\right )}{25-\sqrt {613}+2 x} \, dx\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}+\frac {1}{3} \left (25+\sqrt {613}\right ) e \text {Li}_2\left (-\frac {25-\sqrt {613}+2 x}{2 \sqrt {613}}\right )+\frac {1}{3} \left (25-\sqrt {613}\right ) e \text {Li}_2\left (\frac {25+\sqrt {613}+2 x}{2 \sqrt {613}}\right )+\frac {1}{3} \left (\left (25-\sqrt {613}\right ) e\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (25-\sqrt {613}\right )-2 \left (25+\sqrt {613}\right )}\right )}{x} \, dx,x,25+\sqrt {613}+2 x\right )+\frac {1}{3} \left (\left (25+\sqrt {613}\right ) e\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (25-\sqrt {613}\right )+2 \left (25+\sqrt {613}\right )}\right )}{x} \, dx,x,25-\sqrt {613}+2 x\right )\\ &=\frac {e \log ^2\left (3+25 x+x^2\right )}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 16, normalized size = 0.62 \begin {gather*} \frac {e \log ^2\left (3+25 x+x^2\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 18, normalized size = 0.69
method | result | size |
norman | \(\frac {{\mathrm e} \ln \left (x^{2}+25 x +3\right )^{2}}{x}\) | \(18\) |
risch | \(\frac {{\mathrm e} \ln \left (x^{2}+25 x +3\right )^{2}}{x}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.60, size = 17, normalized size = 0.65 \begin {gather*} \frac {e \log \left (x^{2} + 25 \, x + 3\right )^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 17, normalized size = 0.65 \begin {gather*} \frac {e \log \left (x^{2} + 25 \, x + 3\right )^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 15, normalized size = 0.58 \begin {gather*} \frac {e \log {\left (x^{2} + 25 x + 3 \right )}^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 17, normalized size = 0.65 \begin {gather*} \frac {e \log \left (x^{2} + 25 \, x + 3\right )^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 17, normalized size = 0.65 \begin {gather*} \frac {\mathrm {e}\,{\ln \left (x^2+25\,x+3\right )}^2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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