Optimal. Leaf size=24 \[ \frac {1}{5} x^4 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \]
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Rubi [F] time = 129.88, 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 {\left (-12 x^4-x^5+x^6+e^x \left (6 x^4+2 x^5\right )\right ) \log ^2(3+x)+\log \left (e^{2 x}+e^x (-5+x)\right ) \left (\left (-10 x^4+2 e^x x^4+2 x^5\right ) \log (3+x)+\left (-60 x^3-8 x^4+4 x^5+e^x \left (12 x^3+4 x^4\right )\right ) \log ^2(3+x)\right )}{-75-10 x+5 x^2+e^x (15+5 x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{5} x^3 \log (3+x) \left (\frac {2 x \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}+\log (3+x) \left (\frac {x \left (-4+2 e^x+x\right )}{-5+e^x+x}+4 \log \left (e^x \left (-5+e^x+x\right )\right )\right )\right ) \, dx\\ &=\frac {1}{5} \int x^3 \log (3+x) \left (\frac {2 x \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}+\log (3+x) \left (\frac {x \left (-4+2 e^x+x\right )}{-5+e^x+x}+4 \log \left (e^x \left (-5+e^x+x\right )\right )\right )\right ) \, dx\\ &=\frac {1}{5} \int \left (-\frac {(-6+x) x^4 \log ^2(3+x)}{-5+e^x+x}+\frac {2 x^3 \log (3+x) \left (3 x \log (3+x)+x^2 \log (3+x)+x \log \left (e^x \left (-5+e^x+x\right )\right )+6 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right )}{3+x}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {(-6+x) x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\right )+\frac {2}{5} \int \frac {x^3 \log (3+x) \left (3 x \log (3+x)+x^2 \log (3+x)+x \log \left (e^x \left (-5+e^x+x\right )\right )+6 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right )}{3+x} \, dx\\ &=-\left (\frac {1}{5} \int \left (-\frac {6 x^4 \log ^2(3+x)}{-5+e^x+x}+\frac {x^5 \log ^2(3+x)}{-5+e^x+x}\right ) \, dx\right )+\frac {2}{5} \int \frac {x^3 \log (3+x) \left (x \log \left (e^x \left (-5+e^x+x\right )\right )+(3+x) \log (3+x) \left (x+2 \log \left (e^x \left (-5+e^x+x\right )\right )\right )\right )}{3+x} \, dx\\ &=-\left (\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx\right )+\frac {2}{5} \int \left (x^4 \log ^2(3+x)+\frac {x^3 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\\ &=-\left (\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx\right )+\frac {2}{5} \int x^4 \log ^2(3+x) \, dx+\frac {2}{5} \int \frac {x^3 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\\ &=\frac {2}{25} x^5 \log ^2(3+x)-\frac {4}{25} \int \frac {x^5 \log (3+x)}{3+x} \, dx-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int \frac {x^3 \log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\\ &=\frac {2}{25} x^5 \log ^2(3+x)-\frac {4}{25} \operatorname {Subst}\left (\int \frac {(-3+x)^5 \log (x)}{x} \, dx,x,3+x\right )-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int \left (9 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )-3 x \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )+x^2 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )-\frac {27 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\\ &=-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)+\frac {2}{25} x^5 \log ^2(3+x)+\frac {4}{25} \operatorname {Subst}\left (\int \left (405-135 x+30 x^2-\frac {15 x^3}{4}+\frac {x^4}{5}-\frac {243 \log (x)}{x}\right ) \, dx,x,3+x\right )-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int x^2 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {6}{5} \int x \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {18}{5} \int \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {54}{5} \int \frac {\log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int x^2 \log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {6}{5} \int x \log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {18}{5} \int \log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {54}{5} \int \frac {\log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx-\frac {972}{25} \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,3+x\right )\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)-\frac {486}{25} \log ^2(3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int \left (x^3 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+6 x^2 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x^3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {6}{5} \int \left (x^2 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+6 x \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x^2 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right ) \, dx+\frac {18}{5} \int \left (x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+6 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right ) \, dx-\frac {54}{5} \int \left (\frac {x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}+\frac {6 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}+\frac {2 x \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)-\frac {486}{25} \log ^2(3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int x^3 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {4}{5} \int x^3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {6}{5} \int x^2 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {18}{5} \int x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {54}{5} \int \frac {x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx+\frac {108}{5} \int \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {108}{5} \int \frac {x \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx-\frac {324}{5} \int \frac {\log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5+\frac {9}{5} x^2 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)-\frac {2}{5} x^3 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)+\frac {1}{10} x^4 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)-\frac {486}{25} \log ^2(3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {2}{5} \int \frac {\left (4-2 e^x-x\right ) x^4 \log (3+x)}{4 \left (5-e^x-x\right )} \, dx-\frac {2}{5} \int \frac {x^4 \log \left (e^x \left (-5+e^x+x\right )\right )}{4 (3+x)} \, dx+\frac {4}{5} \int x^3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {\left (4-2 e^x-x\right ) x^3 \log (3+x)}{3 \left (5-e^x-x\right )} \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {6}{5} \int \frac {x^3 \log \left (e^x \left (-5+e^x+x\right )\right )}{9+3 x} \, dx-\frac {18}{5} \int \frac {\left (4-2 e^x-x\right ) x^2 \log (3+x)}{2 \left (5-e^x-x\right )} \, dx-\frac {18}{5} \int \frac {x^2 \log \left (e^x \left (-5+e^x+x\right )\right )}{6+2 x} \, dx-\frac {54}{5} \int \left (\log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )-\frac {3 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx+\frac {108}{5} \int \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {108}{5} \int \left (\log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )-\frac {3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx-\frac {324}{5} \int \frac {\log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5+\frac {9}{5} x^2 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)-\frac {2}{5} x^3 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)+\frac {1}{10} x^4 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)-\frac {486}{25} \log ^2(3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{10} \int \frac {\left (4-2 e^x-x\right ) x^4 \log (3+x)}{5-e^x-x} \, dx-\frac {1}{10} \int \frac {x^4 \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int \frac {\left (4-2 e^x-x\right ) x^3 \log (3+x)}{5-e^x-x} \, dx+\frac {4}{5} \int x^3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {6}{5} \int \left (3 \log \left (e^x \left (-5+e^x+x\right )\right )-x \log \left (e^x \left (-5+e^x+x\right )\right )+\frac {1}{3} x^2 \log \left (e^x \left (-5+e^x+x\right )\right )-\frac {9 \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx-\frac {9}{5} \int \frac {\left (4-2 e^x-x\right ) x^2 \log (3+x)}{5-e^x-x} \, dx-\frac {18}{5} \int \left (-\frac {3}{2} \log \left (e^x \left (-5+e^x+x\right )\right )+\frac {1}{2} x \log \left (e^x \left (-5+e^x+x\right )\right )+\frac {9 \log \left (e^x \left (-5+e^x+x\right )\right )}{2 (3+x)}\right ) \, dx-\frac {54}{5} \int \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {162}{5} \int \frac {\log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.63, size = 43, normalized size = 1.79 \begin {gather*} \frac {1}{5} \left (-162 (3+x)-162 \log \left (-5+e^x+x\right )+\left (162+x^4 \log ^2(3+x)\right ) \log \left (e^x \left (-5+e^x+x\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 23, normalized size = 0.96 \begin {gather*} \frac {1}{5} \, x^{4} \log \left ({\left (x - 5\right )} e^{x} + e^{\left (2 \, x\right )}\right ) \log \left (x + 3\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 29, normalized size = 1.21 \begin {gather*} \frac {1}{5} \, x^{5} \log \left (x + 3\right )^{2} + \frac {1}{5} \, x^{4} \log \left (x + e^{x} - 5\right ) \log \left (x + 3\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 165, normalized size = 6.88
method | result | size |
risch | \(\frac {x^{4} \ln \left (3+x \right )^{2} \ln \left ({\mathrm e}^{x}\right )}{5}+\frac {x^{4} \ln \left (3+x \right )^{2} \ln \left ({\mathrm e}^{x}+x -5\right )}{5}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x -5\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x} \left ({\mathrm e}^{x}+x -5\right )\right ) \ln \left (3+x \right )^{2}}{10}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x -5\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x} \left ({\mathrm e}^{x}+x -5\right )\right )^{2} \ln \left (3+x \right )^{2}}{10}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x} \left ({\mathrm e}^{x}+x -5\right )\right )^{2} \ln \left (3+x \right )^{2}}{10}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{x} \left ({\mathrm e}^{x}+x -5\right )\right )^{3} \ln \left (3+x \right )^{2}}{10}\) | \(165\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 29, normalized size = 1.21 \begin {gather*} \frac {1}{5} \, x^{5} \log \left (x + 3\right )^{2} + \frac {1}{5} \, x^{4} \log \left (x + e^{x} - 5\right ) \log \left (x + 3\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 23, normalized size = 0.96 \begin {gather*} \frac {x^4\,\ln \left ({\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (x-5\right )\right )\,{\ln \left (x+3\right )}^2}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.83, size = 24, normalized size = 1.00 \begin {gather*} \frac {x^{4} \log {\left (x + 3 \right )}^{2} \log {\left (\left (x - 5\right ) e^{x} + e^{2 x} \right )}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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