Optimal. Leaf size=19 \[ \left (\frac {1}{4} (5-x)^2+29^x x\right )^2 \]
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Rubi [B] time = 0.23, antiderivative size = 62, normalized size of antiderivative = 3.26, number of steps used = 29, number of rules used = 5, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 1593, 2196, 2176, 2194} \begin {gather*} \frac {x^4}{16}+\frac {29^x x^3}{2}-\frac {5 x^3}{4}-5\ 29^x x^2+29^{2 x} x^2+\frac {75 x^2}{8}+\frac {25\ 29^x x}{2}-\frac {125 x}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1593
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (-125+75 x-15 x^2+x^3+29^{2 x} \left (8 x+8 x^2 \log (29)\right )+29^x \left (50-40 x+6 x^2+\left (50 x-20 x^2+2 x^3\right ) \log (29)\right )\right ) \, dx\\ &=-\frac {125 x}{4}+\frac {75 x^2}{8}-\frac {5 x^3}{4}+\frac {x^4}{16}+\frac {1}{4} \int 29^{2 x} \left (8 x+8 x^2 \log (29)\right ) \, dx+\frac {1}{4} \int 29^x \left (50-40 x+6 x^2+\left (50 x-20 x^2+2 x^3\right ) \log (29)\right ) \, dx\\ &=-\frac {125 x}{4}+\frac {75 x^2}{8}-\frac {5 x^3}{4}+\frac {x^4}{16}+\frac {1}{4} \int 29^{2 x} x (8+8 x \log (29)) \, dx+\frac {1}{4} \int \left (50\ 29^x-40\ 29^x x+6\ 29^x x^2+2\ 29^x (-5+x)^2 x \log (29)\right ) \, dx\\ &=-\frac {125 x}{4}+\frac {75 x^2}{8}-\frac {5 x^3}{4}+\frac {x^4}{16}+\frac {1}{4} \int \left (8\ 29^{2 x} x+8\ 29^{2 x} x^2 \log (29)\right ) \, dx+\frac {3}{2} \int 29^x x^2 \, dx-10 \int 29^x x \, dx+\frac {25 \int 29^x \, dx}{2}+\frac {1}{2} \log (29) \int 29^x (-5+x)^2 x \, dx\\ &=-\frac {125 x}{4}+\frac {75 x^2}{8}-\frac {5 x^3}{4}+\frac {x^4}{16}+\frac {25\ 29^x}{2 \log (29)}-\frac {10\ 29^x x}{\log (29)}+\frac {3\ 29^x x^2}{2 \log (29)}+2 \int 29^{2 x} x \, dx-\frac {3 \int 29^x x \, dx}{\log (29)}+\frac {10 \int 29^x \, dx}{\log (29)}+\frac {1}{2} \log (29) \int \left (25\ 29^x x-10\ 29^x x^2+29^x x^3\right ) \, dx+(2 \log (29)) \int 29^{2 x} x^2 \, dx\\ &=-\frac {125 x}{4}+\frac {75 x^2}{8}+29^{2 x} x^2-\frac {5 x^3}{4}+\frac {x^4}{16}+\frac {10\ 29^x}{\log ^2(29)}-\frac {3\ 29^x x}{\log ^2(29)}+\frac {25\ 29^x}{2 \log (29)}-\frac {10\ 29^x x}{\log (29)}+\frac {29^{2 x} x}{\log (29)}+\frac {3\ 29^x x^2}{2 \log (29)}-2 \int 29^{2 x} x \, dx+\frac {3 \int 29^x \, dx}{\log ^2(29)}-\frac {\int 29^{2 x} \, dx}{\log (29)}+\frac {1}{2} \log (29) \int 29^x x^3 \, dx-(5 \log (29)) \int 29^x x^2 \, dx+\frac {1}{2} (25 \log (29)) \int 29^x x \, dx\\ &=-\frac {125 x}{4}+\frac {25\ 29^x x}{2}+\frac {75 x^2}{8}-5\ 29^x x^2+29^{2 x} x^2-\frac {5 x^3}{4}+\frac {29^x x^3}{2}+\frac {x^4}{16}+\frac {3\ 29^x}{\log ^3(29)}+\frac {10\ 29^x}{\log ^2(29)}-\frac {29^{2 x}}{2 \log ^2(29)}-\frac {3\ 29^x x}{\log ^2(29)}+\frac {25\ 29^x}{2 \log (29)}-\frac {10\ 29^x x}{\log (29)}+\frac {3\ 29^x x^2}{2 \log (29)}-\frac {3}{2} \int 29^x x^2 \, dx+10 \int 29^x x \, dx-\frac {25 \int 29^x \, dx}{2}+\frac {\int 29^{2 x} \, dx}{\log (29)}\\ &=-\frac {125 x}{4}+\frac {25\ 29^x x}{2}+\frac {75 x^2}{8}-5\ 29^x x^2+29^{2 x} x^2-\frac {5 x^3}{4}+\frac {29^x x^3}{2}+\frac {x^4}{16}+\frac {3\ 29^x}{\log ^3(29)}+\frac {10\ 29^x}{\log ^2(29)}-\frac {3\ 29^x x}{\log ^2(29)}+\frac {3 \int 29^x x \, dx}{\log (29)}-\frac {10 \int 29^x \, dx}{\log (29)}\\ &=-\frac {125 x}{4}+\frac {25\ 29^x x}{2}+\frac {75 x^2}{8}-5\ 29^x x^2+29^{2 x} x^2-\frac {5 x^3}{4}+\frac {29^x x^3}{2}+\frac {x^4}{16}+\frac {3\ 29^x}{\log ^3(29)}-\frac {3 \int 29^x \, dx}{\log ^2(29)}\\ &=-\frac {125 x}{4}+\frac {25\ 29^x x}{2}+\frac {75 x^2}{8}-5\ 29^x x^2+29^{2 x} x^2-\frac {5 x^3}{4}+\frac {29^x x^3}{2}+\frac {x^4}{16}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 21, normalized size = 1.11 \begin {gather*} \frac {1}{16} \left (25+2 \left (-5+2\ 29^x\right ) x+x^2\right )^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 45, normalized size = 2.37 \begin {gather*} \frac {1}{16} \, x^{4} + 29^{2 \, x} x^{2} - \frac {5}{4} \, x^{3} + \frac {1}{2} \, {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} 29^{x} + \frac {75}{8} \, x^{2} - \frac {125}{4} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 62, normalized size = 3.26 \begin {gather*} \frac {1}{16} \, x^{4} + 29^{2 \, x} x^{2} - \frac {5}{4} \, x^{3} + \frac {75}{8} \, x^{2} - \frac {125}{4} \, x + \frac {{\left (x^{3} \log \left (29\right )^{4} - 10 \, x^{2} \log \left (29\right )^{4} + 25 \, x \log \left (29\right )^{4}\right )} 29^{x}}{2 \, \log \left (29\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 48, normalized size = 2.53
method | result | size |
risch | \(29^{2 x} x^{2}+\frac {\left (2 x^{3}-20 x^{2}+50 x \right ) 29^{x}}{4}+\frac {x^{4}}{16}-\frac {5 x^{3}}{4}+\frac {75 x^{2}}{8}-\frac {125 x}{4}\) | \(48\) |
default | \(-\frac {125 x}{4}+\frac {75 x^{2}}{8}-\frac {5 x^{3}}{4}+\frac {x^{4}}{16}+{\mathrm e}^{2 x \ln \left (29\right )} x^{2}+\frac {{\mathrm e}^{x \ln \left (29\right )} x^{3}}{2}-5 \,{\mathrm e}^{x \ln \left (29\right )} x^{2}+\frac {25 \,{\mathrm e}^{x \ln \left (29\right )} x}{2}\) | \(59\) |
norman | \(-\frac {125 x}{4}+\frac {75 x^{2}}{8}-\frac {5 x^{3}}{4}+\frac {x^{4}}{16}+{\mathrm e}^{2 x \ln \left (29\right )} x^{2}+\frac {{\mathrm e}^{x \ln \left (29\right )} x^{3}}{2}-5 \,{\mathrm e}^{x \ln \left (29\right )} x^{2}+\frac {25 \,{\mathrm e}^{x \ln \left (29\right )} x}{2}\) | \(59\) |
derivativedivides | \(\frac {-125 x \ln \left (29\right )+\frac {x^{4} \ln \left (29\right )}{4}+4 \ln \left (29\right ) {\mathrm e}^{2 x \ln \left (29\right )} x^{2}+50 \ln \left (29\right ) {\mathrm e}^{x \ln \left (29\right )} x -20 \ln \left (29\right ) {\mathrm e}^{x \ln \left (29\right )} x^{2}+2 \ln \left (29\right ) {\mathrm e}^{x \ln \left (29\right )} x^{3}+\frac {75 x^{2} \ln \left (29\right )}{2}-5 x^{3} \ln \left (29\right )}{4 \ln \left (29\right )}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 45, normalized size = 2.37 \begin {gather*} \frac {1}{16} \, x^{4} + 29^{2 \, x} x^{2} - \frac {5}{4} \, x^{3} + \frac {1}{2} \, {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} 29^{x} + \frac {75}{8} \, x^{2} - \frac {125}{4} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.90, size = 43, normalized size = 2.26 \begin {gather*} \frac {x\,\left (150\,x+16\,{29}^{2\,x}\,x+8\,{29}^x\,x^2-80\,{29}^x\,x-20\,x^2+x^3+200\,{29}^x-500\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.18, size = 54, normalized size = 2.84 \begin {gather*} \frac {x^{4}}{16} - \frac {5 x^{3}}{4} + x^{2} e^{2 x \log {\left (29 \right )}} + \frac {75 x^{2}}{8} - \frac {125 x}{4} + \frac {\left (x^{3} - 10 x^{2} + 25 x\right ) e^{x \log {\left (29 \right )}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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