Integrand size = 44, antiderivative size = 26 \[ \int \left (150 x^5+e^{-12+10 x} \left (100 x^3+250 x^4\right )+e^{-6+5 x} \left (250 x^4+250 x^5\right )\right ) \, dx=2+25 x^4 \left (e^{-2 (3-2 x)+x}+x\right )^2-\log (4) \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15, number of steps used = 27, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1607, 2227, 2207, 2225} \[ \int \left (150 x^5+e^{-12+10 x} \left (100 x^3+250 x^4\right )+e^{-6+5 x} \left (250 x^4+250 x^5\right )\right ) \, dx=25 x^6+50 e^{5 x-6} x^5+25 e^{10 x-12} x^4 \]
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Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = 25 x^6+\int e^{-12+10 x} \left (100 x^3+250 x^4\right ) \, dx+\int e^{-6+5 x} \left (250 x^4+250 x^5\right ) \, dx \\ & = 25 x^6+\int e^{-12+10 x} x^3 (100+250 x) \, dx+\int e^{-6+5 x} x^4 (250+250 x) \, dx \\ & = 25 x^6+\int \left (100 e^{-12+10 x} x^3+250 e^{-12+10 x} x^4\right ) \, dx+\int \left (250 e^{-6+5 x} x^4+250 e^{-6+5 x} x^5\right ) \, dx \\ & = 25 x^6+100 \int e^{-12+10 x} x^3 \, dx+250 \int e^{-6+5 x} x^4 \, dx+250 \int e^{-12+10 x} x^4 \, dx+250 \int e^{-6+5 x} x^5 \, dx \\ & = 10 e^{-12+10 x} x^3+50 e^{-6+5 x} x^4+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6-30 \int e^{-12+10 x} x^2 \, dx-100 \int e^{-12+10 x} x^3 \, dx-200 \int e^{-6+5 x} x^3 \, dx-250 \int e^{-6+5 x} x^4 \, dx \\ & = -3 e^{-12+10 x} x^2-40 e^{-6+5 x} x^3+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6+6 \int e^{-12+10 x} x \, dx+30 \int e^{-12+10 x} x^2 \, dx+120 \int e^{-6+5 x} x^2 \, dx+200 \int e^{-6+5 x} x^3 \, dx \\ & = \frac {3}{5} e^{-12+10 x} x+24 e^{-6+5 x} x^2+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6-\frac {3}{5} \int e^{-12+10 x} \, dx-6 \int e^{-12+10 x} x \, dx-48 \int e^{-6+5 x} x \, dx-120 \int e^{-6+5 x} x^2 \, dx \\ & = -\frac {3}{50} e^{-12+10 x}-\frac {48}{5} e^{-6+5 x} x+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6+\frac {3}{5} \int e^{-12+10 x} \, dx+\frac {48}{5} \int e^{-6+5 x} \, dx+48 \int e^{-6+5 x} x \, dx \\ & = \frac {48}{25} e^{-6+5 x}+25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6-\frac {48}{5} \int e^{-6+5 x} \, dx \\ & = 25 e^{-12+10 x} x^4+50 e^{-6+5 x} x^5+25 x^6 \\ \end{align*}
Time = 1.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \left (150 x^5+e^{-12+10 x} \left (100 x^3+250 x^4\right )+e^{-6+5 x} \left (250 x^4+250 x^5\right )\right ) \, dx=\frac {25 x^4 \left (e^{5 x}+e^6 x\right )^2}{e^{12}} \]
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Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19
method | result | size |
norman | \(25 x^{6}+50 \,{\mathrm e}^{5 x -6} x^{5}+25 \,{\mathrm e}^{10 x -12} x^{4}\) | \(31\) |
risch | \(25 x^{6}+50 \,{\mathrm e}^{5 x -6} x^{5}+25 \,{\mathrm e}^{10 x -12} x^{4}\) | \(31\) |
parallelrisch | \(25 x^{6}+50 \,{\mathrm e}^{5 x -6} x^{5}+25 \,{\mathrm e}^{10 x -12} x^{4}\) | \(31\) |
derivativedivides | \(25 x^{6}+\frac {1296 \,{\mathrm e}^{10 x -12}}{25}+\frac {864 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )}{25}+\frac {216 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{2}}{25}+\frac {24 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{3}}{25}+\frac {{\mathrm e}^{10 x -12} \left (5 x -6\right )^{4}}{25}+\frac {2592 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )}{25}+\frac {15552 \,{\mathrm e}^{5 x -6}}{125}+\frac {864 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{2}}{25}+\frac {144 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{3}}{25}+\frac {12 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{4}}{25}+\frac {2 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{5}}{125}\) | \(164\) |
default | \(25 x^{6}+\frac {1296 \,{\mathrm e}^{10 x -12}}{25}+\frac {864 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )}{25}+\frac {216 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{2}}{25}+\frac {24 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{3}}{25}+\frac {{\mathrm e}^{10 x -12} \left (5 x -6\right )^{4}}{25}+\frac {2592 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )}{25}+\frac {15552 \,{\mathrm e}^{5 x -6}}{125}+\frac {864 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{2}}{25}+\frac {144 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{3}}{25}+\frac {12 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{4}}{25}+\frac {2 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{5}}{125}\) | \(164\) |
parts | \(25 x^{6}+\frac {1296 \,{\mathrm e}^{10 x -12}}{25}+\frac {864 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )}{25}+\frac {216 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{2}}{25}+\frac {24 \,{\mathrm e}^{10 x -12} \left (5 x -6\right )^{3}}{25}+\frac {{\mathrm e}^{10 x -12} \left (5 x -6\right )^{4}}{25}+\frac {2592 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )}{25}+\frac {15552 \,{\mathrm e}^{5 x -6}}{125}+\frac {864 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{2}}{25}+\frac {144 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{3}}{25}+\frac {12 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{4}}{25}+\frac {2 \,{\mathrm e}^{5 x -6} \left (5 x -6\right )^{5}}{125}\) | \(164\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \left (150 x^5+e^{-12+10 x} \left (100 x^3+250 x^4\right )+e^{-6+5 x} \left (250 x^4+250 x^5\right )\right ) \, dx=25 \, x^{6} + 50 \, x^{5} e^{\left (5 \, x - 6\right )} + 25 \, x^{4} e^{\left (10 \, x - 12\right )} \]
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Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \left (150 x^5+e^{-12+10 x} \left (100 x^3+250 x^4\right )+e^{-6+5 x} \left (250 x^4+250 x^5\right )\right ) \, dx=25 x^{6} + 50 x^{5} e^{5 x - 6} + 25 x^{4} e^{10 x - 12} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (21) = 42\).
Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.00 \[ \int \left (150 x^5+e^{-12+10 x} \left (100 x^3+250 x^4\right )+e^{-6+5 x} \left (250 x^4+250 x^5\right )\right ) \, dx=25 \, x^{6} + 25 \, x^{4} e^{\left (10 \, x - 12\right )} + \frac {2}{25} \, {\left (625 \, x^{5} - 625 \, x^{4} + 500 \, x^{3} - 300 \, x^{2} + 120 \, x - 24\right )} e^{\left (5 \, x - 6\right )} + \frac {2}{25} \, {\left (625 \, x^{4} - 500 \, x^{3} + 300 \, x^{2} - 120 \, x + 24\right )} e^{\left (5 \, x - 6\right )} \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \left (150 x^5+e^{-12+10 x} \left (100 x^3+250 x^4\right )+e^{-6+5 x} \left (250 x^4+250 x^5\right )\right ) \, dx=25 \, x^{6} + 50 \, x^{5} e^{\left (5 \, x - 6\right )} + 25 \, x^{4} e^{\left (10 \, x - 12\right )} \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int \left (150 x^5+e^{-12+10 x} \left (100 x^3+250 x^4\right )+e^{-6+5 x} \left (250 x^4+250 x^5\right )\right ) \, dx=25\,x^4\,{\mathrm {e}}^{-12}\,{\left ({\mathrm {e}}^{5\,x}+x\,{\mathrm {e}}^6\right )}^2 \]
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