Optimal. Leaf size=33 \[ \frac {1}{25} e^2 (-4-x)^2 \left (-\frac {e}{5}+e^{2 x}-x\right )^2 x^2 \]
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Rubi [B] time = 0.64, antiderivative size = 217, normalized size of antiderivative = 6.58, number of steps used = 57, number of rules used = 4, integrand size = 138, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {12, 2196, 2176, 2194} \begin {gather*} \frac {e^2 x^6}{25}-\frac {2}{25} e^{2 x+2} x^5+\frac {2 e^3 x^5}{125}+\frac {8 e^2 x^5}{25}-\frac {16}{25} e^{2 x+2} x^4-\frac {2}{125} e^{2 x+3} x^4+\frac {1}{25} e^{4 x+2} x^4+\frac {e^4 x^4}{625}+\frac {16 e^3 x^4}{125}+\frac {16 e^2 x^4}{25}-\frac {32}{25} e^{2 x+2} x^3-\frac {16}{125} e^{2 x+3} x^3+\frac {8}{25} e^{4 x+2} x^3+\frac {8 e^4 x^3}{625}+\frac {32 e^3 x^3}{125}-\frac {32}{125} e^{2 x+3} x^2+\frac {16}{25} e^{4 x+2} x^2+\frac {16 e^4 x^2}{625} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{625} \int \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx\\ &=\frac {1}{625} \int e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right ) \, dx+\frac {1}{625} \int e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right ) \, dx+\frac {1}{625} e^2 \int \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right ) \, dx\\ &=\frac {16 e^2 x^4}{25}+\frac {8 e^2 x^5}{25}+\frac {e^2 x^6}{25}+\frac {1}{625} \int \left (800 e^{2+4 x} x+2200 e^{2+4 x} x^2+900 e^{2+4 x} x^3+100 e^{2+4 x} x^4\right ) \, dx+\frac {1}{625} \int \left (-2400 e^{2+2 x} x^2-3200 e^{2+2 x} x^3-1050 e^{2+2 x} x^4-100 e^{2+2 x} x^5-20 e^{3+2 x} x \left (16+28 x+10 x^2+x^3\right )\right ) \, dx+\frac {1}{625} e^3 \int \left (480 x^2+320 x^3+50 x^4\right ) \, dx+\frac {1}{625} e^4 \int \left (32 x+24 x^2+4 x^3\right ) \, dx\\ &=\frac {16 e^4 x^2}{625}+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}+\frac {e^2 x^6}{25}-\frac {4}{125} \int e^{3+2 x} x \left (16+28 x+10 x^2+x^3\right ) \, dx+\frac {4}{25} \int e^{2+4 x} x^4 \, dx-\frac {4}{25} \int e^{2+2 x} x^5 \, dx+\frac {32}{25} \int e^{2+4 x} x \, dx+\frac {36}{25} \int e^{2+4 x} x^3 \, dx-\frac {42}{25} \int e^{2+2 x} x^4 \, dx+\frac {88}{25} \int e^{2+4 x} x^2 \, dx-\frac {96}{25} \int e^{2+2 x} x^2 \, dx-\frac {128}{25} \int e^{2+2 x} x^3 \, dx\\ &=\frac {8}{25} e^{2+4 x} x+\frac {16 e^4 x^2}{625}-\frac {48}{25} e^{2+2 x} x^2+\frac {22}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {64}{25} e^{2+2 x} x^3+\frac {9}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {21}{25} e^{2+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}-\frac {4}{125} \int \left (16 e^{3+2 x} x+28 e^{3+2 x} x^2+10 e^{3+2 x} x^3+e^{3+2 x} x^4\right ) \, dx-\frac {4}{25} \int e^{2+4 x} x^3 \, dx-\frac {8}{25} \int e^{2+4 x} \, dx+\frac {2}{5} \int e^{2+2 x} x^4 \, dx-\frac {27}{25} \int e^{2+4 x} x^2 \, dx-\frac {44}{25} \int e^{2+4 x} x \, dx+\frac {84}{25} \int e^{2+2 x} x^3 \, dx+\frac {96}{25} \int e^{2+2 x} x \, dx+\frac {192}{25} \int e^{2+2 x} x^2 \, dx\\ &=-\frac {2}{25} e^{2+4 x}+\frac {48}{25} e^{2+2 x} x-\frac {3}{25} e^{2+4 x} x+\frac {16 e^4 x^2}{625}+\frac {48}{25} e^{2+2 x} x^2+\frac {61}{100} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {22}{25} e^{2+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}-\frac {4}{125} \int e^{3+2 x} x^4 \, dx+\frac {3}{25} \int e^{2+4 x} x^2 \, dx-\frac {8}{25} \int e^{3+2 x} x^3 \, dx+\frac {11}{25} \int e^{2+4 x} \, dx-\frac {64}{125} \int e^{3+2 x} x \, dx+\frac {27}{50} \int e^{2+4 x} x \, dx-\frac {4}{5} \int e^{2+2 x} x^3 \, dx-\frac {112}{125} \int e^{3+2 x} x^2 \, dx-\frac {48}{25} \int e^{2+2 x} \, dx-\frac {126}{25} \int e^{2+2 x} x^2 \, dx-\frac {192}{25} \int e^{2+2 x} x \, dx\\ &=-\frac {24}{25} e^{2+2 x}+\frac {3}{100} e^{2+4 x}-\frac {48}{25} e^{2+2 x} x-\frac {32}{125} e^{3+2 x} x+\frac {3}{200} e^{2+4 x} x+\frac {16 e^4 x^2}{625}-\frac {3}{5} e^{2+2 x} x^2-\frac {56}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {4}{25} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}-\frac {3}{50} \int e^{2+4 x} x \, dx+\frac {8}{125} \int e^{3+2 x} x^3 \, dx-\frac {27}{200} \int e^{2+4 x} \, dx+\frac {32}{125} \int e^{3+2 x} \, dx+\frac {12}{25} \int e^{3+2 x} x^2 \, dx+\frac {112}{125} \int e^{3+2 x} x \, dx+\frac {6}{5} \int e^{2+2 x} x^2 \, dx+\frac {96}{25} \int e^{2+2 x} \, dx+\frac {126}{25} \int e^{2+2 x} x \, dx\\ &=\frac {24}{25} e^{2+2 x}+\frac {16}{125} e^{3+2 x}-\frac {3}{800} e^{2+4 x}+\frac {3}{5} e^{2+2 x} x+\frac {24}{125} e^{3+2 x} x+\frac {16 e^4 x^2}{625}-\frac {26}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {16}{125} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}+\frac {3}{200} \int e^{2+4 x} \, dx-\frac {12}{125} \int e^{3+2 x} x^2 \, dx-\frac {56}{125} \int e^{3+2 x} \, dx-\frac {12}{25} \int e^{3+2 x} x \, dx-\frac {6}{5} \int e^{2+2 x} x \, dx-\frac {63}{25} \int e^{2+2 x} \, dx\\ &=-\frac {3}{10} e^{2+2 x}-\frac {12}{125} e^{3+2 x}-\frac {6}{125} e^{3+2 x} x+\frac {16 e^4 x^2}{625}-\frac {32}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {16}{125} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}+\frac {12}{125} \int e^{3+2 x} x \, dx+\frac {6}{25} \int e^{3+2 x} \, dx+\frac {3}{5} \int e^{2+2 x} \, dx\\ &=\frac {3}{125} e^{3+2 x}+\frac {16 e^4 x^2}{625}-\frac {32}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {16}{125} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}-\frac {6}{125} \int e^{3+2 x} \, dx\\ &=\frac {16 e^4 x^2}{625}-\frac {32}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {16}{125} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{625} e^2 x^2 (4+x)^2 \left (e-5 e^{2 x}+5 x\right )^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 124, normalized size = 3.76 \begin {gather*} \frac {1}{625} \, {\left ({\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{6} + 10 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e^{5} + 25 \, {\left (x^{6} + 8 \, x^{5} + 16 \, x^{4}\right )} e^{4} + 25 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (4 \, x + 4\right )} - 10 \, {\left ({\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{3} + 5 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e^{2}\right )} e^{\left (2 \, x + 2\right )}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 122, normalized size = 3.70 \begin {gather*} \frac {1}{625} \, {\left (25 \, x^{6} + 200 \, x^{5} + 400 \, x^{4} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{2} + 10 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e\right )} e^{2} + \frac {1}{25} \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (4 \, x + 2\right )} - \frac {2}{125} \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (2 \, x + 3\right )} - \frac {2}{25} \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e^{\left (2 \, x + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 144, normalized size = 4.36
method | result | size |
risch | \(\frac {\left (25 x^{4}+200 x^{3}+400 x^{2}\right ) {\mathrm e}^{4 x +2}}{625}+\frac {\left (-10 x^{4} {\mathrm e}-50 x^{5}-80 x^{3} {\mathrm e}-400 x^{4}-160 x^{2} {\mathrm e}-800 x^{3}\right ) {\mathrm e}^{2 x +2}}{625}+\frac {x^{6} {\mathrm e}^{2}}{25}+\frac {2 \,{\mathrm e}^{2} x^{5} {\mathrm e}}{125}+\frac {8 \,{\mathrm e}^{2} x^{5}}{25}+\frac {x^{4} {\mathrm e}^{4}}{625}+\frac {16 \,{\mathrm e}^{2} x^{4} {\mathrm e}}{125}+\frac {16 x^{4} {\mathrm e}^{2}}{25}+\frac {8 x^{3} {\mathrm e}^{4}}{625}+\frac {32 \,{\mathrm e}^{2} x^{3} {\mathrm e}}{125}+\frac {16 x^{2} {\mathrm e}^{4}}{625}\) | \(144\) |
norman | \(\left (\frac {2 \,{\mathrm e} \,{\mathrm e}^{2}}{125}+\frac {8 \,{\mathrm e}^{2}}{25}\right ) x^{5}+\left (\frac {8 \left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {32 \,{\mathrm e} \,{\mathrm e}^{2}}{125}\right ) x^{3}+\left (\frac {\left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {16 \,{\mathrm e} \,{\mathrm e}^{2}}{125}+\frac {16 \,{\mathrm e}^{2}}{25}\right ) x^{4}+\left (-\frac {32 \,{\mathrm e}^{2}}{25}-\frac {16 \,{\mathrm e} \,{\mathrm e}^{2}}{125}\right ) x^{3} {\mathrm e}^{2 x}+\left (-\frac {16 \,{\mathrm e}^{2}}{25}-\frac {2 \,{\mathrm e} \,{\mathrm e}^{2}}{125}\right ) x^{4} {\mathrm e}^{2 x}+\frac {x^{6} {\mathrm e}^{2}}{25}+\frac {16 x^{2} \left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {16 x^{2} {\mathrm e}^{2} {\mathrm e}^{4 x}}{25}+\frac {8 x^{3} {\mathrm e}^{2} {\mathrm e}^{4 x}}{25}+\frac {x^{4} {\mathrm e}^{2} {\mathrm e}^{4 x}}{25}-\frac {2 \,{\mathrm e}^{2} x^{5} {\mathrm e}^{2 x}}{25}-\frac {32 x^{2} {\mathrm e} \,{\mathrm e}^{2} {\mathrm e}^{2 x}}{125}\) | \(172\) |
default | \(\frac {{\mathrm e}^{2} \left ({\mathrm e}^{2} \left (x^{4}+8 x^{3}+16 x^{2}\right )+{\mathrm e} \left (10 x^{5}+80 x^{4}+160 x^{3}\right )+25 x^{6}+200 x^{5}+400 x^{4}\right )}{625}+\frac {{\mathrm e}^{2} \left (-800 \,{\mathrm e}^{2 x} x^{3}-50 x^{5} {\mathrm e}^{2 x}-400 \,{\mathrm e}^{2 x} x^{4}-200 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{4}+\frac {3 x \,{\mathrm e}^{2 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{8}\right )-320 \,{\mathrm e} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )-560 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )-20 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{4}}{2}-{\mathrm e}^{2 x} x^{3}+\frac {3 \,{\mathrm e}^{2 x} x^{2}}{2}-\frac {3 x \,{\mathrm e}^{2 x}}{2}+\frac {3 \,{\mathrm e}^{2 x}}{4}\right )\right )}{625}+\frac {4 \,{\mathrm e}^{2} \left (\frac {x^{4} {\mathrm e}^{4 x}}{4}+2 x^{3} {\mathrm e}^{4 x}+4 x^{2} {\mathrm e}^{4 x}\right )}{25}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 127, normalized size = 3.85 \begin {gather*} \frac {1}{625} \, {\left (25 \, x^{6} + 200 \, x^{5} + 400 \, x^{4} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{2} + 10 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e\right )} e^{2} + \frac {1}{25} \, {\left (x^{4} e^{2} + 8 \, x^{3} e^{2} + 16 \, x^{2} e^{2}\right )} e^{\left (4 \, x\right )} - \frac {2}{125} \, {\left (5 \, x^{5} e^{2} + x^{4} {\left (e^{3} + 40 \, e^{2}\right )} + 8 \, x^{3} {\left (e^{3} + 10 \, e^{2}\right )} + 16 \, x^{2} e^{3}\right )} e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 147, normalized size = 4.45 \begin {gather*} x^5\,\left (\frac {8\,{\mathrm {e}}^2}{25}+\frac {2\,{\mathrm {e}}^3}{125}\right )+x^3\,\left (\frac {32\,{\mathrm {e}}^3}{125}+\frac {8\,{\mathrm {e}}^4}{625}\right )+\frac {16\,x^2\,{\mathrm {e}}^4}{625}+\frac {x^6\,{\mathrm {e}}^2}{25}-\frac {32\,x^2\,{\mathrm {e}}^{2\,x+3}}{125}+\frac {16\,x^2\,{\mathrm {e}}^{4\,x+2}}{25}+\frac {8\,x^3\,{\mathrm {e}}^{4\,x+2}}{25}-\frac {2\,x^5\,{\mathrm {e}}^{2\,x+2}}{25}+\frac {x^4\,{\mathrm {e}}^{4\,x+2}}{25}+x^4\,\left (\frac {16\,{\mathrm {e}}^2}{25}+\frac {16\,{\mathrm {e}}^3}{125}+\frac {{\mathrm {e}}^4}{625}\right )-\frac {x^4\,{\mathrm {e}}^{2\,x+2}\,\left (10\,\mathrm {e}+400\right )}{625}-\frac {x^3\,{\mathrm {e}}^{2\,x+2}\,\left (80\,\mathrm {e}+800\right )}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.33, size = 165, normalized size = 5.00 \begin {gather*} \frac {x^{6} e^{2}}{25} + x^{5} \left (\frac {2 e^{3}}{125} + \frac {8 e^{2}}{25}\right ) + x^{4} \left (\frac {e^{4}}{625} + \frac {16 e^{3}}{125} + \frac {16 e^{2}}{25}\right ) + x^{3} \left (\frac {8 e^{4}}{625} + \frac {32 e^{3}}{125}\right ) + \frac {16 x^{2} e^{4}}{625} + \frac {\left (125 x^{4} e^{2} + 1000 x^{3} e^{2} + 2000 x^{2} e^{2}\right ) e^{4 x}}{3125} + \frac {\left (- 250 x^{5} e^{2} - 2000 x^{4} e^{2} - 50 x^{4} e^{3} - 4000 x^{3} e^{2} - 400 x^{3} e^{3} - 800 x^{2} e^{3}\right ) e^{2 x}}{3125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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