Optimal. Leaf size=26 \[ x \left (4-4 \left (2+e^{\frac {e^5 x}{25}}\right )+x\right ) \left (-x+x^2\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 44, normalized size of antiderivative = 1.69, number of steps used = 18, number of rules used = 4, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 2196, 2176, 2194} \begin {gather*} x^4-4 e^{\frac {e^5 x}{25}} x^3-5 x^3+4 e^{\frac {e^5 x}{25}} x^2+4 x^2 \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}{25} \int \left (200 x-375 x^2+100 x^3+e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right )\right ) \, dx\\ &=4 x^2-5 x^3+x^4+\frac {1}{25} \int e^{\frac {e^5 x}{25}} \left (200 x-300 x^2+e^5 \left (4 x^2-4 x^3\right )\right ) \, dx\\ &=4 x^2-5 x^3+x^4+\frac {1}{25} \int \left (200 e^{\frac {e^5 x}{25}} x-300 e^{\frac {e^5 x}{25}} x^2-4 e^{5+\frac {e^5 x}{25}} (-1+x) x^2\right ) \, dx\\ &=4 x^2-5 x^3+x^4-\frac {4}{25} \int e^{5+\frac {e^5 x}{25}} (-1+x) x^2 \, dx+8 \int e^{\frac {e^5 x}{25}} x \, dx-12 \int e^{\frac {e^5 x}{25}} x^2 \, dx\\ &=200 e^{-5+\frac {e^5 x}{25}} x+4 x^2-300 e^{-5+\frac {e^5 x}{25}} x^2-5 x^3+x^4-\frac {4}{25} \int \left (-e^{5+\frac {e^5 x}{25}} x^2+e^{5+\frac {e^5 x}{25}} x^3\right ) \, dx-\frac {200 \int e^{\frac {e^5 x}{25}} \, dx}{e^5}+\frac {600 \int e^{\frac {e^5 x}{25}} x \, dx}{e^5}\\ &=-5000 e^{-10+\frac {e^5 x}{25}}+15000 e^{-10+\frac {e^5 x}{25}} x+200 e^{-5+\frac {e^5 x}{25}} x+4 x^2-300 e^{-5+\frac {e^5 x}{25}} x^2-5 x^3+x^4+\frac {4}{25} \int e^{5+\frac {e^5 x}{25}} x^2 \, dx-\frac {4}{25} \int e^{5+\frac {e^5 x}{25}} x^3 \, dx-\frac {15000 \int e^{\frac {e^5 x}{25}} \, dx}{e^{10}}\\ &=-375000 e^{-15+\frac {e^5 x}{25}}-5000 e^{-10+\frac {e^5 x}{25}}+15000 e^{-10+\frac {e^5 x}{25}} x+200 e^{-5+\frac {e^5 x}{25}} x+4 x^2+4 e^{\frac {e^5 x}{25}} x^2-300 e^{-5+\frac {e^5 x}{25}} x^2-5 x^3-4 e^{\frac {e^5 x}{25}} x^3+x^4-\frac {8 \int e^{5+\frac {e^5 x}{25}} x \, dx}{e^5}+\frac {12 \int e^{5+\frac {e^5 x}{25}} x^2 \, dx}{e^5}\\ &=-375000 e^{-15+\frac {e^5 x}{25}}-5000 e^{-10+\frac {e^5 x}{25}}+15000 e^{-10+\frac {e^5 x}{25}} x+4 x^2+4 e^{\frac {e^5 x}{25}} x^2-5 x^3-4 e^{\frac {e^5 x}{25}} x^3+x^4+\frac {200 \int e^{5+\frac {e^5 x}{25}} \, dx}{e^{10}}-\frac {600 \int e^{5+\frac {e^5 x}{25}} x \, dx}{e^{10}}\\ &=-375000 e^{-15+\frac {e^5 x}{25}}+4 x^2+4 e^{\frac {e^5 x}{25}} x^2-5 x^3-4 e^{\frac {e^5 x}{25}} x^3+x^4+\frac {15000 \int e^{5+\frac {e^5 x}{25}} \, dx}{e^{15}}\\ &=4 x^2+4 e^{\frac {e^5 x}{25}} x^2-5 x^3-4 e^{\frac {e^5 x}{25}} x^3+x^4\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 22, normalized size = 0.85 \begin {gather*} (-1+x) x^2 \left (-4-4 e^{\frac {e^5 x}{25}}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 31, normalized size = 1.19 \begin {gather*} x^{4} - 5 \, x^{3} + 4 \, x^{2} - 4 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{25} \, x e^{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 93, normalized size = 3.58 \begin {gather*} x^{4} - 5 \, x^{3} + 4 \, x^{2} - 4 \, {\left (x^{3} e^{15} - x^{2} e^{15} - 75 \, x^{2} e^{10} + 50 \, x e^{10} + 3750 \, x e^{5} - 1250 \, e^{5} - 93750\right )} e^{\left (\frac {1}{25} \, x e^{5} - 15\right )} - 100 \, {\left (3 \, x^{2} e^{10} - 2 \, x e^{10} - 150 \, x e^{5} + 50 \, e^{5} + 3750\right )} e^{\left (\frac {1}{25} \, x e^{5} - 15\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 34, normalized size = 1.31
method | result | size |
risch | \(\frac {\left (-100 x^{3}+100 x^{2}\right ) {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}}{25}+x^{4}-5 x^{3}+4 x^{2}\) | \(34\) |
norman | \(x^{4}+4 x^{2}-5 x^{3}+4 x^{2} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}-4 x^{3} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\) | \(37\) |
default | \(x^{4}-5 x^{3}+4 x^{2}+{\mathrm e}^{-5} \left (5000 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-187500 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )+2500 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-62500 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{3} {\mathrm e}^{15}}{15625}-\frac {3 \,{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}+\frac {6 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-6 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )\right )\) | \(185\) |
derivativedivides | \({\mathrm e}^{-5} \left (5000 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-187500 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )+2500 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )-62500 \,{\mathrm e}^{-10} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{3} {\mathrm e}^{15}}{15625}-\frac {3 \,{\mathrm e}^{10} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x^{2}}{625}+\frac {6 \,{\mathrm e}^{5} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}} x}{25}-6 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{25}}\right )+4 x^{2} {\mathrm e}^{5}-5 x^{3} {\mathrm e}^{5}+x^{4} {\mathrm e}^{5}\right )\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 31, normalized size = 1.19 \begin {gather*} x^{4} - 5 \, x^{3} + 4 \, x^{2} - 4 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{25} \, x e^{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 21, normalized size = 0.81 \begin {gather*} -x^2\,\left (x-1\right )\,\left (4\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^5}{25}}-x+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 31, normalized size = 1.19 \begin {gather*} x^{4} - 5 x^{3} + 4 x^{2} + \left (- 4 x^{3} + 4 x^{2}\right ) e^{\frac {x e^{5}}{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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