Optimal. Leaf size=22 \[ \frac {\left (2+\frac {5}{x}-20 x\right )^4}{\left (3+4 e^5\right )^4} \]
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Rubi [B] time = 0.06, antiderivative size = 111, normalized size of antiderivative = 5.05, number of steps used = 7, number of rules used = 3, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6, 12, 14} \begin {gather*} \frac {160000 x^4}{\left (3+4 e^5\right )^4}+\frac {625}{\left (3+4 e^5\right )^4 x^4}-\frac {64000 x^3}{\left (3+4 e^5\right )^4}+\frac {1000}{\left (3+4 e^5\right )^4 x^3}-\frac {150400 x^2}{\left (3+4 e^5\right )^4}-\frac {9400}{\left (3+4 e^5\right )^4 x^2}+\frac {47360 x}{\left (3+4 e^5\right )^4}-\frac {11840}{\left (3+4 e^5\right )^4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 14
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{864 e^{10} x^5+768 e^{15} x^5+256 e^{20} x^5+\left (81+432 e^5\right ) x^5} \, dx\\ &=\int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{256 e^{20} x^5+\left (81+432 e^5\right ) x^5+\left (864 e^{10}+768 e^{15}\right ) x^5} \, dx\\ &=\int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{\left (864 e^{10}+768 e^{15}\right ) x^5+\left (81+432 e^5+256 e^{20}\right ) x^5} \, dx\\ &=\int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{\left (81+432 e^5+864 e^{10}+768 e^{15}+256 e^{20}\right ) x^5} \, dx\\ &=\frac {\int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{x^5} \, dx}{\left (3+4 e^5\right )^4}\\ &=\frac {\int \left (47360-\frac {2500}{x^5}-\frac {3000}{x^4}+\frac {18800}{x^3}+\frac {11840}{x^2}-300800 x-192000 x^2+640000 x^3\right ) \, dx}{\left (3+4 e^5\right )^4}\\ &=\frac {625}{\left (3+4 e^5\right )^4 x^4}+\frac {1000}{\left (3+4 e^5\right )^4 x^3}-\frac {9400}{\left (3+4 e^5\right )^4 x^2}-\frac {11840}{\left (3+4 e^5\right )^4 x}+\frac {47360 x}{\left (3+4 e^5\right )^4}-\frac {150400 x^2}{\left (3+4 e^5\right )^4}-\frac {64000 x^3}{\left (3+4 e^5\right )^4}+\frac {160000 x^4}{\left (3+4 e^5\right )^4}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.01, size = 52, normalized size = 2.36 \begin {gather*} \frac {20 \left (\frac {125}{4 x^4}+\frac {50}{x^3}-\frac {470}{x^2}-\frac {592}{x}+2368 x-7520 x^2-3200 x^3+8000 x^4\right )}{\left (3+4 e^5\right )^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 73, normalized size = 3.32 \begin {gather*} \frac {5 \, {\left (32000 \, x^{8} - 12800 \, x^{7} - 30080 \, x^{6} + 9472 \, x^{5} - 2368 \, x^{3} - 1880 \, x^{2} + 200 \, x + 125\right )}}{256 \, x^{4} e^{20} + 768 \, x^{4} e^{15} + 864 \, x^{4} e^{10} + 432 \, x^{4} e^{5} + 81 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.08, size = 442, normalized size = 20.09 \begin {gather*} \frac {640 \, {\left (4194304000 \, x^{4} e^{60} + 37748736000 \, x^{4} e^{55} + 155713536000 \, x^{4} e^{50} + 389283840000 \, x^{4} e^{45} + 656916480000 \, x^{4} e^{40} + 788299776000 \, x^{4} e^{35} + 689762304000 \, x^{4} e^{30} + 443418624000 \, x^{4} e^{25} + 207852480000 \, x^{4} e^{20} + 69284160000 \, x^{4} e^{15} + 15588936000 \, x^{4} e^{10} + 2125764000 \, x^{4} e^{5} + 132860250 \, x^{4} - 1677721600 \, x^{3} e^{60} - 15099494400 \, x^{3} e^{55} - 62285414400 \, x^{3} e^{50} - 155713536000 \, x^{3} e^{45} - 262766592000 \, x^{3} e^{40} - 315319910400 \, x^{3} e^{35} - 275904921600 \, x^{3} e^{30} - 177367449600 \, x^{3} e^{25} - 83140992000 \, x^{3} e^{20} - 27713664000 \, x^{3} e^{15} - 6235574400 \, x^{3} e^{10} - 850305600 \, x^{3} e^{5} - 53144100 \, x^{3} - 3942645760 \, x^{2} e^{60} - 35483811840 \, x^{2} e^{55} - 146370723840 \, x^{2} e^{50} - 365926809600 \, x^{2} e^{45} - 617501491200 \, x^{2} e^{40} - 741001789440 \, x^{2} e^{35} - 648376565760 \, x^{2} e^{30} - 416813506560 \, x^{2} e^{25} - 195381331200 \, x^{2} e^{20} - 65127110400 \, x^{2} e^{15} - 14653599840 \, x^{2} e^{10} - 1998218160 \, x^{2} e^{5} - 124888635 \, x^{2} + 1241513984 \, x e^{60} + 11173625856 \, x e^{55} + 46091206656 \, x e^{50} + 115228016640 \, x e^{45} + 194447278080 \, x e^{40} + 233336733696 \, x e^{35} + 204169641984 \, x e^{30} + 131251912704 \, x e^{25} + 61524334080 \, x e^{20} + 20508111360 \, x e^{15} + 4614325056 \, x e^{10} + 629226144 \, x e^{5} + 39326634 \, x\right )}}{4294967296 \, e^{80} + 51539607552 \, e^{75} + 289910292480 \, e^{70} + 1014686023680 \, e^{65} + 2473297182720 \, e^{60} + 4451934928896 \, e^{55} + 6121410527232 \, e^{50} + 6558654136320 \, e^{45} + 5533864427520 \, e^{40} + 3689242951680 \, e^{35} + 1936852549632 \, e^{30} + 792348770304 \, e^{25} + 247608990720 \, e^{20} + 57140536320 \, e^{15} + 9183300480 \, e^{10} + 918330048 \, e^{5} + 43046721} - \frac {5 \, {\left (2368 \, x^{3} + 1880 \, x^{2} - 200 \, x - 125\right )}}{x^{4} {\left (256 \, e^{20} + 768 \, e^{15} + 864 \, e^{10} + 432 \, e^{5} + 81\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 67, normalized size = 3.05
method | result | size |
gosper | \(\frac {160000 x^{8}-64000 x^{7}-150400 x^{6}+47360 x^{5}-11840 x^{3}-9400 x^{2}+1000 x +625}{x^{4} \left (256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81\right )}\) | \(67\) |
default | \(\frac {160000 x^{4}-64000 x^{3}-150400 x^{2}+47360 x -\frac {9400}{x^{2}}+\frac {1000}{x^{3}}-\frac {11840}{x}+\frac {625}{x^{4}}}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}\) | \(68\) |
norman | \(\frac {\frac {625}{4 \,{\mathrm e}^{5}+3}+\frac {1000 x}{4 \,{\mathrm e}^{5}+3}-\frac {9400 x^{2}}{4 \,{\mathrm e}^{5}+3}-\frac {11840 x^{3}}{4 \,{\mathrm e}^{5}+3}+\frac {47360 x^{5}}{4 \,{\mathrm e}^{5}+3}-\frac {150400 x^{6}}{4 \,{\mathrm e}^{5}+3}-\frac {64000 x^{7}}{4 \,{\mathrm e}^{5}+3}+\frac {160000 x^{8}}{4 \,{\mathrm e}^{5}+3}}{x^{4} \left (4 \,{\mathrm e}^{5}+3\right )^{3}}\) | \(113\) |
risch | \(\frac {160000 x^{4}}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}-\frac {64000 x^{3}}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}-\frac {150400 x^{2}}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}+\frac {47360 x}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}+\frac {256 \left (-11840 \,{\mathrm e}^{20}-35520 \,{\mathrm e}^{15}-39960 \,{\mathrm e}^{10}-19980 \,{\mathrm e}^{5}-\frac {14985}{4}\right ) x^{3}+256 \left (-9400 \,{\mathrm e}^{20}-28200 \,{\mathrm e}^{15}-31725 \,{\mathrm e}^{10}-\frac {31725 \,{\mathrm e}^{5}}{2}-\frac {95175}{32}\right ) x^{2}+256 \left (1000 \,{\mathrm e}^{20}+3000 \,{\mathrm e}^{15}+3375 \,{\mathrm e}^{10}+\frac {3375 \,{\mathrm e}^{5}}{2}+\frac {10125}{32}\right ) x +160000 \,{\mathrm e}^{20}+480000 \,{\mathrm e}^{15}+540000 \,{\mathrm e}^{10}+270000 \,{\mathrm e}^{5}+50625}{\left (256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81\right )^{2} x^{4}}\) | \(207\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 82, normalized size = 3.73 \begin {gather*} \frac {640 \, {\left (250 \, x^{4} - 100 \, x^{3} - 235 \, x^{2} + 74 \, x\right )}}{256 \, e^{20} + 768 \, e^{15} + 864 \, e^{10} + 432 \, e^{5} + 81} - \frac {5 \, {\left (2368 \, x^{3} + 1880 \, x^{2} - 200 \, x - 125\right )}}{x^{4} {\left (256 \, e^{20} + 768 \, e^{15} + 864 \, e^{10} + 432 \, e^{5} + 81\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 90, normalized size = 4.09 \begin {gather*} \frac {47360\,x}{{\left (4\,{\mathrm {e}}^5+3\right )}^4}-\frac {150400\,x^2}{{\left (4\,{\mathrm {e}}^5+3\right )}^4}-\frac {64000\,x^3}{{\left (4\,{\mathrm {e}}^5+3\right )}^4}+\frac {160000\,x^4}{{\left (4\,{\mathrm {e}}^5+3\right )}^4}+\frac {-11840\,x^3-9400\,x^2+1000\,x+625}{x^4\,\left (432\,{\mathrm {e}}^5+864\,{\mathrm {e}}^{10}+768\,{\mathrm {e}}^{15}+256\,{\mathrm {e}}^{20}+81\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 58, normalized size = 2.64 \begin {gather*} \frac {160000 x^{4} - 64000 x^{3} - 150400 x^{2} + 47360 x + \frac {- 11840 x^{3} - 9400 x^{2} + 1000 x + 625}{x^{4}}}{81 + 432 e^{5} + 864 e^{10} + 768 e^{15} + 256 e^{20}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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