Optimal. Leaf size=360 \[ \frac {x^3 \left (1000 d^2+180 d e+111 e^2\right )}{3 e^6}-\frac {x^2 \left (2000 d^3+450 d^2 e+444 d e^2+37 e^3\right )}{2 e^7}-\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{3 e^9 (d+e x)^3}+\frac {2 x \left (1750 d^4+450 d^3 e+555 d^2 e^2+74 d e^3+74 e^4\right )}{e^8}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right )}{2 e^9 (d+e x)^2}-\frac {\left (5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right ) \log (d+e x)}{e^9}-\frac {2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6}{e^9 (d+e x)}-\frac {5 x^4 (80 d+9 e)}{4 e^5}+\frac {20 x^5}{e^4} \]
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Rubi [A] time = 0.36, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {1628} \[ \frac {x^3 \left (1000 d^2+180 d e+111 e^2\right )}{3 e^6}-\frac {x^2 \left (450 d^2 e+2000 d^3+444 d e^2+37 e^3\right )}{2 e^7}+\frac {2 x \left (555 d^2 e^2+450 d^3 e+1750 d^4+74 d e^3+74 e^4\right )}{e^8}-\frac {1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6}{e^9 (d+e x)}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right )}{2 e^9 (d+e x)^2}-\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{3 e^9 (d+e x)^3}-\frac {\left (2220 d^3 e^2+370 d^2 e^3+1575 d^4 e+5600 d^5+592 d e^4-65 e^5\right ) \log (d+e x)}{e^9}-\frac {5 x^4 (80 d+9 e)}{4 e^5}+\frac {20 x^5}{e^4} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin {align*} \int \frac {\left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^4} \, dx &=\int \left (\frac {2 \left (1750 d^4+450 d^3 e+555 d^2 e^2+74 d e^3+74 e^4\right )}{e^8}-\frac {\left (2000 d^3+450 d^2 e+444 d e^2+37 e^3\right ) x}{e^7}+\frac {\left (1000 d^2+180 d e+111 e^2\right ) x^2}{e^6}-\frac {5 (80 d+9 e) x^3}{e^5}+\frac {100 x^4}{e^4}+\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{e^8 (d+e x)^4}+\frac {-800 d^7-315 d^6 e-666 d^5 e^2-185 d^4 e^3-592 d^3 e^4+195 d^2 e^5-214 d e^6+33 e^7}{e^8 (d+e x)^3}+\frac {2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6}{e^8 (d+e x)^2}+\frac {-5600 d^5-1575 d^4 e-2220 d^3 e^2-370 d^2 e^3-592 d e^4+65 e^5}{e^8 (d+e x)}\right ) \, dx\\ &=\frac {2 \left (1750 d^4+450 d^3 e+555 d^2 e^2+74 d e^3+74 e^4\right ) x}{e^8}-\frac {\left (2000 d^3+450 d^2 e+444 d e^2+37 e^3\right ) x^2}{2 e^7}+\frac {\left (1000 d^2+180 d e+111 e^2\right ) x^3}{3 e^6}-\frac {5 (80 d+9 e) x^4}{4 e^5}+\frac {20 x^5}{e^4}-\frac {\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{3 e^9 (d+e x)^3}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right )}{2 e^9 (d+e x)^2}-\frac {2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6}{e^9 (d+e x)}-\frac {\left (5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right ) \log (d+e x)}{e^9}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 344, normalized size = 0.96 \[ \frac {4 e^3 x^3 \left (1000 d^2+180 d e+111 e^2\right )-6 e^2 x^2 \left (2000 d^3+450 d^2 e+444 d e^2+37 e^3\right )+24 e x \left (1750 d^4+450 d^3 e+555 d^2 e^2+74 d e^3+74 e^4\right )-\frac {4 \left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{(d+e x)^3}-12 \left (5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right ) \log (d+e x)-\frac {12 \left (2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right )}{d+e x}+\frac {6 \left (800 d^7+315 d^6 e+666 d^5 e^2+185 d^4 e^3+592 d^3 e^4-195 d^2 e^5+214 d e^6-33 e^7\right )}{(d+e x)^2}-15 e^4 x^4 (80 d+9 e)+240 e^5 x^5}{12 e^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 587, normalized size = 1.63 \[ \frac {240 \, e^{8} x^{8} - 29200 \, d^{8} - 9630 \, d^{7} e - 16428 \, d^{6} e^{2} - 3478 \, d^{5} e^{3} - 7696 \, d^{4} e^{4} + 1430 \, d^{3} e^{5} - 428 \, d^{2} e^{6} - 66 \, d e^{7} - 72 \, e^{8} - 15 \, {\left (32 \, d e^{7} + 9 \, e^{8}\right )} x^{7} + {\left (1120 \, d^{2} e^{6} + 315 \, d e^{7} + 444 \, e^{8}\right )} x^{6} - 3 \, {\left (1120 \, d^{3} e^{5} + 315 \, d^{2} e^{6} + 444 \, d e^{7} + 74 \, e^{8}\right )} x^{5} + 3 \, {\left (5600 \, d^{4} e^{4} + 1575 \, d^{3} e^{5} + 2220 \, d^{2} e^{6} + 370 \, d e^{7} + 592 \, e^{8}\right )} x^{4} + 2 \, {\left (47000 \, d^{5} e^{3} + 12510 \, d^{4} e^{4} + 16206 \, d^{3} e^{5} + 2331 \, d^{2} e^{6} + 2664 \, d e^{7}\right )} x^{3} + 6 \, {\left (13400 \, d^{6} e^{2} + 3060 \, d^{5} e^{3} + 2886 \, d^{4} e^{4} + 111 \, d^{3} e^{5} - 888 \, d^{2} e^{6} + 390 \, d e^{7} - 214 \, e^{8}\right )} x^{2} - 6 \, {\left (3400 \, d^{7} e + 1665 \, d^{6} e^{2} + 3774 \, d^{5} e^{3} + 999 \, d^{4} e^{4} + 2664 \, d^{3} e^{5} - 585 \, d^{2} e^{6} + 214 \, d e^{7} + 33 \, e^{8}\right )} x - 12 \, {\left (5600 \, d^{8} + 1575 \, d^{7} e + 2220 \, d^{6} e^{2} + 370 \, d^{5} e^{3} + 592 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + {\left (5600 \, d^{5} e^{3} + 1575 \, d^{4} e^{4} + 2220 \, d^{3} e^{5} + 370 \, d^{2} e^{6} + 592 \, d e^{7} - 65 \, e^{8}\right )} x^{3} + 3 \, {\left (5600 \, d^{6} e^{2} + 1575 \, d^{5} e^{3} + 2220 \, d^{4} e^{4} + 370 \, d^{3} e^{5} + 592 \, d^{2} e^{6} - 65 \, d e^{7}\right )} x^{2} + 3 \, {\left (5600 \, d^{7} e + 1575 \, d^{6} e^{2} + 2220 \, d^{5} e^{3} + 370 \, d^{4} e^{4} + 592 \, d^{3} e^{5} - 65 \, d^{2} e^{6}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{12} x^{3} + 3 \, d e^{11} x^{2} + 3 \, d^{2} e^{10} x + d^{3} e^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 345, normalized size = 0.96 \[ -{\left (5600 \, d^{5} + 1575 \, d^{4} e + 2220 \, d^{3} e^{2} + 370 \, d^{2} e^{3} + 592 \, d e^{4} - 65 \, e^{5}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (240 \, x^{5} e^{16} - 1200 \, d x^{4} e^{15} + 4000 \, d^{2} x^{3} e^{14} - 12000 \, d^{3} x^{2} e^{13} + 42000 \, d^{4} x e^{12} - 135 \, x^{4} e^{16} + 720 \, d x^{3} e^{15} - 2700 \, d^{2} x^{2} e^{14} + 10800 \, d^{3} x e^{13} + 444 \, x^{3} e^{16} - 2664 \, d x^{2} e^{15} + 13320 \, d^{2} x e^{14} - 222 \, x^{2} e^{16} + 1776 \, d x e^{15} + 1776 \, x e^{16}\right )} e^{\left (-20\right )} - \frac {{\left (14600 \, d^{8} + 4815 \, d^{7} e + 8214 \, d^{6} e^{2} + 1739 \, d^{5} e^{3} + 3848 \, d^{4} e^{4} - 715 \, d^{3} e^{5} + 6 \, {\left (2800 \, d^{6} e^{2} + 945 \, d^{5} e^{3} + 1665 \, d^{4} e^{4} + 370 \, d^{3} e^{5} + 888 \, d^{2} e^{6} - 195 \, d e^{7} + 107 \, e^{8}\right )} x^{2} + 214 \, d^{2} e^{6} + 3 \, {\left (10400 \, d^{7} e + 3465 \, d^{6} e^{2} + 5994 \, d^{5} e^{3} + 1295 \, d^{4} e^{4} + 2960 \, d^{3} e^{5} - 585 \, d^{2} e^{6} + 214 \, d e^{7} + 33 \, e^{8}\right )} x + 33 \, d e^{7} + 36 \, e^{8}\right )} e^{\left (-9\right )}}{6 \, {\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 558, normalized size = 1.55 \[ \frac {20 x^{5}}{e^{4}}-\frac {100 d \,x^{4}}{e^{5}}-\frac {45 x^{4}}{4 e^{4}}-\frac {100 d^{8}}{3 \left (e x +d \right )^{3} e^{9}}-\frac {15 d^{7}}{\left (e x +d \right )^{3} e^{8}}-\frac {37 d^{6}}{\left (e x +d \right )^{3} e^{7}}-\frac {37 d^{5}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {148 d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {65 d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {107 d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {1000 d^{2} x^{3}}{3 e^{6}}+\frac {11 d}{\left (e x +d \right )^{3} e^{2}}+\frac {60 d \,x^{3}}{e^{5}}-\frac {6}{\left (e x +d \right )^{3} e}+\frac {37 x^{3}}{e^{4}}+\frac {400 d^{7}}{\left (e x +d \right )^{2} e^{9}}+\frac {315 d^{6}}{2 \left (e x +d \right )^{2} e^{8}}+\frac {333 d^{5}}{\left (e x +d \right )^{2} e^{7}}+\frac {185 d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {296 d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {1000 d^{3} x^{2}}{e^{7}}-\frac {195 d^{2}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {225 d^{2} x^{2}}{e^{6}}+\frac {107 d}{\left (e x +d \right )^{2} e^{3}}-\frac {222 d \,x^{2}}{e^{5}}-\frac {33}{2 \left (e x +d \right )^{2} e^{2}}-\frac {37 x^{2}}{2 e^{4}}-\frac {2800 d^{6}}{\left (e x +d \right ) e^{9}}-\frac {945 d^{5}}{\left (e x +d \right ) e^{8}}-\frac {5600 d^{5} \ln \left (e x +d \right )}{e^{9}}-\frac {1665 d^{4}}{\left (e x +d \right ) e^{7}}+\frac {3500 d^{4} x}{e^{8}}-\frac {1575 d^{4} \ln \left (e x +d \right )}{e^{8}}-\frac {370 d^{3}}{\left (e x +d \right ) e^{6}}+\frac {900 d^{3} x}{e^{7}}-\frac {2220 d^{3} \ln \left (e x +d \right )}{e^{7}}-\frac {888 d^{2}}{\left (e x +d \right ) e^{5}}+\frac {1110 d^{2} x}{e^{6}}-\frac {370 d^{2} \ln \left (e x +d \right )}{e^{6}}+\frac {195 d}{\left (e x +d \right ) e^{4}}+\frac {148 d x}{e^{5}}-\frac {592 d \ln \left (e x +d \right )}{e^{5}}-\frac {107}{\left (e x +d \right ) e^{3}}+\frac {148 x}{e^{4}}+\frac {65 \ln \left (e x +d \right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 390, normalized size = 1.08 \[ -\frac {14600 \, d^{8} + 4815 \, d^{7} e + 8214 \, d^{6} e^{2} + 1739 \, d^{5} e^{3} + 3848 \, d^{4} e^{4} - 715 \, d^{3} e^{5} + 214 \, d^{2} e^{6} + 33 \, d e^{7} + 36 \, e^{8} + 6 \, {\left (2800 \, d^{6} e^{2} + 945 \, d^{5} e^{3} + 1665 \, d^{4} e^{4} + 370 \, d^{3} e^{5} + 888 \, d^{2} e^{6} - 195 \, d e^{7} + 107 \, e^{8}\right )} x^{2} + 3 \, {\left (10400 \, d^{7} e + 3465 \, d^{6} e^{2} + 5994 \, d^{5} e^{3} + 1295 \, d^{4} e^{4} + 2960 \, d^{3} e^{5} - 585 \, d^{2} e^{6} + 214 \, d e^{7} + 33 \, e^{8}\right )} x}{6 \, {\left (e^{12} x^{3} + 3 \, d e^{11} x^{2} + 3 \, d^{2} e^{10} x + d^{3} e^{9}\right )}} + \frac {240 \, e^{4} x^{5} - 15 \, {\left (80 \, d e^{3} + 9 \, e^{4}\right )} x^{4} + 4 \, {\left (1000 \, d^{2} e^{2} + 180 \, d e^{3} + 111 \, e^{4}\right )} x^{3} - 6 \, {\left (2000 \, d^{3} e + 450 \, d^{2} e^{2} + 444 \, d e^{3} + 37 \, e^{4}\right )} x^{2} + 24 \, {\left (1750 \, d^{4} + 450 \, d^{3} e + 555 \, d^{2} e^{2} + 74 \, d e^{3} + 74 \, e^{4}\right )} x}{12 \, e^{8}} - \frac {{\left (5600 \, d^{5} + 1575 \, d^{4} e + 2220 \, d^{3} e^{2} + 370 \, d^{2} e^{3} + 592 \, d e^{4} - 65 \, e^{5}\right )} \log \left (e x + d\right )}{e^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.28, size = 560, normalized size = 1.56 \[ x^3\,\left (\frac {37}{e^4}-\frac {200\,d^2}{e^6}+\frac {4\,d\,\left (\frac {400\,d}{e^5}+\frac {45}{e^4}\right )}{3\,e}\right )-x^2\,\left (\frac {37}{2\,e^4}+\frac {200\,d^3}{e^7}+\frac {2\,d\,\left (\frac {111}{e^4}-\frac {600\,d^2}{e^6}+\frac {4\,d\,\left (\frac {400\,d}{e^5}+\frac {45}{e^4}\right )}{e}\right )}{e}-\frac {3\,d^2\,\left (\frac {400\,d}{e^5}+\frac {45}{e^4}\right )}{e^2}\right )-\frac {x\,\left (5200\,d^7+\frac {3465\,d^6\,e}{2}+2997\,d^5\,e^2+\frac {1295\,d^4\,e^3}{2}+1480\,d^3\,e^4-\frac {585\,d^2\,e^5}{2}+107\,d\,e^6+\frac {33\,e^7}{2}\right )+\frac {14600\,d^8+4815\,d^7\,e+8214\,d^6\,e^2+1739\,d^5\,e^3+3848\,d^4\,e^4-715\,d^3\,e^5+214\,d^2\,e^6+33\,d\,e^7+36\,e^8}{6\,e}+x^2\,\left (2800\,d^6\,e+945\,d^5\,e^2+1665\,d^4\,e^3+370\,d^3\,e^4+888\,d^2\,e^5-195\,d\,e^6+107\,e^7\right )}{d^3\,e^8+3\,d^2\,e^9\,x+3\,d\,e^{10}\,x^2+e^{11}\,x^3}-x^4\,\left (\frac {100\,d}{e^5}+\frac {45}{4\,e^4}\right )+x\,\left (\frac {148}{e^4}-\frac {100\,d^4}{e^8}+\frac {4\,d\,\left (\frac {37}{e^4}+\frac {400\,d^3}{e^7}+\frac {4\,d\,\left (\frac {111}{e^4}-\frac {600\,d^2}{e^6}+\frac {4\,d\,\left (\frac {400\,d}{e^5}+\frac {45}{e^4}\right )}{e}\right )}{e}-\frac {6\,d^2\,\left (\frac {400\,d}{e^5}+\frac {45}{e^4}\right )}{e^2}\right )}{e}-\frac {6\,d^2\,\left (\frac {111}{e^4}-\frac {600\,d^2}{e^6}+\frac {4\,d\,\left (\frac {400\,d}{e^5}+\frac {45}{e^4}\right )}{e}\right )}{e^2}+\frac {4\,d^3\,\left (\frac {400\,d}{e^5}+\frac {45}{e^4}\right )}{e^3}\right )+\frac {20\,x^5}{e^4}-\frac {\ln \left (d+e\,x\right )\,\left (5600\,d^5+1575\,d^4\,e+2220\,d^3\,e^2+370\,d^2\,e^3+592\,d\,e^4-65\,e^5\right )}{e^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.09, size = 401, normalized size = 1.11 \[ x^{4} \left (- \frac {100 d}{e^{5}} - \frac {45}{4 e^{4}}\right ) + x^{3} \left (\frac {1000 d^{2}}{3 e^{6}} + \frac {60 d}{e^{5}} + \frac {37}{e^{4}}\right ) + x^{2} \left (- \frac {1000 d^{3}}{e^{7}} - \frac {225 d^{2}}{e^{6}} - \frac {222 d}{e^{5}} - \frac {37}{2 e^{4}}\right ) + x \left (\frac {3500 d^{4}}{e^{8}} + \frac {900 d^{3}}{e^{7}} + \frac {1110 d^{2}}{e^{6}} + \frac {148 d}{e^{5}} + \frac {148}{e^{4}}\right ) + \frac {- 14600 d^{8} - 4815 d^{7} e - 8214 d^{6} e^{2} - 1739 d^{5} e^{3} - 3848 d^{4} e^{4} + 715 d^{3} e^{5} - 214 d^{2} e^{6} - 33 d e^{7} - 36 e^{8} + x^{2} \left (- 16800 d^{6} e^{2} - 5670 d^{5} e^{3} - 9990 d^{4} e^{4} - 2220 d^{3} e^{5} - 5328 d^{2} e^{6} + 1170 d e^{7} - 642 e^{8}\right ) + x \left (- 31200 d^{7} e - 10395 d^{6} e^{2} - 17982 d^{5} e^{3} - 3885 d^{4} e^{4} - 8880 d^{3} e^{5} + 1755 d^{2} e^{6} - 642 d e^{7} - 99 e^{8}\right )}{6 d^{3} e^{9} + 18 d^{2} e^{10} x + 18 d e^{11} x^{2} + 6 e^{12} x^{3}} + \frac {20 x^{5}}{e^{4}} - \frac {\left (5600 d^{5} + 1575 d^{4} e + 2220 d^{3} e^{2} + 370 d^{2} e^{3} + 592 d e^{4} - 65 e^{5}\right ) \log {\left (d + e x \right )}}{e^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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