Optimal. Leaf size=360 \[ \frac{x^3 \left (1000 d^2+180 d e+111 e^2\right )}{3 e^6}-\frac{x^2 \left (450 d^2 e+2000 d^3+444 d e^2+37 e^3\right )}{2 e^7}+\frac{2 x \left (555 d^2 e^2+450 d^3 e+1750 d^4+74 d e^3+74 e^4\right )}{e^8}-\frac{1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6}{e^9 (d+e x)}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right )}{2 e^9 (d+e x)^2}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{3 e^9 (d+e x)^3}-\frac{\left (2220 d^3 e^2+370 d^2 e^3+1575 d^4 e+5600 d^5+592 d e^4-65 e^5\right ) \log (d+e x)}{e^9}-\frac{5 x^4 (80 d+9 e)}{4 e^5}+\frac{20 x^5}{e^4} \]
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Rubi [A] time = 0.358292, antiderivative size = 360, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.121456, size = 344, normalized size = 0.96 \[ \frac{4 e^3 x^3 \left (1000 d^2+180 d e+111 e^2\right )-6 e^2 x^2 \left (450 d^2 e+2000 d^3+444 d e^2+37 e^3\right )+24 e x \left (555 d^2 e^2+450 d^3 e+1750 d^4+74 d e^3+74 e^4\right )-\frac{12 \left (1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6\right )}{d+e x}+\frac{6 \left (666 d^5 e^2+185 d^4 e^3+592 d^3 e^4-195 d^2 e^5+315 d^6 e+800 d^7+214 d e^6-33 e^7\right )}{(d+e x)^2}-\frac{4 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{(d+e x)^3}-12 \left (2220 d^3 e^2+370 d^2 e^3+1575 d^4 e+5600 d^5+592 d e^4-65 e^5\right ) \log (d+e x)-15 e^4 x^4 (80 d+9 e)+240 e^5 x^5}{12 e^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 558, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0135, size = 527, normalized size = 1.46 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02185, size = 1434, normalized size = 3.98 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.23281, size = 391, normalized size = 1.09 \begin{align*} - \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{x^{4} \left (400 d + 45 e\right )}{4 e^{5}} + \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{x \left (3500 d^{4} + 900 d^{3} e + 1110 d^{2} e^{2} + 148 d e^{3} + 148 e^{4}\right )}{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 (d + e x \right )}}{e^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1417, size = 466, normalized size = 1.29 \begin{align*} -{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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