Optimal. Leaf size=353 \[ \frac{3 x^5 \left (100 d^2+30 d e+37 e^2\right )}{5 e^4}-\frac{x^4 \left (135 d^2 e+400 d^3+222 d e^2+37 e^3\right )}{4 e^5}+\frac{x^3 \left (333 d^2 e^2+180 d^3 e+500 d^4+74 d e^3+148 e^4\right )}{3 e^6}-\frac{x^2 \left (444 d^3 e^2+111 d^2 e^3+225 d^4 e+600 d^5+296 d e^4-65 e^5\right )}{2 e^7}+\frac{x \left (555 d^4 e^2+148 d^3 e^3+444 d^2 e^4+270 d^5 e+700 d^6-130 d e^5+107 e^6\right )}{e^8}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{e^9 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right ) \log (d+e x)}{e^9}-\frac{5 x^6 (40 d+9 e)}{6 e^3}+\frac{100 x^7}{7 e^2} \]
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Rubi [A] time = 0.32805, antiderivative size = 353, 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} \[ \frac{3 x^5 \left (100 d^2+30 d e+37 e^2\right )}{5 e^4}-\frac{x^4 \left (135 d^2 e+400 d^3+222 d e^2+37 e^3\right )}{4 e^5}+\frac{x^3 \left (333 d^2 e^2+180 d^3 e+500 d^4+74 d e^3+148 e^4\right )}{3 e^6}-\frac{x^2 \left (444 d^3 e^2+111 d^2 e^3+225 d^4 e+600 d^5+296 d e^4-65 e^5\right )}{2 e^7}+\frac{x \left (555 d^4 e^2+148 d^3 e^3+444 d^2 e^4+270 d^5 e+700 d^6-130 d e^5+107 e^6\right )}{e^8}-\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{e^9 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right ) \log (d+e x)}{e^9}-\frac{5 x^6 (40 d+9 e)}{6 e^3}+\frac{100 x^7}{7 e^2} \]
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)^2} \, dx &=\int \left (\frac{700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6}{e^8}+\frac{\left (-600 d^5-225 d^4 e-444 d^3 e^2-111 d^2 e^3-296 d e^4+65 e^5\right ) x}{e^7}+\frac{\left (500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right ) x^2}{e^6}-\frac{\left (400 d^3+135 d^2 e+222 d e^2+37 e^3\right ) x^3}{e^5}+\frac{3 \left (100 d^2+30 d e+37 e^2\right ) x^4}{e^4}-\frac{5 (40 d+9 e) x^5}{e^3}+\frac{100 x^6}{e^2}+\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)^2}+\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)}\right ) \, dx\\ &=\frac{\left (700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6\right ) x}{e^8}-\frac{\left (600 d^5+225 d^4 e+444 d^3 e^2+111 d^2 e^3+296 d e^4-65 e^5\right ) x^2}{2 e^7}+\frac{\left (500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right ) x^3}{3 e^6}-\frac{\left (400 d^3+135 d^2 e+222 d e^2+37 e^3\right ) x^4}{4 e^5}+\frac{3 \left (100 d^2+30 d e+37 e^2\right ) x^5}{5 e^4}-\frac{5 (40 d+9 e) x^6}{6 e^3}+\frac{100 x^7}{7 e^2}-\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^9 (d+e x)}-\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 ) \log (d+e x)}{e^9}\\ \end{align*}
Mathematica [A] time = 0.136448, size = 342, normalized size = 0.97 \[ \frac{252 e^5 x^5 \left (100 d^2+30 d e+37 e^2\right )-105 e^4 x^4 \left (135 d^2 e+400 d^3+222 d e^2+37 e^3\right )+140 e^3 x^3 \left (333 d^2 e^2+180 d^3 e+500 d^4+74 d e^3+148 e^4\right )-210 e^2 x^2 \left (444 d^3 e^2+111 d^2 e^3+225 d^4 e+600 d^5+296 d e^4-65 e^5\right )+420 e x \left (555 d^4 e^2+148 d^3 e^3+444 d^2 e^4+270 d^5 e+700 d^6-130 d e^5+107 e^6\right )-\frac{420 \left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{d+e x}-420 \left (666 d^5 e^2+185 d^4 e^3+592 d^3 e^4-195 d^2 e^5+315 d^6 e+800 d^7+214 d e^6-33 e^7\right ) \log (d+e x)-350 e^6 x^6 (40 d+9 e)+6000 e^7 x^7}{420 e^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 500, normalized size = 1.4 \begin{align*} -666\,{\frac{\ln \left ( ex+d \right ){d}^{5}}{{e}^{7}}}-{\frac{15\,{x}^{6}}{2\,{e}^{2}}}-{\frac{37\,{x}^{4}}{4\,{e}^{2}}}+{\frac{148\,{x}^{3}}{3\,{e}^{2}}}+{\frac{65\,{x}^{2}}{2\,{e}^{2}}}+33\,{\frac{\ln \left ( ex+d \right ) }{{e}^{2}}}-18\,{\frac{1}{e \left ( ex+d \right ) }}-185\,{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{6}}}-592\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{{e}^{5}}}+195\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{{e}^{4}}}-214\,{\frac{\ln \left ( ex+d \right ) d}{{e}^{3}}}+555\,{\frac{{d}^{4}x}{{e}^{6}}}+148\,{\frac{{d}^{3}x}{{e}^{5}}}+444\,{\frac{{d}^{2}x}{{e}^{4}}}-130\,{\frac{dx}{{e}^{3}}}-{\frac{111\,d{x}^{4}}{2\,{e}^{3}}}+111\,{\frac{{x}^{3}{d}^{2}}{{e}^{4}}}+{\frac{74\,d{x}^{3}}{3\,{e}^{3}}}-222\,{\frac{{x}^{2}{d}^{3}}{{e}^{5}}}-{\frac{111\,{x}^{2}{d}^{2}}{2\,{e}^{4}}}-148\,{\frac{d{x}^{2}}{{e}^{3}}}-111\,{\frac{{d}^{6}}{{e}^{7} \left ( ex+d \right ) }}-37\,{\frac{{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}+270\,{\frac{x{d}^{5}}{{e}^{7}}}+18\,{\frac{{x}^{5}d}{{e}^{3}}}-{\frac{135\,{x}^{4}{d}^{2}}{4\,{e}^{4}}}+60\,{\frac{{x}^{5}{d}^{2}}{{e}^{4}}}-{\frac{100\,d{x}^{6}}{3\,{e}^{3}}}+60\,{\frac{{x}^{3}{d}^{3}}{{e}^{5}}}-100\,{\frac{{x}^{4}{d}^{3}}{{e}^{5}}}-{\frac{225\,{x}^{2}{d}^{4}}{2\,{e}^{6}}}-300\,{\frac{{x}^{2}{d}^{5}}{{e}^{7}}}-100\,{\frac{{d}^{8}}{{e}^{9} \left ( ex+d \right ) }}-45\,{\frac{{d}^{7}}{{e}^{8} \left ( ex+d \right ) }}-800\,{\frac{\ln \left ( ex+d \right ){d}^{7}}{{e}^{9}}}-315\,{\frac{\ln \left ( ex+d \right ){d}^{6}}{{e}^{8}}}+{\frac{500\,{x}^{3}{d}^{4}}{3\,{e}^{6}}}+700\,{\frac{{d}^{6}x}{{e}^{8}}}-148\,{\frac{{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+65\,{\frac{{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-107\,{\frac{{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+33\,{\frac{d}{{e}^{2} \left ( ex+d \right ) }}+107\,{\frac{x}{{e}^{2}}}+{\frac{100\,{x}^{7}}{7\,{e}^{2}}}+{\frac{111\,{x}^{5}}{5\,{e}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996031, size = 502, normalized size = 1.42 \begin{align*} -\frac{100 \, d^{8} + 45 \, d^{7} e + 111 \, d^{6} e^{2} + 37 \, d^{5} e^{3} + 148 \, d^{4} e^{4} - 65 \, d^{3} e^{5} + 107 \, d^{2} e^{6} - 33 \, d e^{7} + 18 \, e^{8}}{e^{10} x + d e^{9}} + \frac{6000 \, e^{6} x^{7} - 350 \,{\left (40 \, d e^{5} + 9 \, e^{6}\right )} x^{6} + 252 \,{\left (100 \, d^{2} e^{4} + 30 \, d e^{5} + 37 \, e^{6}\right )} x^{5} - 105 \,{\left (400 \, d^{3} e^{3} + 135 \, d^{2} e^{4} + 222 \, d e^{5} + 37 \, e^{6}\right )} x^{4} + 140 \,{\left (500 \, d^{4} e^{2} + 180 \, d^{3} e^{3} + 333 \, d^{2} e^{4} + 74 \, d e^{5} + 148 \, e^{6}\right )} x^{3} - 210 \,{\left (600 \, d^{5} e + 225 \, d^{4} e^{2} + 444 \, d^{3} e^{3} + 111 \, d^{2} e^{4} + 296 \, d e^{5} - 65 \, e^{6}\right )} x^{2} + 420 \,{\left (700 \, d^{6} + 270 \, d^{5} e + 555 \, d^{4} e^{2} + 148 \, d^{3} e^{3} + 444 \, d^{2} e^{4} - 130 \, d e^{5} + 107 \, e^{6}\right )} x}{420 \, e^{8}} - \frac{{\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 )} \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 = 0.984773, size = 1211, normalized size = 3.43 \begin{align*} \frac{6000 \, e^{8} x^{8} - 42000 \, d^{8} - 18900 \, d^{7} e - 46620 \, d^{6} e^{2} - 15540 \, d^{5} e^{3} - 62160 \, d^{4} e^{4} + 27300 \, d^{3} e^{5} - 44940 \, d^{2} e^{6} + 13860 \, d e^{7} - 7560 \, e^{8} - 50 \,{\left (160 \, d e^{7} + 63 \, e^{8}\right )} x^{7} + 14 \,{\left (800 \, d^{2} e^{6} + 315 \, d e^{7} + 666 \, e^{8}\right )} x^{6} - 21 \,{\left (800 \, d^{3} e^{5} + 315 \, d^{2} e^{6} + 666 \, d e^{7} + 185 \, e^{8}\right )} x^{5} + 35 \,{\left (800 \, d^{4} e^{4} + 315 \, d^{3} e^{5} + 666 \, d^{2} e^{6} + 185 \, d e^{7} + 592 \, e^{8}\right )} x^{4} - 70 \,{\left (800 \, d^{5} e^{3} + 315 \, d^{4} e^{4} + 666 \, d^{3} e^{5} + 185 \, d^{2} e^{6} + 592 \, d e^{7} - 195 \, e^{8}\right )} x^{3} + 210 \,{\left (800 \, d^{6} e^{2} + 315 \, d^{5} e^{3} + 666 \, d^{4} e^{4} + 185 \, d^{3} e^{5} + 592 \, d^{2} e^{6} - 195 \, d e^{7} + 214 \, e^{8}\right )} x^{2} + 420 \,{\left (700 \, d^{7} e + 270 \, d^{6} e^{2} + 555 \, d^{5} e^{3} + 148 \, d^{4} e^{4} + 444 \, d^{3} e^{5} - 130 \, d^{2} e^{6} + 107 \, d e^{7}\right )} x - 420 \,{\left (800 \, d^{8} + 315 \, d^{7} e + 666 \, d^{6} e^{2} + 185 \, d^{5} e^{3} + 592 \, d^{4} e^{4} - 195 \, d^{3} e^{5} + 214 \, d^{2} e^{6} - 33 \, d e^{7} +{\left (800 \, d^{7} e + 315 \, d^{6} e^{2} + 666 \, d^{5} e^{3} + 185 \, d^{4} e^{4} + 592 \, d^{3} e^{5} - 195 \, d^{2} e^{6} + 214 \, d e^{7} - 33 \, e^{8}\right )} x\right )} \log \left (e x + d\right )}{420 \,{\left (e^{10} x + d e^{9}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.45614, size = 367, normalized size = 1.04 \begin{align*} - \frac{100 d^{8} + 45 d^{7} e + 111 d^{6} e^{2} + 37 d^{5} e^{3} + 148 d^{4} e^{4} - 65 d^{3} e^{5} + 107 d^{2} e^{6} - 33 d e^{7} + 18 e^{8}}{d e^{9} + e^{10} x} + \frac{100 x^{7}}{7 e^{2}} - \frac{x^{6} \left (200 d + 45 e\right )}{6 e^{3}} + \frac{x^{5} \left (300 d^{2} + 90 d e + 111 e^{2}\right )}{5 e^{4}} - \frac{x^{4} \left (400 d^{3} + 135 d^{2} e + 222 d e^{2} + 37 e^{3}\right )}{4 e^{5}} + \frac{x^{3} \left (500 d^{4} + 180 d^{3} e + 333 d^{2} e^{2} + 74 d e^{3} + 148 e^{4}\right )}{3 e^{6}} - \frac{x^{2} \left (600 d^{5} + 225 d^{4} e + 444 d^{3} e^{2} + 111 d^{2} e^{3} + 296 d e^{4} - 65 e^{5}\right )}{2 e^{7}} + \frac{x \left (700 d^{6} + 270 d^{5} e + 555 d^{4} e^{2} + 148 d^{3} e^{3} + 444 d^{2} e^{4} - 130 d e^{5} + 107 e^{6}\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 ) \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 = 1.16612, size = 620, normalized size = 1.76 \begin{align*} -\frac{1}{420} \,{\left (x e + d\right )}^{7}{\left (\frac{350 \,{\left (160 \, d e + 9 \, e^{2}\right )} e^{\left (-1\right )}}{x e + d} - \frac{84 \,{\left (2800 \, d^{2} e^{2} + 315 \, d e^{3} + 111 \, e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} + \frac{105 \,{\left (5600 \, d^{3} e^{3} + 945 \, d^{2} e^{4} + 666 \, d e^{5} + 37 \, e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} - \frac{140 \,{\left (7000 \, d^{4} e^{4} + 1575 \, d^{3} e^{5} + 1665 \, d^{2} e^{6} + 185 \, d e^{7} + 148 \, e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} + \frac{210 \,{\left (5600 \, d^{5} e^{5} + 1575 \, d^{4} e^{6} + 2220 \, d^{3} e^{7} + 370 \, d^{2} e^{8} + 592 \, d e^{9} - 65 \, e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} - \frac{420 \,{\left (2800 \, d^{6} e^{6} + 945 \, d^{5} e^{7} + 1665 \, d^{4} e^{8} + 370 \, d^{3} e^{9} + 888 \, d^{2} e^{10} - 195 \, d e^{11} + 107 \, e^{12}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{6}} - 6000\right )} e^{\left (-9\right )} +{\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 )} e^{\left (-9\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{100 \, d^{8} e^{7}}{x e + d} + \frac{45 \, d^{7} e^{8}}{x e + d} + \frac{111 \, d^{6} e^{9}}{x e + d} + \frac{37 \, d^{5} e^{10}}{x e + d} + \frac{148 \, d^{4} e^{11}}{x e + d} - \frac{65 \, d^{3} e^{12}}{x e + d} + \frac{107 \, d^{2} e^{13}}{x e + d} - \frac{33 \, d e^{14}}{x e + d} + \frac{18 \, e^{15}}{x e + d}\right )} e^{\left (-16\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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