Optimal. Leaf size=352 \[ \frac{x^6 \left (100 d^2+45 d e+111 e^2\right )}{6 e^3}-\frac{x^5 \left (45 d^2 e+100 d^3+111 d e^2+37 e^3\right )}{5 e^4}+\frac{x^4 \left (111 d^2 e^2+45 d^3 e+100 d^4+37 d e^3+148 e^4\right )}{4 e^5}-\frac{x^3 \left (111 d^3 e^2+37 d^2 e^3+45 d^4 e+100 d^5+148 d e^4-65 e^5\right )}{3 e^6}+\frac{x^2 \left (111 d^4 e^2+37 d^3 e^3+148 d^2 e^4+45 d^5 e+100 d^6-65 d e^5+107 e^6\right )}{2 e^7}-\frac{x \left (111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+45 d^6 e+100 d^7+107 d e^6-33 e^7\right )}{e^8}+\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^9}-\frac{5 x^7 (20 d+9 e)}{7 e^2}+\frac{25 x^8}{2 e} \]
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Rubi [A] time = 0.316476, antiderivative size = 352, 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} \[ \frac{x^6 \left (100 d^2+45 d e+111 e^2\right )}{6 e^3}-\frac{x^5 \left (45 d^2 e+100 d^3+111 d e^2+37 e^3\right )}{5 e^4}+\frac{x^4 \left (111 d^2 e^2+45 d^3 e+100 d^4+37 d e^3+148 e^4\right )}{4 e^5}-\frac{x^3 \left (111 d^3 e^2+37 d^2 e^3+45 d^4 e+100 d^5+148 d e^4-65 e^5\right )}{3 e^6}+\frac{x^2 \left (111 d^4 e^2+37 d^3 e^3+148 d^2 e^4+45 d^5 e+100 d^6-65 d e^5+107 e^6\right )}{2 e^7}-\frac{x \left (111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+45 d^6 e+100 d^7+107 d e^6-33 e^7\right )}{e^8}+\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \log (d+e x)}{e^9}-\frac{5 x^7 (20 d+9 e)}{7 e^2}+\frac{25 x^8}{2 e} \]
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} \, dx &=\int \left (\frac{-100 d^7-45 d^6 e-111 d^5 e^2-37 d^4 e^3-148 d^3 e^4+65 d^2 e^5-107 d e^6+33 e^7}{e^8}+\frac{\left (100 d^6+45 d^5 e+111 d^4 e^2+37 d^3 e^3+148 d^2 e^4-65 d e^5+107 e^6\right ) x}{e^7}+\frac{\left (-100 d^5-45 d^4 e-111 d^3 e^2-37 d^2 e^3-148 d e^4+65 e^5\right ) x^2}{e^6}+\frac{\left (100 d^4+45 d^3 e+111 d^2 e^2+37 d e^3+148 e^4\right ) x^3}{e^5}-\frac{\left (100 d^3+45 d^2 e+111 d e^2+37 e^3\right ) x^4}{e^4}+\frac{\left (100 d^2+45 d e+111 e^2\right ) x^5}{e^3}-\frac{5 (20 d+9 e) x^6}{e^2}+\frac{100 x^7}{e}+\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)}\right ) \, dx\\ &=-\frac{\left (100 d^7+45 d^6 e+111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+107 d e^6-33 e^7\right ) x}{e^8}+\frac{\left (100 d^6+45 d^5 e+111 d^4 e^2+37 d^3 e^3+148 d^2 e^4-65 d e^5+107 e^6\right ) x^2}{2 e^7}-\frac{\left (100 d^5+45 d^4 e+111 d^3 e^2+37 d^2 e^3+148 d e^4-65 e^5\right ) x^3}{3 e^6}+\frac{\left (100 d^4+45 d^3 e+111 d^2 e^2+37 d e^3+148 e^4\right ) x^4}{4 e^5}-\frac{\left (100 d^3+45 d^2 e+111 d e^2+37 e^3\right ) x^5}{5 e^4}+\frac{\left (100 d^2+45 d e+111 e^2\right ) x^6}{6 e^3}-\frac{5 (20 d+9 e) x^7}{7 e^2}+\frac{25 x^8}{2 e}+\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 ) \log (d+e x)}{e^9}\\ \end{align*}
Mathematica [A] time = 0.1217, size = 262, normalized size = 0.74 \[ \frac{x \left (-70 d^5 e^2 \left (200 x^2-135 x+666\right )+210 d^4 e^3 \left (50 x^3-30 x^2+111 x-74\right )-105 d^3 e^4 \left (80 x^4-45 x^3+148 x^2-74 x+592\right )+35 d^2 e^5 \left (200 x^5-108 x^4+333 x^3-148 x^2+888 x+780\right )+2100 d^6 e (10 x-9)-42000 d^7-d e^6 \left (6000 x^6-3150 x^5+9324 x^4-3885 x^3+20720 x^2+13650 x+44940\right )+2 e^7 \left (2625 x^7-1350 x^6+3885 x^5-1554 x^4+7770 x^3+4550 x^2+11235 x+6930\right )\right )}{420 e^8}+\frac{\left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )^2 \log (d+e x)}{e^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 465, normalized size = 1.3 \begin{align*} -{\frac{45\,{x}^{7}}{7\,e}}+33\,{\frac{x}{e}}+18\,{\frac{\ln \left ( ex+d \right ) }{e}}+{\frac{65\,{x}^{3}}{3\,e}}+37\,{\frac{{x}^{4}}{e}}-{\frac{37\,{x}^{5}}{5\,e}}-{\frac{148\,d{x}^{3}}{3\,{e}^{2}}}-{\frac{37\,{x}^{3}{d}^{2}}{3\,{e}^{3}}}-{\frac{111\,{x}^{5}d}{5\,{e}^{2}}}+{\frac{111\,{x}^{4}{d}^{2}}{4\,{e}^{3}}}+{\frac{37\,d{x}^{4}}{4\,{e}^{2}}}+{\frac{37\,{x}^{2}{d}^{3}}{2\,{e}^{4}}}+65\,{\frac{{d}^{2}x}{{e}^{3}}}+{\frac{111\,{x}^{2}{d}^{4}}{2\,{e}^{5}}}-37\,{\frac{x{d}^{4}}{{e}^{5}}}-148\,{\frac{{d}^{3}x}{{e}^{4}}}-{\frac{65\,d{x}^{2}}{2\,{e}^{2}}}-111\,{\frac{{d}^{5}x}{{e}^{6}}}+100\,{\frac{\ln \left ( ex+d \right ){d}^{8}}{{e}^{9}}}-107\,{\frac{dx}{{e}^{2}}}+74\,{\frac{{x}^{2}{d}^{2}}{{e}^{3}}}+37\,{\frac{\ln \left ( ex+d \right ){d}^{5}}{{e}^{6}}}+148\,{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{5}}}+111\,{\frac{\ln \left ( ex+d \right ){d}^{6}}{{e}^{7}}}-65\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{{e}^{4}}}+107\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{{e}^{3}}}-33\,{\frac{\ln \left ( ex+d \right ) d}{{e}^{2}}}+25\,{\frac{{x}^{4}{d}^{4}}{{e}^{5}}}-15\,{\frac{{x}^{3}{d}^{4}}{{e}^{5}}}-100\,{\frac{{d}^{7}x}{{e}^{8}}}+{\frac{15\,d{x}^{6}}{2\,{e}^{2}}}+45\,{\frac{\ln \left ( ex+d \right ){d}^{7}}{{e}^{8}}}+{\frac{50\,{x}^{6}{d}^{2}}{3\,{e}^{3}}}-45\,{\frac{{d}^{6}x}{{e}^{7}}}+50\,{\frac{{x}^{2}{d}^{6}}{{e}^{7}}}-{\frac{100\,{x}^{3}{d}^{5}}{3\,{e}^{6}}}-20\,{\frac{{x}^{5}{d}^{3}}{{e}^{4}}}-37\,{\frac{{x}^{3}{d}^{3}}{{e}^{4}}}+{\frac{45\,{x}^{2}{d}^{5}}{2\,{e}^{6}}}-9\,{\frac{{x}^{5}{d}^{2}}{{e}^{3}}}-{\frac{100\,{x}^{7}d}{7\,{e}^{2}}}+{\frac{45\,{x}^{4}{d}^{3}}{4\,{e}^{4}}}+{\frac{107\,{x}^{2}}{2\,e}}+{\frac{25\,{x}^{8}}{2\,e}}+{\frac{37\,{x}^{6}}{2\,e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01593, size = 494, normalized size = 1.4 \begin{align*} \frac{5250 \, e^{7} x^{8} - 300 \,{\left (20 \, d e^{6} + 9 \, e^{7}\right )} x^{7} + 70 \,{\left (100 \, d^{2} e^{5} + 45 \, d e^{6} + 111 \, e^{7}\right )} x^{6} - 84 \,{\left (100 \, d^{3} e^{4} + 45 \, d^{2} e^{5} + 111 \, d e^{6} + 37 \, e^{7}\right )} x^{5} + 105 \,{\left (100 \, d^{4} e^{3} + 45 \, d^{3} e^{4} + 111 \, d^{2} e^{5} + 37 \, d e^{6} + 148 \, e^{7}\right )} x^{4} - 140 \,{\left (100 \, d^{5} e^{2} + 45 \, d^{4} e^{3} + 111 \, d^{3} e^{4} + 37 \, d^{2} e^{5} + 148 \, d e^{6} - 65 \, e^{7}\right )} x^{3} + 210 \,{\left (100 \, d^{6} e + 45 \, d^{5} e^{2} + 111 \, d^{4} e^{3} + 37 \, d^{3} e^{4} + 148 \, d^{2} e^{5} - 65 \, d e^{6} + 107 \, e^{7}\right )} x^{2} - 420 \,{\left (100 \, d^{7} + 45 \, d^{6} e + 111 \, d^{5} e^{2} + 37 \, d^{4} e^{3} + 148 \, d^{3} e^{4} - 65 \, d^{2} e^{5} + 107 \, d e^{6} - 33 \, e^{7}\right )} x}{420 \, e^{8}} + \frac{{\left (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}\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 = 1.05444, size = 875, normalized size = 2.49 \begin{align*} \frac{5250 \, e^{8} x^{8} - 300 \,{\left (20 \, d e^{7} + 9 \, e^{8}\right )} x^{7} + 70 \,{\left (100 \, d^{2} e^{6} + 45 \, d e^{7} + 111 \, e^{8}\right )} x^{6} - 84 \,{\left (100 \, d^{3} e^{5} + 45 \, d^{2} e^{6} + 111 \, d e^{7} + 37 \, e^{8}\right )} x^{5} + 105 \,{\left (100 \, d^{4} e^{4} + 45 \, d^{3} e^{5} + 111 \, d^{2} e^{6} + 37 \, d e^{7} + 148 \, e^{8}\right )} x^{4} - 140 \,{\left (100 \, d^{5} e^{3} + 45 \, d^{4} e^{4} + 111 \, d^{3} e^{5} + 37 \, d^{2} e^{6} + 148 \, d e^{7} - 65 \, e^{8}\right )} x^{3} + 210 \,{\left (100 \, d^{6} e^{2} + 45 \, d^{5} e^{3} + 111 \, d^{4} e^{4} + 37 \, d^{3} e^{5} + 148 \, d^{2} e^{6} - 65 \, d e^{7} + 107 \, e^{8}\right )} x^{2} - 420 \,{\left (100 \, d^{7} e + 45 \, d^{6} e^{2} + 111 \, d^{5} e^{3} + 37 \, d^{4} e^{4} + 148 \, d^{3} e^{5} - 65 \, d^{2} e^{6} + 107 \, d e^{7} - 33 \, e^{8}\right )} x + 420 \,{\left (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}\right )} \log \left (e x + d\right )}{420 \, e^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.717569, size = 347, normalized size = 0.99 \begin{align*} \frac{25 x^{8}}{2 e} - \frac{x^{7} \left (100 d + 45 e\right )}{7 e^{2}} + \frac{x^{6} \left (100 d^{2} + 45 d e + 111 e^{2}\right )}{6 e^{3}} - \frac{x^{5} \left (100 d^{3} + 45 d^{2} e + 111 d e^{2} + 37 e^{3}\right )}{5 e^{4}} + \frac{x^{4} \left (100 d^{4} + 45 d^{3} e + 111 d^{2} e^{2} + 37 d e^{3} + 148 e^{4}\right )}{4 e^{5}} - \frac{x^{3} \left (100 d^{5} + 45 d^{4} e + 111 d^{3} e^{2} + 37 d^{2} e^{3} + 148 d e^{4} - 65 e^{5}\right )}{3 e^{6}} + \frac{x^{2} \left (100 d^{6} + 45 d^{5} e + 111 d^{4} e^{2} + 37 d^{3} e^{3} + 148 d^{2} e^{4} - 65 d e^{5} + 107 e^{6}\right )}{2 e^{7}} - \frac{x \left (100 d^{7} + 45 d^{6} e + 111 d^{5} e^{2} + 37 d^{4} e^{3} + 148 d^{3} e^{4} - 65 d^{2} e^{5} + 107 d e^{6} - 33 e^{7}\right )}{e^{8}} + \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 ) \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 = 1.14073, size = 510, normalized size = 1.45 \begin{align*}{\left (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}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{420} \,{\left (5250 \, x^{8} e^{7} - 6000 \, d x^{7} e^{6} + 7000 \, d^{2} x^{6} e^{5} - 8400 \, d^{3} x^{5} e^{4} + 10500 \, d^{4} x^{4} e^{3} - 14000 \, d^{5} x^{3} e^{2} + 21000 \, d^{6} x^{2} e - 42000 \, d^{7} x - 2700 \, x^{7} e^{7} + 3150 \, d x^{6} e^{6} - 3780 \, d^{2} x^{5} e^{5} + 4725 \, d^{3} x^{4} e^{4} - 6300 \, d^{4} x^{3} e^{3} + 9450 \, d^{5} x^{2} e^{2} - 18900 \, d^{6} x e + 7770 \, x^{6} e^{7} - 9324 \, d x^{5} e^{6} + 11655 \, d^{2} x^{4} e^{5} - 15540 \, d^{3} x^{3} e^{4} + 23310 \, d^{4} x^{2} e^{3} - 46620 \, d^{5} x e^{2} - 3108 \, x^{5} e^{7} + 3885 \, d x^{4} e^{6} - 5180 \, d^{2} x^{3} e^{5} + 7770 \, d^{3} x^{2} e^{4} - 15540 \, d^{4} x e^{3} + 15540 \, x^{4} e^{7} - 20720 \, d x^{3} e^{6} + 31080 \, d^{2} x^{2} e^{5} - 62160 \, d^{3} x e^{4} + 9100 \, x^{3} e^{7} - 13650 \, d x^{2} e^{6} + 27300 \, d^{2} x e^{5} + 22470 \, x^{2} e^{7} - 44940 \, d x e^{6} + 13860 \, x e^{7}\right )} e^{\left (-8\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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