Optimal. Leaf size=231 \[ \frac{x^2 \left (120 d^2+51 d e+17 e^2\right )}{2 e^5}-\frac{x \left (102 d^2 e+200 d^3+51 d e^2+4 e^3\right )}{e^6}+\frac{68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5}{e^7 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{2 e^7 (d+e x)^2}+\frac{\left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) \log (d+e x)}{e^7}-\frac{x^3 (60 d+17 e)}{3 e^4}+\frac{5 x^4}{e^3} \]
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Rubi [A] time = 0.203744, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028, Rules used = {1628} \[ \frac{x^2 \left (120 d^2+51 d e+17 e^2\right )}{2 e^5}-\frac{x \left (102 d^2 e+200 d^3+51 d e^2+4 e^3\right )}{e^6}+\frac{68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5}{e^7 (d+e x)}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right )}{2 e^7 (d+e x)^2}+\frac{\left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) \log (d+e x)}{e^7}-\frac{x^3 (60 d+17 e)}{3 e^4}+\frac{5 x^4}{e^3} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int \frac{\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{(d+e x)^3} \, dx &=\int \left (\frac{-200 d^3-102 d^2 e-51 d e^2-4 e^3}{e^6}+\frac{\left (120 d^2+51 d e+17 e^2\right ) x}{e^5}-\frac{(60 d+17 e) x^2}{e^4}+\frac{20 x^3}{e^3}+\frac{20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6}{e^6 (d+e x)^3}+\frac{-120 d^5-85 d^4 e-68 d^3 e^2-12 d^2 e^3-42 d e^4+7 e^5}{e^6 (d+e x)^2}+\frac{300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{\left (200 d^3+102 d^2 e+51 d e^2+4 e^3\right ) x}{e^6}+\frac{\left (120 d^2+51 d e+17 e^2\right ) x^2}{2 e^5}-\frac{(60 d+17 e) x^3}{3 e^4}+\frac{5 x^4}{e^3}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right )}{2 e^7 (d+e x)^2}+\frac{120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5}{e^7 (d+e x)}+\frac{\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.0682934, size = 204, normalized size = 0.88 \[ \frac{-51 d^4 e^2 \left (40 x^2+2 x-7\right )-3 d^3 e^3 \left (200 x^3+357 x^2-34 x-20\right )+d^2 e^4 \left (150 x^4-340 x^3-561 x^2+48 x+189\right )+6 \left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) (d+e x)^2 \log (d+e x)+d^5 e (459-480 x)+660 d^6-d e^5 \left (60 x^5-85 x^4+204 x^3+48 x^2-252 x+21\right )+e^6 \left (30 x^6-34 x^5+51 x^4-24 x^3-42 x-18\right )}{6 e^7 (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 336, normalized size = 1.5 \begin{align*} 21\,{\frac{\ln \left ( ex+d \right ) }{{e}^{3}}}-7\,{\frac{1}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{1}{e \left ( ex+d \right ) ^{2}}}+{\frac{17\,{x}^{2}}{2\,{e}^{3}}}-4\,{\frac{x}{{e}^{3}}}-{\frac{17\,{x}^{3}}{3\,{e}^{3}}}+120\,{\frac{{d}^{5}}{{e}^{7} \left ( ex+d \right ) }}+85\,{\frac{{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+300\,{\frac{\ln \left ( ex+d \right ){d}^{4}}{{e}^{7}}}+170\,{\frac{\ln \left ( ex+d \right ){d}^{3}}{{e}^{6}}}+102\,{\frac{\ln \left ( ex+d \right ){d}^{2}}{{e}^{5}}}+60\,{\frac{{x}^{2}{d}^{2}}{{e}^{5}}}+{\frac{51\,d{x}^{2}}{2\,{e}^{4}}}+68\,{\frac{{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}+12\,{\frac{{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+42\,{\frac{d}{{e}^{3} \left ( ex+d \right ) }}-10\,{\frac{{d}^{6}}{{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{17\,{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{17\,{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-2\,{\frac{{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{21\,{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{7\,d}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+12\,{\frac{\ln \left ( ex+d \right ) d}{{e}^{4}}}-200\,{\frac{{d}^{3}x}{{e}^{6}}}-102\,{\frac{{d}^{2}x}{{e}^{5}}}-51\,{\frac{dx}{{e}^{4}}}-20\,{\frac{d{x}^{3}}{{e}^{4}}}+5\,{\frac{{x}^{4}}{{e}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99386, size = 324, normalized size = 1.4 \begin{align*} \frac{220 \, d^{6} + 153 \, d^{5} e + 119 \, d^{4} e^{2} + 20 \, d^{3} e^{3} + 63 \, d^{2} e^{4} - 7 \, d e^{5} - 6 \, e^{6} + 2 \,{\left (120 \, d^{5} e + 85 \, d^{4} e^{2} + 68 \, d^{3} e^{3} + 12 \, d^{2} e^{4} + 42 \, d e^{5} - 7 \, e^{6}\right )} x}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{30 \, e^{3} x^{4} - 2 \,{\left (60 \, d e^{2} + 17 \, e^{3}\right )} x^{3} + 3 \,{\left (120 \, d^{2} e + 51 \, d e^{2} + 17 \, e^{3}\right )} x^{2} - 6 \,{\left (200 \, d^{3} + 102 \, d^{2} e + 51 \, d e^{2} + 4 \, e^{3}\right )} x}{6 \, e^{6}} + \frac{{\left (300 \, d^{4} + 170 \, d^{3} e + 102 \, d^{2} e^{2} + 12 \, d e^{3} + 21 \, e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.923654, size = 819, normalized size = 3.55 \begin{align*} \frac{30 \, e^{6} x^{6} + 660 \, d^{6} + 459 \, d^{5} e + 357 \, d^{4} e^{2} + 60 \, d^{3} e^{3} + 189 \, d^{2} e^{4} - 21 \, d e^{5} - 18 \, e^{6} - 2 \,{\left (30 \, d e^{5} + 17 \, e^{6}\right )} x^{5} +{\left (150 \, d^{2} e^{4} + 85 \, d e^{5} + 51 \, e^{6}\right )} x^{4} - 4 \,{\left (150 \, d^{3} e^{3} + 85 \, d^{2} e^{4} + 51 \, d e^{5} + 6 \, e^{6}\right )} x^{3} - 3 \,{\left (680 \, d^{4} e^{2} + 357 \, d^{3} e^{3} + 187 \, d^{2} e^{4} + 16 \, d e^{5}\right )} x^{2} - 6 \,{\left (80 \, d^{5} e + 17 \, d^{4} e^{2} - 17 \, d^{3} e^{3} - 8 \, d^{2} e^{4} - 42 \, d e^{5} + 7 \, e^{6}\right )} x + 6 \,{\left (300 \, d^{6} + 170 \, d^{5} e + 102 \, d^{4} e^{2} + 12 \, d^{3} e^{3} + 21 \, d^{2} e^{4} +{\left (300 \, d^{4} e^{2} + 170 \, d^{3} e^{3} + 102 \, d^{2} e^{4} + 12 \, d e^{5} + 21 \, e^{6}\right )} x^{2} + 2 \,{\left (300 \, d^{5} e + 170 \, d^{4} e^{2} + 102 \, d^{3} e^{3} + 12 \, d^{2} e^{4} + 21 \, d e^{5}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.82195, size = 238, normalized size = 1.03 \begin{align*} \frac{220 d^{6} + 153 d^{5} e + 119 d^{4} e^{2} + 20 d^{3} e^{3} + 63 d^{2} e^{4} - 7 d e^{5} - 6 e^{6} + x \left (240 d^{5} e + 170 d^{4} e^{2} + 136 d^{3} e^{3} + 24 d^{2} e^{4} + 84 d e^{5} - 14 e^{6}\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{5 x^{4}}{e^{3}} - \frac{x^{3} \left (60 d + 17 e\right )}{3 e^{4}} + \frac{x^{2} \left (120 d^{2} + 51 d e + 17 e^{2}\right )}{2 e^{5}} - \frac{x \left (200 d^{3} + 102 d^{2} e + 51 d e^{2} + 4 e^{3}\right )}{e^{6}} + \frac{\left (300 d^{4} + 170 d^{3} e + 102 d^{2} e^{2} + 12 d e^{3} + 21 e^{4}\right ) \log{\left (d + e x \right )}}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16683, size = 292, normalized size = 1.26 \begin{align*}{\left (300 \, d^{4} + 170 \, d^{3} e + 102 \, d^{2} e^{2} + 12 \, d e^{3} + 21 \, e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (30 \, x^{4} e^{9} - 120 \, d x^{3} e^{8} + 360 \, d^{2} x^{2} e^{7} - 1200 \, d^{3} x e^{6} - 34 \, x^{3} e^{9} + 153 \, d x^{2} e^{8} - 612 \, d^{2} x e^{7} + 51 \, x^{2} e^{9} - 306 \, d x e^{8} - 24 \, x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (220 \, d^{6} + 153 \, d^{5} e + 119 \, d^{4} e^{2} + 20 \, d^{3} e^{3} + 63 \, d^{2} e^{4} + 2 \,{\left (120 \, d^{5} e + 85 \, d^{4} e^{2} + 68 \, d^{3} e^{3} + 12 \, d^{2} e^{4} + 42 \, d e^{5} - 7 \, e^{6}\right )} x - 7 \, d e^{5} - 6 \, e^{6}\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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