Optimal. Leaf size=258 \[ \frac{\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac{2 \left (85 d^2 e+200 d^3+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac{\left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) (d+e x)^6}{6 e^7}-\frac{\left (68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5\right ) (d+e x)^5}{5 e^7}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) (d+e x)^4}{4 e^7}+\frac{2 (d+e x)^{10}}{e^7}-\frac{(120 d+17 e) (d+e x)^9}{9 e^7} \]
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Rubi [A] time = 0.256623, antiderivative size = 258, 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} \[ \frac{\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac{2 \left (85 d^2 e+200 d^3+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac{\left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) (d+e x)^6}{6 e^7}-\frac{\left (68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5\right ) (d+e x)^5}{5 e^7}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) (d+e x)^4}{4 e^7}+\frac{2 (d+e x)^{10}}{e^7}-\frac{(120 d+17 e) (d+e x)^9}{9 e^7} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int (d+e x)^3 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx &=\int \left (\frac{\left (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\right ) (d+e x)^3}{e^6}+\frac{\left (-120 d^5-85 d^4 e-68 d^3 e^2-12 d^2 e^3-42 d e^4+7 e^5\right ) (d+e x)^4}{e^6}+\frac{\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^5}{e^6}-\frac{2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^6}{e^6}+\frac{\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^7}{e^6}+\frac{(-120 d-17 e) (d+e x)^8}{e^6}+\frac{20 (d+e x)^9}{e^6}\right ) \, dx\\ &=\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 ) (d+e x)^4}{4 e^7}-\frac{\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)^5}{5 e^7}+\frac{\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^6}{6 e^7}-\frac{2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac{\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac{(120 d+17 e) (d+e x)^9}{9 e^7}+\frac{2 (d+e x)^{10}}{e^7}\\ \end{align*}
Mathematica [A] time = 0.0406716, size = 212, normalized size = 0.82 \[ \frac{1}{8} e x^8 \left (60 d^2-51 d e+17 e^2\right )+\frac{1}{7} x^7 \left (-51 d^2 e+20 d^3+51 d e^2-4 e^3\right )+\frac{1}{6} x^6 \left (51 d^2 e-17 d^3-12 d e^2+21 e^3\right )+\frac{1}{5} x^5 \left (-12 d^2 e+17 d^3+63 d e^2+7 e^3\right )+\frac{1}{4} x^4 \left (63 d^2 e-4 d^3+21 d e^2+6 e^3\right )+d x^3 \left (7 d^2+7 d e+6 e^2\right )+\frac{1}{2} d^2 x^2 (7 d+18 e)+6 d^3 x+\frac{1}{9} e^2 x^9 (60 d-17 e)+2 e^3 x^{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 208, normalized size = 0.8 \begin{align*} 2\,{e}^{3}{x}^{10}+{\frac{ \left ( 60\,d{e}^{2}-17\,{e}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 60\,{d}^{2}e-51\,d{e}^{2}+17\,{e}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 20\,{d}^{3}-51\,{d}^{2}e+51\,d{e}^{2}-4\,{e}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( -17\,{d}^{3}+51\,{d}^{2}e-12\,d{e}^{2}+21\,{e}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 17\,{d}^{3}-12\,{d}^{2}e+63\,d{e}^{2}+7\,{e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( -4\,{d}^{3}+63\,{d}^{2}e+21\,d{e}^{2}+6\,{e}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{d}^{3}+21\,{d}^{2}e+18\,d{e}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{d}^{3}+18\,{d}^{2}e \right ){x}^{2}}{2}}+6\,{d}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984523, size = 278, normalized size = 1.08 \begin{align*} 2 \, e^{3} x^{10} + \frac{1}{9} \,{\left (60 \, d e^{2} - 17 \, e^{3}\right )} x^{9} + \frac{1}{8} \,{\left (60 \, d^{2} e - 51 \, d e^{2} + 17 \, e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (20 \, d^{3} - 51 \, d^{2} e + 51 \, d e^{2} - 4 \, e^{3}\right )} x^{7} - \frac{1}{6} \,{\left (17 \, d^{3} - 51 \, d^{2} e + 12 \, d e^{2} - 21 \, e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (17 \, d^{3} - 12 \, d^{2} e + 63 \, d e^{2} + 7 \, e^{3}\right )} x^{5} - \frac{1}{4} \,{\left (4 \, d^{3} - 63 \, d^{2} e - 21 \, d e^{2} - 6 \, e^{3}\right )} x^{4} + 6 \, d^{3} x +{\left (7 \, d^{3} + 7 \, d^{2} e + 6 \, d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (7 \, d^{3} + 18 \, d^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.831358, size = 560, normalized size = 2.17 \begin{align*} 2 x^{10} e^{3} - \frac{17}{9} x^{9} e^{3} + \frac{20}{3} x^{9} e^{2} d + \frac{17}{8} x^{8} e^{3} - \frac{51}{8} x^{8} e^{2} d + \frac{15}{2} x^{8} e d^{2} - \frac{4}{7} x^{7} e^{3} + \frac{51}{7} x^{7} e^{2} d - \frac{51}{7} x^{7} e d^{2} + \frac{20}{7} x^{7} d^{3} + \frac{7}{2} x^{6} e^{3} - 2 x^{6} e^{2} d + \frac{17}{2} x^{6} e d^{2} - \frac{17}{6} x^{6} d^{3} + \frac{7}{5} x^{5} e^{3} + \frac{63}{5} x^{5} e^{2} d - \frac{12}{5} x^{5} e d^{2} + \frac{17}{5} x^{5} d^{3} + \frac{3}{2} x^{4} e^{3} + \frac{21}{4} x^{4} e^{2} d + \frac{63}{4} x^{4} e d^{2} - x^{4} d^{3} + 6 x^{3} e^{2} d + 7 x^{3} e d^{2} + 7 x^{3} d^{3} + 9 x^{2} e d^{2} + \frac{7}{2} x^{2} d^{3} + 6 x d^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.104844, size = 230, normalized size = 0.89 \begin{align*} 6 d^{3} x + 2 e^{3} x^{10} + x^{9} \left (\frac{20 d e^{2}}{3} - \frac{17 e^{3}}{9}\right ) + x^{8} \left (\frac{15 d^{2} e}{2} - \frac{51 d e^{2}}{8} + \frac{17 e^{3}}{8}\right ) + x^{7} \left (\frac{20 d^{3}}{7} - \frac{51 d^{2} e}{7} + \frac{51 d e^{2}}{7} - \frac{4 e^{3}}{7}\right ) + x^{6} \left (- \frac{17 d^{3}}{6} + \frac{17 d^{2} e}{2} - 2 d e^{2} + \frac{7 e^{3}}{2}\right ) + x^{5} \left (\frac{17 d^{3}}{5} - \frac{12 d^{2} e}{5} + \frac{63 d e^{2}}{5} + \frac{7 e^{3}}{5}\right ) + x^{4} \left (- d^{3} + \frac{63 d^{2} e}{4} + \frac{21 d e^{2}}{4} + \frac{3 e^{3}}{2}\right ) + x^{3} \left (7 d^{3} + 7 d^{2} e + 6 d e^{2}\right ) + x^{2} \left (\frac{7 d^{3}}{2} + 9 d^{2} e\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12525, size = 311, normalized size = 1.21 \begin{align*} 2 \, x^{10} e^{3} + \frac{20}{3} \, d x^{9} e^{2} + \frac{15}{2} \, d^{2} x^{8} e + \frac{20}{7} \, d^{3} x^{7} - \frac{17}{9} \, x^{9} e^{3} - \frac{51}{8} \, d x^{8} e^{2} - \frac{51}{7} \, d^{2} x^{7} e - \frac{17}{6} \, d^{3} x^{6} + \frac{17}{8} \, x^{8} e^{3} + \frac{51}{7} \, d x^{7} e^{2} + \frac{17}{2} \, d^{2} x^{6} e + \frac{17}{5} \, d^{3} x^{5} - \frac{4}{7} \, x^{7} e^{3} - 2 \, d x^{6} e^{2} - \frac{12}{5} \, d^{2} x^{5} e - d^{3} x^{4} + \frac{7}{2} \, x^{6} e^{3} + \frac{63}{5} \, d x^{5} e^{2} + \frac{63}{4} \, d^{2} x^{4} e + 7 \, d^{3} x^{3} + \frac{7}{5} \, x^{5} e^{3} + \frac{21}{4} \, d x^{4} e^{2} + 7 \, d^{2} x^{3} e + \frac{7}{2} \, d^{3} x^{2} + \frac{3}{2} \, x^{4} e^{3} + 6 \, d x^{3} e^{2} + 9 \, d^{2} x^{2} e + 6 \, d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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