Optimal. Leaf size=131 \[ \frac{1}{18} (x+1)^{18} (d-8 e)-\frac{7}{17} (x+1)^{17} (d-4 e)+\frac{7}{16} (x+1)^{16} (3 d-8 e)-\frac{7}{3} (x+1)^{15} (d-2 e)+\frac{1}{2} (x+1)^{14} (5 d-8 e)-\frac{7}{13} (x+1)^{13} (3 d-4 e)+\frac{1}{12} (x+1)^{12} (7 d-8 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{19} e (x+1)^{19} \]
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Rubi [A] time = 0.127674, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {27, 76} \[ \frac{1}{18} (x+1)^{18} (d-8 e)-\frac{7}{17} (x+1)^{17} (d-4 e)+\frac{7}{16} (x+1)^{16} (3 d-8 e)-\frac{7}{3} (x+1)^{15} (d-2 e)+\frac{1}{2} (x+1)^{14} (5 d-8 e)-\frac{7}{13} (x+1)^{13} (3 d-4 e)+\frac{1}{12} (x+1)^{12} (7 d-8 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{19} e (x+1)^{19} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^7 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((-d+e) (1+x)^{10}+(7 d-8 e) (1+x)^{11}-7 (3 d-4 e) (1+x)^{12}+7 (5 d-8 e) (1+x)^{13}-35 (d-2 e) (1+x)^{14}+7 (3 d-8 e) (1+x)^{15}-7 (d-4 e) (1+x)^{16}+(d-8 e) (1+x)^{17}+e (1+x)^{18}\right ) \, dx\\ &=-\frac{1}{11} (d-e) (1+x)^{11}+\frac{1}{12} (7 d-8 e) (1+x)^{12}-\frac{7}{13} (3 d-4 e) (1+x)^{13}+\frac{1}{2} (5 d-8 e) (1+x)^{14}-\frac{7}{3} (d-2 e) (1+x)^{15}+\frac{7}{16} (3 d-8 e) (1+x)^{16}-\frac{7}{17} (d-4 e) (1+x)^{17}+\frac{1}{18} (d-8 e) (1+x)^{18}+\frac{1}{19} e (1+x)^{19}\\ \end{align*}
Mathematica [A] time = 0.0253752, size = 149, normalized size = 1.14 \[ \frac{1}{18} x^{18} (d+10 e)+\frac{5}{17} x^{17} (2 d+9 e)+\frac{15}{16} x^{16} (3 d+8 e)+2 x^{15} (4 d+7 e)+3 x^{14} (5 d+6 e)+\frac{42}{13} x^{13} (6 d+5 e)+\frac{5}{2} x^{12} (7 d+4 e)+\frac{15}{11} x^{11} (8 d+3 e)+\frac{1}{2} x^{10} (9 d+2 e)+\frac{1}{9} x^9 (10 d+e)+\frac{d x^8}{8}+\frac{e x^{19}}{19} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 130, normalized size = 1. \begin{align*}{\frac{e{x}^{19}}{19}}+{\frac{ \left ( d+10\,e \right ){x}^{18}}{18}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{17}}{17}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{16}}{16}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{15}}{15}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{13}}{13}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,d+e \right ){x}^{9}}{9}}+{\frac{d{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992084, size = 174, normalized size = 1.33 \begin{align*} \frac{1}{19} \, e x^{19} + \frac{1}{18} \,{\left (d + 10 \, e\right )} x^{18} + \frac{5}{17} \,{\left (2 \, d + 9 \, e\right )} x^{17} + \frac{15}{16} \,{\left (3 \, d + 8 \, e\right )} x^{16} + 2 \,{\left (4 \, d + 7 \, e\right )} x^{15} + 3 \,{\left (5 \, d + 6 \, e\right )} x^{14} + \frac{42}{13} \,{\left (6 \, d + 5 \, e\right )} x^{13} + \frac{5}{2} \,{\left (7 \, d + 4 \, e\right )} x^{12} + \frac{15}{11} \,{\left (8 \, d + 3 \, e\right )} x^{11} + \frac{1}{2} \,{\left (9 \, d + 2 \, e\right )} x^{10} + \frac{1}{9} \,{\left (10 \, d + e\right )} x^{9} + \frac{1}{8} \, d x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08753, size = 397, normalized size = 3.03 \begin{align*} \frac{1}{19} x^{19} e + \frac{5}{9} x^{18} e + \frac{1}{18} x^{18} d + \frac{45}{17} x^{17} e + \frac{10}{17} x^{17} d + \frac{15}{2} x^{16} e + \frac{45}{16} x^{16} d + 14 x^{15} e + 8 x^{15} d + 18 x^{14} e + 15 x^{14} d + \frac{210}{13} x^{13} e + \frac{252}{13} x^{13} d + 10 x^{12} e + \frac{35}{2} x^{12} d + \frac{45}{11} x^{11} e + \frac{120}{11} x^{11} d + x^{10} e + \frac{9}{2} x^{10} d + \frac{1}{9} x^{9} e + \frac{10}{9} x^{9} d + \frac{1}{8} x^{8} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.208979, size = 133, normalized size = 1.02 \begin{align*} \frac{d x^{8}}{8} + \frac{e x^{19}}{19} + x^{18} \left (\frac{d}{18} + \frac{5 e}{9}\right ) + x^{17} \left (\frac{10 d}{17} + \frac{45 e}{17}\right ) + x^{16} \left (\frac{45 d}{16} + \frac{15 e}{2}\right ) + x^{15} \left (8 d + 14 e\right ) + x^{14} \left (15 d + 18 e\right ) + x^{13} \left (\frac{252 d}{13} + \frac{210 e}{13}\right ) + x^{12} \left (\frac{35 d}{2} + 10 e\right ) + x^{11} \left (\frac{120 d}{11} + \frac{45 e}{11}\right ) + x^{10} \left (\frac{9 d}{2} + e\right ) + x^{9} \left (\frac{10 d}{9} + \frac{e}{9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14572, size = 193, normalized size = 1.47 \begin{align*} \frac{1}{19} \, x^{19} e + \frac{1}{18} \, d x^{18} + \frac{5}{9} \, x^{18} e + \frac{10}{17} \, d x^{17} + \frac{45}{17} \, x^{17} e + \frac{45}{16} \, d x^{16} + \frac{15}{2} \, x^{16} e + 8 \, d x^{15} + 14 \, x^{15} e + 15 \, d x^{14} + 18 \, x^{14} e + \frac{252}{13} \, d x^{13} + \frac{210}{13} \, x^{13} e + \frac{35}{2} \, d x^{12} + 10 \, x^{12} e + \frac{120}{11} \, d x^{11} + \frac{45}{11} \, x^{11} e + \frac{9}{2} \, d x^{10} + x^{10} e + \frac{10}{9} \, d x^{9} + \frac{1}{9} \, x^{9} e + \frac{1}{8} \, d x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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