Optimal. Leaf size=87 \[ \frac{1}{15} (x+1)^{15} (d-5 e)-\frac{1}{7} (x+1)^{14} (2 d-5 e)+\frac{2}{13} (x+1)^{13} (3 d-5 e)-\frac{1}{12} (x+1)^{12} (4 d-5 e)+\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{16} e (x+1)^{16} \]
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Rubi [A] time = 0.0567202, antiderivative size = 87, 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} \[ \frac{1}{15} (x+1)^{15} (d-5 e)-\frac{1}{7} (x+1)^{14} (2 d-5 e)+\frac{2}{13} (x+1)^{13} (3 d-5 e)-\frac{1}{12} (x+1)^{12} (4 d-5 e)+\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{16} e (x+1)^{16} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^4 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((d-e) (1+x)^{10}+(-4 d+5 e) (1+x)^{11}+2 (3 d-5 e) (1+x)^{12}-2 (2 d-5 e) (1+x)^{13}+(d-5 e) (1+x)^{14}+e (1+x)^{15}\right ) \, dx\\ &=\frac{1}{11} (d-e) (1+x)^{11}-\frac{1}{12} (4 d-5 e) (1+x)^{12}+\frac{2}{13} (3 d-5 e) (1+x)^{13}-\frac{1}{7} (2 d-5 e) (1+x)^{14}+\frac{1}{15} (d-5 e) (1+x)^{15}+\frac{1}{16} e (1+x)^{16}\\ \end{align*}
Mathematica [A] time = 0.0182847, size = 153, normalized size = 1.76 \[ \frac{1}{15} x^{15} (d+10 e)+\frac{5}{14} x^{14} (2 d+9 e)+\frac{15}{13} x^{13} (3 d+8 e)+\frac{5}{2} x^{12} (4 d+7 e)+\frac{42}{11} x^{11} (5 d+6 e)+\frac{21}{5} x^{10} (6 d+5 e)+\frac{10}{3} x^9 (7 d+4 e)+\frac{15}{8} x^8 (8 d+3 e)+\frac{5}{7} x^7 (9 d+2 e)+\frac{1}{6} x^6 (10 d+e)+\frac{d x^5}{5}+\frac{e x^{16}}{16} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 130, normalized size = 1.5 \begin{align*}{\frac{e{x}^{16}}{16}}+{\frac{ \left ( d+10\,e \right ){x}^{15}}{15}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{13}}{13}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{9}}{9}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{8}}{8}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{7}}{7}}+{\frac{ \left ( 10\,d+e \right ){x}^{6}}{6}}+{\frac{d{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966038, size = 174, normalized size = 2. \begin{align*} \frac{1}{16} \, e x^{16} + \frac{1}{15} \,{\left (d + 10 \, e\right )} x^{15} + \frac{5}{14} \,{\left (2 \, d + 9 \, e\right )} x^{14} + \frac{15}{13} \,{\left (3 \, d + 8 \, e\right )} x^{13} + \frac{5}{2} \,{\left (4 \, d + 7 \, e\right )} x^{12} + \frac{42}{11} \,{\left (5 \, d + 6 \, e\right )} x^{11} + \frac{21}{5} \,{\left (6 \, d + 5 \, e\right )} x^{10} + \frac{10}{3} \,{\left (7 \, d + 4 \, e\right )} x^{9} + \frac{15}{8} \,{\left (8 \, d + 3 \, e\right )} x^{8} + \frac{5}{7} \,{\left (9 \, d + 2 \, e\right )} x^{7} + \frac{1}{6} \,{\left (10 \, d + e\right )} x^{6} + \frac{1}{5} \, d x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10661, size = 400, normalized size = 4.6 \begin{align*} \frac{1}{16} x^{16} e + \frac{2}{3} x^{15} e + \frac{1}{15} x^{15} d + \frac{45}{14} x^{14} e + \frac{5}{7} x^{14} d + \frac{120}{13} x^{13} e + \frac{45}{13} x^{13} d + \frac{35}{2} x^{12} e + 10 x^{12} d + \frac{252}{11} x^{11} e + \frac{210}{11} x^{11} d + 21 x^{10} e + \frac{126}{5} x^{10} d + \frac{40}{3} x^{9} e + \frac{70}{3} x^{9} d + \frac{45}{8} x^{8} e + 15 x^{8} d + \frac{10}{7} x^{7} e + \frac{45}{7} x^{7} d + \frac{1}{6} x^{6} e + \frac{5}{3} x^{6} d + \frac{1}{5} x^{5} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.137223, size = 139, normalized size = 1.6 \begin{align*} \frac{d x^{5}}{5} + \frac{e x^{16}}{16} + x^{15} \left (\frac{d}{15} + \frac{2 e}{3}\right ) + x^{14} \left (\frac{5 d}{7} + \frac{45 e}{14}\right ) + x^{13} \left (\frac{45 d}{13} + \frac{120 e}{13}\right ) + x^{12} \left (10 d + \frac{35 e}{2}\right ) + x^{11} \left (\frac{210 d}{11} + \frac{252 e}{11}\right ) + x^{10} \left (\frac{126 d}{5} + 21 e\right ) + x^{9} \left (\frac{70 d}{3} + \frac{40 e}{3}\right ) + x^{8} \left (15 d + \frac{45 e}{8}\right ) + x^{7} \left (\frac{45 d}{7} + \frac{10 e}{7}\right ) + x^{6} \left (\frac{5 d}{3} + \frac{e}{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10664, size = 194, normalized size = 2.23 \begin{align*} \frac{1}{16} \, x^{16} e + \frac{1}{15} \, d x^{15} + \frac{2}{3} \, x^{15} e + \frac{5}{7} \, d x^{14} + \frac{45}{14} \, x^{14} e + \frac{45}{13} \, d x^{13} + \frac{120}{13} \, x^{13} e + 10 \, d x^{12} + \frac{35}{2} \, x^{12} e + \frac{210}{11} \, d x^{11} + \frac{252}{11} \, x^{11} e + \frac{126}{5} \, d x^{10} + 21 \, x^{10} e + \frac{70}{3} \, d x^{9} + \frac{40}{3} \, x^{9} e + 15 \, d x^{8} + \frac{45}{8} \, x^{8} e + \frac{45}{7} \, d x^{7} + \frac{10}{7} \, x^{7} e + \frac{5}{3} \, d x^{6} + \frac{1}{6} \, x^{6} e + \frac{1}{5} \, d x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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