Optimal. Leaf size=63 \[ \frac{1}{26} \left (x^2+1\right )^{13} (d-3 e)-\frac{1}{24} \left (x^2+1\right )^{12} (2 d-3 e)+\frac{1}{22} \left (x^2+1\right )^{11} (d-e)+\frac{1}{28} e \left (x^2+1\right )^{14} \]
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Rubi [A] time = 0.196701, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {28, 446, 76} \[ \frac{1}{26} \left (x^2+1\right )^{13} (d-3 e)-\frac{1}{24} \left (x^2+1\right )^{12} (2 d-3 e)+\frac{1}{22} \left (x^2+1\right )^{11} (d-e)+\frac{1}{28} e \left (x^2+1\right )^{14} \]
Antiderivative was successfully verified.
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Rule 28
Rule 446
Rule 76
Rubi steps
\begin{align*} \int x^5 \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^5 \left (1+x^2\right )^{10} \left (d+e x^2\right ) \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (1+x)^{10} (d+e x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left ((d-e) (1+x)^{10}+(-2 d+3 e) (1+x)^{11}+(d-3 e) (1+x)^{12}+e (1+x)^{13}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{22} (d-e) \left (1+x^2\right )^{11}-\frac{1}{24} (2 d-3 e) \left (1+x^2\right )^{12}+\frac{1}{26} (d-3 e) \left (1+x^2\right )^{13}+\frac{1}{28} e \left (1+x^2\right )^{14}\\ \end{align*}
Mathematica [B] time = 0.0243831, size = 153, normalized size = 2.43 \[ \frac{1}{26} x^{26} (d+10 e)+\frac{5}{24} x^{24} (2 d+9 e)+\frac{15}{22} x^{22} (3 d+8 e)+\frac{3}{2} x^{20} (4 d+7 e)+\frac{7}{3} x^{18} (5 d+6 e)+\frac{21}{8} x^{16} (6 d+5 e)+\frac{15}{7} x^{14} (7 d+4 e)+\frac{5}{4} x^{12} (8 d+3 e)+\frac{1}{2} x^{10} (9 d+2 e)+\frac{1}{8} x^8 (10 d+e)+\frac{d x^6}{6}+\frac{e x^{28}}{28} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 130, normalized size = 2.1 \begin{align*}{\frac{e{x}^{28}}{28}}+{\frac{ \left ( d+10\,e \right ){x}^{26}}{26}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{24}}{24}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{22}}{22}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{20}}{20}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{18}}{18}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{16}}{16}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,d+e \right ){x}^{8}}{8}}+{\frac{d{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.941249, size = 174, normalized size = 2.76 \begin{align*} \frac{1}{28} \, e x^{28} + \frac{1}{26} \,{\left (d + 10 \, e\right )} x^{26} + \frac{5}{24} \,{\left (2 \, d + 9 \, e\right )} x^{24} + \frac{15}{22} \,{\left (3 \, d + 8 \, e\right )} x^{22} + \frac{3}{2} \,{\left (4 \, d + 7 \, e\right )} x^{20} + \frac{7}{3} \,{\left (5 \, d + 6 \, e\right )} x^{18} + \frac{21}{8} \,{\left (6 \, d + 5 \, e\right )} x^{16} + \frac{15}{7} \,{\left (7 \, d + 4 \, e\right )} x^{14} + \frac{5}{4} \,{\left (8 \, d + 3 \, e\right )} x^{12} + \frac{1}{2} \,{\left (9 \, d + 2 \, e\right )} x^{10} + \frac{1}{8} \,{\left (10 \, d + e\right )} x^{8} + \frac{1}{6} \, d x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.27951, size = 390, normalized size = 6.19 \begin{align*} \frac{1}{28} x^{28} e + \frac{5}{13} x^{26} e + \frac{1}{26} x^{26} d + \frac{15}{8} x^{24} e + \frac{5}{12} x^{24} d + \frac{60}{11} x^{22} e + \frac{45}{22} x^{22} d + \frac{21}{2} x^{20} e + 6 x^{20} d + 14 x^{18} e + \frac{35}{3} x^{18} d + \frac{105}{8} x^{16} e + \frac{63}{4} x^{16} d + \frac{60}{7} x^{14} e + 15 x^{14} d + \frac{15}{4} x^{12} e + 10 x^{12} d + x^{10} e + \frac{9}{2} x^{10} d + \frac{1}{8} x^{8} e + \frac{5}{4} x^{8} d + \frac{1}{6} x^{6} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.09754, size = 134, normalized size = 2.13 \begin{align*} \frac{d x^{6}}{6} + \frac{e x^{28}}{28} + x^{26} \left (\frac{d}{26} + \frac{5 e}{13}\right ) + x^{24} \left (\frac{5 d}{12} + \frac{15 e}{8}\right ) + x^{22} \left (\frac{45 d}{22} + \frac{60 e}{11}\right ) + x^{20} \left (6 d + \frac{21 e}{2}\right ) + x^{18} \left (\frac{35 d}{3} + 14 e\right ) + x^{16} \left (\frac{63 d}{4} + \frac{105 e}{8}\right ) + x^{14} \left (15 d + \frac{60 e}{7}\right ) + x^{12} \left (10 d + \frac{15 e}{4}\right ) + x^{10} \left (\frac{9 d}{2} + e\right ) + x^{8} \left (\frac{5 d}{4} + \frac{e}{8}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11646, size = 193, normalized size = 3.06 \begin{align*} \frac{1}{28} \, x^{28} e + \frac{1}{26} \, d x^{26} + \frac{5}{13} \, x^{26} e + \frac{5}{12} \, d x^{24} + \frac{15}{8} \, x^{24} e + \frac{45}{22} \, d x^{22} + \frac{60}{11} \, x^{22} e + 6 \, d x^{20} + \frac{21}{2} \, x^{20} e + \frac{35}{3} \, d x^{18} + 14 \, x^{18} e + \frac{63}{4} \, d x^{16} + \frac{105}{8} \, x^{16} e + 15 \, d x^{14} + \frac{60}{7} \, x^{14} e + 10 \, d x^{12} + \frac{15}{4} \, x^{12} e + \frac{9}{2} \, d x^{10} + x^{10} e + \frac{5}{4} \, d x^{8} + \frac{1}{8} \, x^{8} e + \frac{1}{6} \, d x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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