Optimal. Leaf size=34 \[ \frac{1}{28} \left (x^2+1\right )^{14}-\frac{1}{13} \left (x^2+1\right )^{13}+\frac{1}{24} \left (x^2+1\right )^{12} \]
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Rubi [A] time = 0.0467126, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {28, 266, 43} \[ \frac{1}{28} \left (x^2+1\right )^{14}-\frac{1}{13} \left (x^2+1\right )^{13}+\frac{1}{24} \left (x^2+1\right )^{12} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^5 \left (1+x^2\right )^{11} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (1+x)^{11} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left ((1+x)^{11}-2 (1+x)^{12}+(1+x)^{13}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{24} \left (1+x^2\right )^{12}-\frac{1}{13} \left (1+x^2\right )^{13}+\frac{1}{28} \left (1+x^2\right )^{14}\\ \end{align*}
Mathematica [B] time = 0.0019485, size = 85, normalized size = 2.5 \[ \frac{x^{28}}{28}+\frac{11 x^{26}}{26}+\frac{55 x^{24}}{24}+\frac{15 x^{22}}{2}+\frac{33 x^{20}}{2}+\frac{77 x^{18}}{3}+\frac{231 x^{16}}{8}+\frac{165 x^{14}}{7}+\frac{55 x^{12}}{4}+\frac{11 x^{10}}{2}+\frac{11 x^8}{8}+\frac{x^6}{6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 62, normalized size = 1.8 \begin{align*}{\frac{{x}^{28}}{28}}+{\frac{11\,{x}^{26}}{26}}+{\frac{55\,{x}^{24}}{24}}+{\frac{15\,{x}^{22}}{2}}+{\frac{33\,{x}^{20}}{2}}+{\frac{77\,{x}^{18}}{3}}+{\frac{231\,{x}^{16}}{8}}+{\frac{165\,{x}^{14}}{7}}+{\frac{55\,{x}^{12}}{4}}+{\frac{11\,{x}^{10}}{2}}+{\frac{11\,{x}^{8}}{8}}+{\frac{{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.953037, size = 82, normalized size = 2.41 \begin{align*} \frac{1}{28} \, x^{28} + \frac{11}{26} \, x^{26} + \frac{55}{24} \, x^{24} + \frac{15}{2} \, x^{22} + \frac{33}{2} \, x^{20} + \frac{77}{3} \, x^{18} + \frac{231}{8} \, x^{16} + \frac{165}{7} \, x^{14} + \frac{55}{4} \, x^{12} + \frac{11}{2} \, x^{10} + \frac{11}{8} \, x^{8} + \frac{1}{6} \, x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.25211, size = 194, normalized size = 5.71 \begin{align*} \frac{1}{28} x^{28} + \frac{11}{26} x^{26} + \frac{55}{24} x^{24} + \frac{15}{2} x^{22} + \frac{33}{2} x^{20} + \frac{77}{3} x^{18} + \frac{231}{8} x^{16} + \frac{165}{7} x^{14} + \frac{55}{4} x^{12} + \frac{11}{2} x^{10} + \frac{11}{8} x^{8} + \frac{1}{6} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.068427, size = 76, normalized size = 2.24 \begin{align*} \frac{x^{28}}{28} + \frac{11 x^{26}}{26} + \frac{55 x^{24}}{24} + \frac{15 x^{22}}{2} + \frac{33 x^{20}}{2} + \frac{77 x^{18}}{3} + \frac{231 x^{16}}{8} + \frac{165 x^{14}}{7} + \frac{55 x^{12}}{4} + \frac{11 x^{10}}{2} + \frac{11 x^{8}}{8} + \frac{x^{6}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11737, size = 82, normalized size = 2.41 \begin{align*} \frac{1}{28} \, x^{28} + \frac{11}{26} \, x^{26} + \frac{55}{24} \, x^{24} + \frac{15}{2} \, x^{22} + \frac{33}{2} \, x^{20} + \frac{77}{3} \, x^{18} + \frac{231}{8} \, x^{16} + \frac{165}{7} \, x^{14} + \frac{55}{4} \, x^{12} + \frac{11}{2} \, x^{10} + \frac{11}{8} \, x^{8} + \frac{1}{6} \, x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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