Optimal. Leaf size=23 \[ \frac{1}{26} \left (x^2+1\right )^{13}-\frac{1}{24} \left (x^2+1\right )^{12} \]
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Rubi [A] time = 0.0219489, antiderivative size = 23, 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} \[ \frac{1}{26} \left (x^2+1\right )^{13}-\frac{1}{24} \left (x^2+1\right )^{12} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^3 \left (1+x^2\right )^{11} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x (1+x)^{11} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-(1+x)^{11}+(1+x)^{12}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{24} \left (1+x^2\right )^{12}+\frac{1}{26} \left (1+x^2\right )^{13}\\ \end{align*}
Mathematica [B] time = 0.0016193, size = 83, normalized size = 3.61 \[ \frac{x^{26}}{26}+\frac{11 x^{24}}{24}+\frac{5 x^{22}}{2}+\frac{33 x^{20}}{4}+\frac{55 x^{18}}{3}+\frac{231 x^{16}}{8}+33 x^{14}+\frac{55 x^{12}}{2}+\frac{33 x^{10}}{2}+\frac{55 x^8}{8}+\frac{11 x^6}{6}+\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 62, normalized size = 2.7 \begin{align*}{\frac{{x}^{26}}{26}}+{\frac{11\,{x}^{24}}{24}}+{\frac{5\,{x}^{22}}{2}}+{\frac{33\,{x}^{20}}{4}}+{\frac{55\,{x}^{18}}{3}}+{\frac{231\,{x}^{16}}{8}}+33\,{x}^{14}+{\frac{55\,{x}^{12}}{2}}+{\frac{33\,{x}^{10}}{2}}+{\frac{55\,{x}^{8}}{8}}+{\frac{11\,{x}^{6}}{6}}+{\frac{{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.940523, size = 82, normalized size = 3.57 \begin{align*} \frac{1}{26} \, x^{26} + \frac{11}{24} \, x^{24} + \frac{5}{2} \, x^{22} + \frac{33}{4} \, x^{20} + \frac{55}{3} \, x^{18} + \frac{231}{8} \, x^{16} + 33 \, x^{14} + \frac{55}{2} \, x^{12} + \frac{33}{2} \, x^{10} + \frac{55}{8} \, x^{8} + \frac{11}{6} \, x^{6} + \frac{1}{4} \, x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.25516, size = 186, normalized size = 8.09 \begin{align*} \frac{1}{26} x^{26} + \frac{11}{24} x^{24} + \frac{5}{2} x^{22} + \frac{33}{4} x^{20} + \frac{55}{3} x^{18} + \frac{231}{8} x^{16} + 33 x^{14} + \frac{55}{2} x^{12} + \frac{33}{2} x^{10} + \frac{55}{8} x^{8} + \frac{11}{6} x^{6} + \frac{1}{4} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.07234, size = 75, normalized size = 3.26 \begin{align*} \frac{x^{26}}{26} + \frac{11 x^{24}}{24} + \frac{5 x^{22}}{2} + \frac{33 x^{20}}{4} + \frac{55 x^{18}}{3} + \frac{231 x^{16}}{8} + 33 x^{14} + \frac{55 x^{12}}{2} + \frac{33 x^{10}}{2} + \frac{55 x^{8}}{8} + \frac{11 x^{6}}{6} + \frac{x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11224, size = 82, normalized size = 3.57 \begin{align*} \frac{1}{26} \, x^{26} + \frac{11}{24} \, x^{24} + \frac{5}{2} \, x^{22} + \frac{33}{4} \, x^{20} + \frac{55}{3} \, x^{18} + \frac{231}{8} \, x^{16} + 33 \, x^{14} + \frac{55}{2} \, x^{12} + \frac{33}{2} \, x^{10} + \frac{55}{8} \, x^{8} + \frac{11}{6} \, x^{6} + \frac{1}{4} \, x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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