Optimal. Leaf size=40 \[ \frac{1}{7} \left (x^2+1\right )^{7/2}-\frac{2}{5} \left (x^2+1\right )^{5/2}+\frac{1}{3} \left (x^2+1\right )^{3/2} \]
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Rubi [A] time = 0.0135769, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{1}{7} \left (x^2+1\right )^{7/2}-\frac{2}{5} \left (x^2+1\right )^{5/2}+\frac{1}{3} \left (x^2+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \sqrt{1+x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sqrt{1+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\sqrt{1+x}-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{3} \left (1+x^2\right )^{3/2}-\frac{2}{5} \left (1+x^2\right )^{5/2}+\frac{1}{7} \left (1+x^2\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0074132, size = 25, normalized size = 0.62 \[ \frac{1}{105} \left (x^2+1\right )^{3/2} \left (15 x^4-12 x^2+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 22, normalized size = 0.6 \begin{align*}{\frac{15\,{x}^{4}-12\,{x}^{2}+8}{105} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40403, size = 46, normalized size = 1.15 \begin{align*} \frac{1}{7} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} x^{4} - \frac{4}{35} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} x^{2} + \frac{8}{105} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82029, size = 68, normalized size = 1.7 \begin{align*} \frac{1}{105} \,{\left (15 \, x^{6} + 3 \, x^{4} - 4 \, x^{2} + 8\right )} \sqrt{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.02928, size = 53, normalized size = 1.32 \begin{align*} \frac{x^{6} \sqrt{x^{2} + 1}}{7} + \frac{x^{4} \sqrt{x^{2} + 1}}{35} - \frac{4 x^{2} \sqrt{x^{2} + 1}}{105} + \frac{8 \sqrt{x^{2} + 1}}{105} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05398, size = 38, normalized size = 0.95 \begin{align*} \frac{1}{7} \,{\left (x^{2} + 1\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{2} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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