Optimal. Leaf size=27 \[ \frac{1}{5} \left (\sqrt{x}+1\right )^{10}-\frac{2}{9} \left (\sqrt{x}+1\right )^9 \]
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Rubi [A] time = 0.0062772, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {190, 43} \[ \frac{1}{5} \left (\sqrt{x}+1\right )^{10}-\frac{2}{9} \left (\sqrt{x}+1\right )^9 \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (1+\sqrt{x}\right )^8 \, dx &=2 \operatorname{Subst}\left (\int x (1+x)^8 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-(1+x)^8+(1+x)^9\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2}{9} \left (1+\sqrt{x}\right )^9+\frac{1}{5} \left (1+\sqrt{x}\right )^{10}\\ \end{align*}
Mathematica [A] time = 0.0172791, size = 22, normalized size = 0.81 \[ \frac{1}{45} \left (\sqrt{x}+1\right )^9 \left (9 \sqrt{x}-1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 43, normalized size = 1.6 \begin{align*}{\frac{{x}^{5}}{5}}+{\frac{16}{9}{x}^{{\frac{9}{2}}}}+7\,{x}^{4}+16\,{x}^{7/2}+{\frac{70\,{x}^{3}}{3}}+{\frac{112}{5}{x}^{{\frac{5}{2}}}}+14\,{x}^{2}+{\frac{16}{3}{x}^{{\frac{3}{2}}}}+x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.929772, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{5} \,{\left (\sqrt{x} + 1\right )}^{10} - \frac{2}{9} \,{\left (\sqrt{x} + 1\right )}^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81559, size = 122, normalized size = 4.52 \begin{align*} \frac{1}{5} \, x^{5} + 7 \, x^{4} + \frac{70}{3} \, x^{3} + 14 \, x^{2} + \frac{16}{45} \,{\left (5 \, x^{4} + 45 \, x^{3} + 63 \, x^{2} + 15 \, x\right )} \sqrt{x} + x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.808166, size = 54, normalized size = 2. \begin{align*} \frac{16 x^{\frac{9}{2}}}{9} + 16 x^{\frac{7}{2}} + \frac{112 x^{\frac{5}{2}}}{5} + \frac{16 x^{\frac{3}{2}}}{3} + \frac{x^{5}}{5} + 7 x^{4} + \frac{70 x^{3}}{3} + 14 x^{2} + x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09836, size = 57, normalized size = 2.11 \begin{align*} \frac{1}{5} \, x^{5} + \frac{16}{9} \, x^{\frac{9}{2}} + 7 \, x^{4} + 16 \, x^{\frac{7}{2}} + \frac{70}{3} \, x^{3} + \frac{112}{5} \, x^{\frac{5}{2}} + 14 \, x^{2} + \frac{16}{3} \, x^{\frac{3}{2}} + x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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