Optimal. Leaf size=62 \[ 2 a^2 \sqrt{x}+\frac{2}{9} x^{9/2} \left (2 a c+b^2\right )+\frac{4}{5} a b x^{5/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2} \]
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Rubi [A] time = 0.022, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1108} \[ 2 a^2 \sqrt{x}+\frac{2}{9} x^{9/2} \left (2 a c+b^2\right )+\frac{4}{5} a b x^{5/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2} \]
Antiderivative was successfully verified.
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Rule 1108
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^2}{\sqrt{x}} \, dx &=\int \left (\frac{a^2}{\sqrt{x}}+2 a b x^{3/2}+\left (b^2+2 a c\right ) x^{7/2}+2 b c x^{11/2}+c^2 x^{15/2}\right ) \, dx\\ &=2 a^2 \sqrt{x}+\frac{4}{5} a b x^{5/2}+\frac{2}{9} \left (b^2+2 a c\right ) x^{9/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2}\\ \end{align*}
Mathematica [A] time = 0.042085, size = 63, normalized size = 1.02 \[ 2 \left (a^2 \sqrt{x}+\frac{1}{9} x^{9/2} \left (2 a c+b^2\right )+\frac{2}{5} a b x^{5/2}+\frac{2}{13} b c x^{13/2}+\frac{1}{17} c^2 x^{17/2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 49, normalized size = 0.8 \begin{align*}{\frac{1170\,{c}^{2}{x}^{8}+3060\,bc{x}^{6}+4420\,{x}^{4}ac+2210\,{b}^{2}{x}^{4}+7956\,ab{x}^{2}+19890\,{a}^{2}}{9945}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964708, size = 65, normalized size = 1.05 \begin{align*} \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b c x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} x^{\frac{9}{2}} + 2 \, a^{2} \sqrt{x} + \frac{4}{45} \,{\left (5 \, c x^{\frac{9}{2}} + 9 \, b x^{\frac{5}{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31232, size = 130, normalized size = 2.1 \begin{align*} \frac{2}{9945} \,{\left (585 \, c^{2} x^{8} + 1530 \, b c x^{6} + 1105 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 3978 \, a b x^{2} + 9945 \, a^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.93246, size = 68, normalized size = 1.1 \begin{align*} 2 a^{2} \sqrt{x} + \frac{4 a b x^{\frac{5}{2}}}{5} + \frac{4 a c x^{\frac{9}{2}}}{9} + \frac{2 b^{2} x^{\frac{9}{2}}}{9} + \frac{4 b c x^{\frac{13}{2}}}{13} + \frac{2 c^{2} x^{\frac{17}{2}}}{17} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16505, size = 62, normalized size = 1. \begin{align*} \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b c x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} x^{\frac{9}{2}} + \frac{4}{9} \, a c x^{\frac{9}{2}} + \frac{4}{5} \, a b x^{\frac{5}{2}} + 2 \, a^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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