Optimal. Leaf size=36 \[ \frac{2}{11} b^2 x^{11/2}+\frac{4}{15} b c x^{15/2}+\frac{2}{19} c^2 x^{19/2} \]
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Rubi [A] time = 0.0139267, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1584, 270} \[ \frac{2}{11} b^2 x^{11/2}+\frac{4}{15} b c x^{15/2}+\frac{2}{19} c^2 x^{19/2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 270
Rubi steps
\begin{align*} \int \sqrt{x} \left (b x^2+c x^4\right )^2 \, dx &=\int x^{9/2} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 x^{9/2}+2 b c x^{13/2}+c^2 x^{17/2}\right ) \, dx\\ &=\frac{2}{11} b^2 x^{11/2}+\frac{4}{15} b c x^{15/2}+\frac{2}{19} c^2 x^{19/2}\\ \end{align*}
Mathematica [A] time = 0.0079015, size = 30, normalized size = 0.83 \[ \frac{2 x^{11/2} \left (285 b^2+418 b c x^2+165 c^2 x^4\right )}{3135} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 27, normalized size = 0.8 \begin{align*}{\frac{330\,{c}^{2}{x}^{4}+836\,bc{x}^{2}+570\,{b}^{2}}{3135}{x}^{{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989824, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{19} \, c^{2} x^{\frac{19}{2}} + \frac{4}{15} \, b c x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24831, size = 78, normalized size = 2.17 \begin{align*} \frac{2}{3135} \,{\left (165 \, c^{2} x^{9} + 418 \, b c x^{7} + 285 \, b^{2} x^{5}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.47159, size = 34, normalized size = 0.94 \begin{align*} \frac{2 b^{2} x^{\frac{11}{2}}}{11} + \frac{4 b c x^{\frac{15}{2}}}{15} + \frac{2 c^{2} x^{\frac{19}{2}}}{19} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11144, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{19} \, c^{2} x^{\frac{19}{2}} + \frac{4}{15} \, b c x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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