Optimal. Leaf size=36 \[ \frac{2}{13} b^2 x^{13/2}+\frac{4}{17} b c x^{17/2}+\frac{2}{21} c^2 x^{21/2} \]
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Rubi [A] time = 0.0149835, 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} \[ \frac{2}{13} b^2 x^{13/2}+\frac{4}{17} b c x^{17/2}+\frac{2}{21} c^2 x^{21/2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 270
Rubi steps
\begin{align*} \int x^{3/2} \left (b x^2+c x^4\right )^2 \, dx &=\int x^{11/2} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 x^{11/2}+2 b c x^{15/2}+c^2 x^{19/2}\right ) \, dx\\ &=\frac{2}{13} b^2 x^{13/2}+\frac{4}{17} b c x^{17/2}+\frac{2}{21} c^2 x^{21/2}\\ \end{align*}
Mathematica [A] time = 0.0082043, size = 30, normalized size = 0.83 \[ \frac{2 x^{13/2} \left (357 b^2+546 b c x^2+221 c^2 x^4\right )}{4641} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 27, normalized size = 0.8 \begin{align*}{\frac{442\,{c}^{2}{x}^{4}+1092\,bc{x}^{2}+714\,{b}^{2}}{4641}{x}^{{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980056, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{21} \, c^{2} x^{\frac{21}{2}} + \frac{4}{17} \, b c x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} x^{\frac{13}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24746, size = 80, normalized size = 2.22 \begin{align*} \frac{2}{4641} \,{\left (221 \, c^{2} x^{10} + 546 \, b c x^{8} + 357 \, b^{2} x^{6}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.1359, size = 34, normalized size = 0.94 \begin{align*} \frac{2 b^{2} x^{\frac{13}{2}}}{13} + \frac{4 b c x^{\frac{17}{2}}}{17} + \frac{2 c^{2} x^{\frac{21}{2}}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11707, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{21} \, c^{2} x^{\frac{21}{2}} + \frac{4}{17} \, b c x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} x^{\frac{13}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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