Optimal. Leaf size=36 \[ \frac{2}{3} b^2 x^{3/2}+\frac{4}{7} b c x^{7/2}+\frac{2}{11} c^2 x^{11/2} \]
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Rubi [A] time = 0.0162743, 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} \[ \frac{2}{3} b^2 x^{3/2}+\frac{4}{7} b c x^{7/2}+\frac{2}{11} c^2 x^{11/2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx &=\int \sqrt{x} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 \sqrt{x}+2 b c x^{5/2}+c^2 x^{9/2}\right ) \, dx\\ &=\frac{2}{3} b^2 x^{3/2}+\frac{4}{7} b c x^{7/2}+\frac{2}{11} c^2 x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0079873, size = 30, normalized size = 0.83 \[ \frac{2}{231} x^{3/2} \left (77 b^2+66 b c x^2+21 c^2 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 27, normalized size = 0.8 \begin{align*}{\frac{42\,{c}^{2}{x}^{4}+132\,bc{x}^{2}+154\,{b}^{2}}{231}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990744, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{11} \, c^{2} x^{\frac{11}{2}} + \frac{4}{7} \, b c x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24509, size = 70, normalized size = 1.94 \begin{align*} \frac{2}{231} \,{\left (21 \, c^{2} x^{5} + 66 \, b c x^{3} + 77 \, b^{2} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.0807, size = 34, normalized size = 0.94 \begin{align*} \frac{2 b^{2} x^{\frac{3}{2}}}{3} + \frac{4 b c x^{\frac{7}{2}}}{7} + \frac{2 c^{2} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14193, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{11} \, c^{2} x^{\frac{11}{2}} + \frac{4}{7} \, b c x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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