Optimal. Leaf size=31 \[ \frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2} \]
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Rubi [A] time = 0.0061682, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {14} \[ \frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2} \]
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin{align*} \int x^{5/2} \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a x^{5/2}+b x^{9/2}+c x^{13/2}\right ) \, dx\\ &=\frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.006172, size = 25, normalized size = 0.81 \[ \frac{2 x^{7/2} \left (165 a+105 b x^2+77 c x^4\right )}{1155} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 22, normalized size = 0.7 \begin{align*}{\frac{154\,c{x}^{4}+210\,b{x}^{2}+330\,a}{1155}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990398, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{15} \, c x^{\frac{15}{2}} + \frac{2}{11} \, b x^{\frac{11}{2}} + \frac{2}{7} \, a x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22206, size = 69, normalized size = 2.23 \begin{align*} \frac{2}{1155} \,{\left (77 \, c x^{7} + 105 \, b x^{5} + 165 \, a x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.76865, size = 29, normalized size = 0.94 \begin{align*} \frac{2 a x^{\frac{7}{2}}}{7} + \frac{2 b x^{\frac{11}{2}}}{11} + \frac{2 c x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1642, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{15} \, c x^{\frac{15}{2}} + \frac{2}{11} \, b x^{\frac{11}{2}} + \frac{2}{7} \, a x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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