Optimal. Leaf size=36 \[ \frac {2}{5} b^2 x^{5/2}+\frac {4}{9} b c x^{9/2}+\frac {2}{13} c^2 x^{13/2} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, 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} \[ \frac {2}{5} b^2 x^{5/2}+\frac {4}{9} b c x^{9/2}+\frac {2}{13} c^2 x^{13/2} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^2}{x^{5/2}} \, dx &=\int x^{3/2} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 x^{3/2}+2 b c x^{7/2}+c^2 x^{11/2}\right ) \, dx\\ &=\frac {2}{5} b^2 x^{5/2}+\frac {4}{9} b c x^{9/2}+\frac {2}{13} c^2 x^{13/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.83 \[ \frac {2}{585} x^{5/2} \left (117 b^2+130 b c x^2+45 c^2 x^4\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 29, normalized size = 0.81 \[ \frac {2}{585} \, {\left (45 \, c^{2} x^{6} + 130 \, b c x^{4} + 117 \, b^{2} x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 24, normalized size = 0.67 \[ \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b c x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.75 \[ \frac {2 \left (45 c^{2} x^{4}+130 b c \,x^{2}+117 b^{2}\right ) x^{\frac {5}{2}}}{585} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 24, normalized size = 0.67 \[ \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b c x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 26, normalized size = 0.72 \[ \frac {2\,x^{5/2}\,\left (117\,b^2+130\,b\,c\,x^2+45\,c^2\,x^4\right )}{585} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.99, size = 34, normalized size = 0.94 \[ \frac {2 b^{2} x^{\frac {5}{2}}}{5} + \frac {4 b c x^{\frac {9}{2}}}{9} + \frac {2 c^{2} x^{\frac {13}{2}}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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