Optimal. Leaf size=36 \[ \frac {2}{15} b^2 x^{15/2}+\frac {4}{19} b c x^{19/2}+\frac {2}{23} c^2 x^{23/2} \]
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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} \begin {gather*} \frac {2}{15} b^2 x^{15/2}+\frac {4}{19} b c x^{19/2}+\frac {2}{23} c^2 x^{23/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1584
Rubi steps
\begin {align*} \int x^{5/2} \left (b x^2+c x^4\right )^2 \, dx &=\int x^{13/2} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 x^{13/2}+2 b c x^{17/2}+c^2 x^{21/2}\right ) \, dx\\ &=\frac {2}{15} b^2 x^{15/2}+\frac {4}{19} b c x^{19/2}+\frac {2}{23} c^2 x^{23/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.83 \begin {gather*} \frac {2 x^{15/2} \left (437 b^2+690 b c x^2+285 c^2 x^4\right )}{6555} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 \left (437 b^2 x^{15/2}+690 b c x^{19/2}+285 c^2 x^{23/2}\right )}{6555} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 29, normalized size = 0.81 \begin {gather*} \frac {2}{6555} \, {\left (285 \, c^{2} x^{11} + 690 \, b c x^{9} + 437 \, b^{2} x^{7}\right )} \sqrt {x} \end {gather*}
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 \begin {gather*} \frac {2}{23} \, c^{2} x^{\frac {23}{2}} + \frac {4}{19} \, b c x^{\frac {19}{2}} + \frac {2}{15} \, b^{2} x^{\frac {15}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.75 \begin {gather*} \frac {2 \left (285 c^{2} x^{4}+690 b c \,x^{2}+437 b^{2}\right ) x^{\frac {15}{2}}}{6555} \end {gather*}
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 \begin {gather*} \frac {2}{23} \, c^{2} x^{\frac {23}{2}} + \frac {4}{19} \, b c x^{\frac {19}{2}} + \frac {2}{15} \, b^{2} x^{\frac {15}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 25, normalized size = 0.69 \begin {gather*} x^{15/2}\,\left (\frac {2\,b^2}{15}+\frac {4\,b\,c\,x^2}{19}+\frac {2\,c^2\,x^4}{23}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.81, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 b^{2} x^{\frac {15}{2}}}{15} + \frac {4 b c x^{\frac {19}{2}}}{19} + \frac {2 c^{2} x^{\frac {23}{2}}}{23} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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