Optimal. Leaf size=39 \[ \frac {2}{9} x^{9/2} (A c+b B)+\frac {2}{5} A b x^{5/2}+\frac {2}{13} B c x^{13/2} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1584, 448} \begin {gather*} \frac {2}{9} x^{9/2} (A c+b B)+\frac {2}{5} A b x^{5/2}+\frac {2}{13} B c x^{13/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 448
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{\sqrt {x}} \, dx &=\int x^{3/2} \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b x^{3/2}+(b B+A c) x^{7/2}+B c x^{11/2}\right ) \, dx\\ &=\frac {2}{5} A b x^{5/2}+\frac {2}{9} (b B+A c) x^{9/2}+\frac {2}{13} B c x^{13/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.85 \begin {gather*} \frac {2}{585} x^{5/2} \left (65 x^2 (A c+b B)+117 A b+45 B c x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 41, normalized size = 1.05 \begin {gather*} \frac {2}{585} \left (117 A b x^{5/2}+65 A c x^{9/2}+65 b B x^{9/2}+45 B c x^{13/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 32, normalized size = 0.82 \begin {gather*} \frac {2}{585} \, {\left (45 \, B c x^{6} + 65 \, {\left (B b + A c\right )} x^{4} + 117 \, A b x^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{13} \, B c x^{\frac {13}{2}} + \frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{9} \, A c x^{\frac {9}{2}} + \frac {2}{5} \, A b x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 0.82 \begin {gather*} \frac {2 \left (45 B c \,x^{4}+65 A c \,x^{2}+65 B b \,x^{2}+117 A b \right ) x^{\frac {5}{2}}}{585} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{13} \, B c x^{\frac {13}{2}} + \frac {2}{9} \, {\left (B b + A c\right )} x^{\frac {9}{2}} + \frac {2}{5} \, A b x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 31, normalized size = 0.79 \begin {gather*} \frac {2\,x^{5/2}\,\left (117\,A\,b+65\,A\,c\,x^2+65\,B\,b\,x^2+45\,B\,c\,x^4\right )}{585} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.06, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A b x^{\frac {5}{2}}}{5} + \frac {2 A c x^{\frac {9}{2}}}{9} + \frac {2 B b x^{\frac {9}{2}}}{9} + \frac {2 B c x^{\frac {13}{2}}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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