Optimal. Leaf size=39 \[ \frac {2}{9} x^{9/2} (A c+b B)+\frac {2}{7} A b x^{7/2}+\frac {2}{11} B c x^{11/2} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} \frac {2}{9} x^{9/2} (A c+b B)+\frac {2}{7} A b x^{7/2}+\frac {2}{11} B c x^{11/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int x^{3/2} (A+B x) \left (b x+c x^2\right ) \, dx &=\int \left (A b x^{5/2}+(b B+A c) x^{7/2}+B c x^{9/2}\right ) \, dx\\ &=\frac {2}{7} A b x^{7/2}+\frac {2}{9} (b B+A c) x^{9/2}+\frac {2}{11} B c x^{11/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.85 \begin {gather*} \frac {2}{693} x^{7/2} (11 A (9 b+7 c x)+7 B x (11 b+9 c x)) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 41, normalized size = 1.05 \begin {gather*} \frac {2}{693} \left (99 A b x^{7/2}+77 A c x^{9/2}+77 b B x^{9/2}+63 B c x^{11/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 32, normalized size = 0.82 \begin {gather*} \frac {2}{693} \, {\left (63 \, B c x^{5} + 99 \, A b x^{3} + 77 \, {\left (B b + A c\right )} x^{4}\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 = 29, normalized size = 0.74 \begin {gather*} \frac {2}{11} \, B c x^{\frac {11}{2}} + \frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{9} \, A c x^{\frac {9}{2}} + \frac {2}{7} \, A b x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 0.72 \begin {gather*} \frac {2 \left (63 B c \,x^{2}+77 A c x +77 B b x +99 A b \right ) x^{\frac {7}{2}}}{693} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{11} \, B c x^{\frac {11}{2}} + \frac {2}{7} \, A b x^{\frac {7}{2}} + \frac {2}{9} \, {\left (B b + A c\right )} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 27, normalized size = 0.69 \begin {gather*} \frac {2\,x^{7/2}\,\left (99\,A\,b+77\,A\,c\,x+77\,B\,b\,x+63\,B\,c\,x^2\right )}{693} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.62, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A b x^{\frac {7}{2}}}{7} + \frac {2 A c x^{\frac {9}{2}}}{9} + \frac {2 B b x^{\frac {9}{2}}}{9} + \frac {2 B c x^{\frac {11}{2}}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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