Optimal. Leaf size=39 \[ \frac {2}{7} x^{7/2} (a B+A b)+\frac {2}{5} a A x^{5/2}+\frac {2}{9} b B x^{9/2} \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \begin {gather*} \frac {2}{7} x^{7/2} (a B+A b)+\frac {2}{5} a A x^{5/2}+\frac {2}{9} b B x^{9/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int x^{3/2} (a+b x) (A+B x) \, dx &=\int \left (a A x^{3/2}+(A b+a B) x^{5/2}+b B x^{7/2}\right ) \, dx\\ &=\frac {2}{5} a A x^{5/2}+\frac {2}{7} (A b+a B) x^{7/2}+\frac {2}{9} b B x^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.85 \begin {gather*} \frac {2}{315} x^{5/2} (9 a (7 A+5 B x)+5 b x (9 A+7 B 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}{315} \left (63 a A x^{5/2}+45 a B x^{7/2}+45 A b x^{7/2}+35 b B x^{9/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 32, normalized size = 0.82 \begin {gather*} \frac {2}{315} \, {\left (35 \, B b x^{4} + 63 \, A a x^{2} + 45 \, {\left (B a + A b\right )} x^{3}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{7} \, B a x^{\frac {7}{2}} + \frac {2}{7} \, A b x^{\frac {7}{2}} + \frac {2}{5} \, A a x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.72 \begin {gather*} \frac {2 \left (35 B b \,x^{2}+45 A b x +45 B a x +63 A a \right ) x^{\frac {5}{2}}}{315} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{5} \, A a x^{\frac {5}{2}} + \frac {2}{7} \, {\left (B a + A b\right )} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 27, normalized size = 0.69 \begin {gather*} \frac {2\,x^{5/2}\,\left (63\,A\,a+45\,A\,b\,x+45\,B\,a\,x+35\,B\,b\,x^2\right )}{315} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.01, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A a x^{\frac {5}{2}}}{5} + \frac {2 A b x^{\frac {7}{2}}}{7} + \frac {2 B a x^{\frac {7}{2}}}{7} + \frac {2 B b x^{\frac {9}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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