Optimal. Leaf size=39 \[ \frac {2}{11} x^{11/2} (a B+A b)+\frac {2}{9} a A x^{9/2}+\frac {2}{13} b B x^{13/2} \]
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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} \[ \frac {2}{11} x^{11/2} (a B+A b)+\frac {2}{9} a A x^{9/2}+\frac {2}{13} b B x^{13/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int x^{7/2} (a+b x) (A+B x) \, dx &=\int \left (a A x^{7/2}+(A b+a B) x^{9/2}+b B x^{11/2}\right ) \, dx\\ &=\frac {2}{9} a A x^{9/2}+\frac {2}{11} (A b+a B) x^{11/2}+\frac {2}{13} b B x^{13/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.85 \[ \frac {2 x^{9/2} (13 a (11 A+9 B x)+9 b x (13 A+11 B x))}{1287} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 32, normalized size = 0.82 \[ \frac {2}{1287} \, {\left (99 \, B b x^{6} + 143 \, A a x^{4} + 117 \, {\left (B a + A b\right )} x^{5}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 29, normalized size = 0.74 \[ \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{11} \, B a x^{\frac {11}{2}} + \frac {2}{11} \, A b x^{\frac {11}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \]
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 \[ \frac {2 \left (99 B b \,x^{2}+117 A b x +117 B a x +143 A a \right ) x^{\frac {9}{2}}}{1287} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 27, normalized size = 0.69 \[ \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} + \frac {2}{11} \, {\left (B a + A b\right )} x^{\frac {11}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 27, normalized size = 0.69 \[ \frac {2\,x^{9/2}\,\left (143\,A\,a+117\,A\,b\,x+117\,B\,a\,x+99\,B\,b\,x^2\right )}{1287} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.52, size = 46, normalized size = 1.18 \[ \frac {2 A a x^{\frac {9}{2}}}{9} + \frac {2 A b x^{\frac {11}{2}}}{11} + \frac {2 B a x^{\frac {11}{2}}}{11} + \frac {2 B b x^{\frac {13}{2}}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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