Optimal. Leaf size=39 \[ \frac{2}{5} x^{5/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{7} b B x^{7/2} \]
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Rubi [A] time = 0.0135763, antiderivative size = 39, normalized size of antiderivative = 1., 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} \[ \frac{2}{5} x^{5/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{7} b B x^{7/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \sqrt{x} (a+b x) (A+B x) \, dx &=\int \left (a A \sqrt{x}+(A b+a B) x^{3/2}+b B x^{5/2}\right ) \, dx\\ &=\frac{2}{3} a A x^{3/2}+\frac{2}{5} (A b+a B) x^{5/2}+\frac{2}{7} b B x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0109327, size = 33, normalized size = 0.85 \[ \frac{2}{105} x^{3/2} (7 a (5 A+3 B x)+3 b x (7 A+5 B x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.7 \begin{align*}{\frac{30\,bB{x}^{2}+42\,Abx+42\,Bax+70\,Aa}{105}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09683, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} + \frac{2}{5} \,{\left (B a + A b\right )} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54123, size = 81, normalized size = 2.08 \begin{align*} \frac{2}{105} \,{\left (15 \, B b x^{3} + 35 \, A a x + 21 \,{\left (B a + A b\right )} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.57499, size = 37, normalized size = 0.95 \begin{align*} \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{7}{2}}}{7} + \frac{2 x^{\frac{5}{2}} \left (A b + B a\right )}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1671, size = 39, normalized size = 1. \begin{align*} \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{5} \, A b x^{\frac{5}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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