Optimal. Leaf size=37 \[ \frac{2}{3} x^{3/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{5} b B x^{5/2} \]
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Rubi [A] time = 0.0135054, antiderivative size = 37, 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} \[ \frac{2}{3} x^{3/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{5} b B x^{5/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x) (A+B x)}{\sqrt{x}} \, dx &=\int \left (\frac{a A}{\sqrt{x}}+(A b+a B) \sqrt{x}+b B x^{3/2}\right ) \, dx\\ &=2 a A \sqrt{x}+\frac{2}{3} (A b+a B) x^{3/2}+\frac{2}{5} b B x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0110247, size = 31, normalized size = 0.84 \[ \frac{2}{15} \sqrt{x} (5 a (3 A+B x)+b x (5 A+3 B x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.8 \begin{align*}{\frac{6\,bB{x}^{2}+10\,Abx+10\,Bax+30\,Aa}{15}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05287, size = 36, normalized size = 0.97 \begin{align*} \frac{2}{5} \, B b x^{\frac{5}{2}} + 2 \, A a \sqrt{x} + \frac{2}{3} \,{\left (B a + A b\right )} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48563, size = 72, normalized size = 1.95 \begin{align*} \frac{2}{15} \,{\left (3 \, B b x^{2} + 15 \, A a + 5 \,{\left (B a + A b\right )} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.411172, size = 44, normalized size = 1.19 \begin{align*} 2 A a \sqrt{x} + \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19712, size = 39, normalized size = 1.05 \begin{align*} \frac{2}{5} \, B b x^{\frac{5}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} + 2 \, A a \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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