Optimal. Leaf size=35 \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} b B x^{3/2} \]
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Rubi [A] time = 0.0127751, antiderivative size = 35, 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} \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} b B x^{3/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x) (A+B x)}{x^{3/2}} \, dx &=\int \left (\frac{a A}{x^{3/2}}+\frac{A b+a B}{\sqrt{x}}+b B \sqrt{x}\right ) \, dx\\ &=-\frac{2 a A}{\sqrt{x}}+2 (A b+a B) \sqrt{x}+\frac{2}{3} b B x^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0120495, size = 29, normalized size = 0.83 \[ \frac{2 (b x (3 A+B x)-3 a (A-B x))}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.8 \begin{align*} -{\frac{-2\,bB{x}^{2}-6\,Abx-6\,Bax+6\,Aa}{3}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12629, size = 36, normalized size = 1.03 \begin{align*} \frac{2}{3} \, B b x^{\frac{3}{2}} - \frac{2 \, A a}{\sqrt{x}} + 2 \,{\left (B a + A b\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.62197, size = 66, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (B b x^{2} - 3 \, A a + 3 \,{\left (B a + A b\right )} x\right )}}{3 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.581993, size = 41, normalized size = 1.17 \begin{align*} - \frac{2 A a}{\sqrt{x}} + 2 A b \sqrt{x} + 2 B a \sqrt{x} + \frac{2 B b x^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22355, size = 39, normalized size = 1.11 \begin{align*} \frac{2}{3} \, B b x^{\frac{3}{2}} + 2 \, B a \sqrt{x} + 2 \, A b \sqrt{x} - \frac{2 \, A a}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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