Optimal. Leaf size=35 \[ 2 \sqrt{x} (A c+b B)-\frac{2 A b}{\sqrt{x}}+\frac{2}{3} B c x^{3/2} \]
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Rubi [A] time = 0.0160369, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ 2 \sqrt{x} (A c+b B)-\frac{2 A b}{\sqrt{x}}+\frac{2}{3} B c x^{3/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac{A b}{x^{3/2}}+\frac{b B+A c}{\sqrt{x}}+B c \sqrt{x}\right ) \, dx\\ &=-\frac{2 A b}{\sqrt{x}}+2 (b B+A c) \sqrt{x}+\frac{2}{3} B c x^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0114129, size = 29, normalized size = 0.83 \[ \frac{2 (B x (3 b+c x)-3 A (b-c x))}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 28, normalized size = 0.8 \begin{align*} -{\frac{-2\,Bc{x}^{2}-6\,Acx-6\,bBx+6\,Ab}{3}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03003, size = 36, normalized size = 1.03 \begin{align*} \frac{2}{3} \, B c x^{\frac{3}{2}} - \frac{2 \, A b}{\sqrt{x}} + 2 \,{\left (B b + A c\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77157, size = 66, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (B c x^{2} - 3 \, A b + 3 \,{\left (B b + A c\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 = 0.889824, size = 41, normalized size = 1.17 \begin{align*} - \frac{2 A b}{\sqrt{x}} + 2 A c \sqrt{x} + 2 B b \sqrt{x} + \frac{2 B c x^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16114, size = 39, normalized size = 1.11 \begin{align*} \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \, B b \sqrt{x} + 2 \, A c \sqrt{x} - \frac{2 \, A b}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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