Optimal. Leaf size=41 \[ -\frac{2 a A}{3 x^{3/2}}-\frac{2 a B}{\sqrt{x}}+2 A c \sqrt{x}+\frac{2}{3} B c x^{3/2} \]
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Rubi [A] time = 0.012395, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ -\frac{2 a A}{3 x^{3/2}}-\frac{2 a B}{\sqrt{x}}+2 A c \sqrt{x}+\frac{2}{3} B c x^{3/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac{a A}{x^{5/2}}+\frac{a B}{x^{3/2}}+\frac{A c}{\sqrt{x}}+B c \sqrt{x}\right ) \, dx\\ &=-\frac{2 a A}{3 x^{3/2}}-\frac{2 a B}{\sqrt{x}}+2 A c \sqrt{x}+\frac{2}{3} B c x^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0113768, size = 32, normalized size = 0.78 \[ \frac{2 c x^2 (3 A+B x)-2 a (A+3 B x)}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 29, normalized size = 0.7 \begin{align*} -{\frac{-2\,Bc{x}^{3}-6\,Ac{x}^{2}+6\,aBx+2\,aA}{3}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00301, size = 39, normalized size = 0.95 \begin{align*} \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \, A c \sqrt{x} - \frac{2 \,{\left (3 \, B a x + A a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33047, size = 69, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (B c x^{3} + 3 \, A c x^{2} - 3 \, B a x - A a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.02441, size = 42, normalized size = 1.02 \begin{align*} - \frac{2 A a}{3 x^{\frac{3}{2}}} + 2 A c \sqrt{x} - \frac{2 B a}{\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 = 1.221, size = 39, normalized size = 0.95 \begin{align*} \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \, A c \sqrt{x} - \frac{2 \,{\left (3 \, B a x + A a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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