Optimal. Leaf size=35 \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{3 x^{3/2}}+2 b B \sqrt{x} \]
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Rubi [A] time = 0.0131018, 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} \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{3 x^{3/2}}+2 b B \sqrt{x} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x) (A+B x)}{x^{5/2}} \, dx &=\int \left (\frac{a A}{x^{5/2}}+\frac{A b+a B}{x^{3/2}}+\frac{b B}{\sqrt{x}}\right ) \, dx\\ &=-\frac{2 a A}{3 x^{3/2}}-\frac{2 (A b+a B)}{\sqrt{x}}+2 b B \sqrt{x}\\ \end{align*}
Mathematica [A] time = 0.0101252, size = 28, normalized size = 0.8 \[ -\frac{2 (a (A+3 B x)+3 b x (A-B x))}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-6\,bB{x}^{2}+6\,Abx+6\,Bax+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 = 1.08111, size = 36, normalized size = 1.03 \begin{align*} 2 \, B b \sqrt{x} - \frac{2 \,{\left (A a + 3 \,{\left (B a + A b\right )} x\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 = 2.58216, size = 66, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (3 \, B b x^{2} - A a - 3 \,{\left (B a + A b\right )} x\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 = 0.832327, size = 41, normalized size = 1.17 \begin{align*} - \frac{2 A a}{3 x^{\frac{3}{2}}} - \frac{2 A b}{\sqrt{x}} - \frac{2 B a}{\sqrt{x}} + 2 B b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21156, size = 36, normalized size = 1.03 \begin{align*} 2 \, B b \sqrt{x} - \frac{2 \,{\left (3 \, B a x + 3 \, A b 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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