Optimal. Leaf size=63 \[ \frac{2 \sqrt{a+b x} (A b-2 a B)}{b^3}+\frac{2 a (A b-a B)}{b^3 \sqrt{a+b x}}+\frac{2 B (a+b x)^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.0238114, antiderivative size = 63, 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 = {77} \[ \frac{2 \sqrt{a+b x} (A b-2 a B)}{b^3}+\frac{2 a (A b-a B)}{b^3 \sqrt{a+b x}}+\frac{2 B (a+b x)^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x (A+B x)}{(a+b x)^{3/2}} \, dx &=\int \left (\frac{a (-A b+a B)}{b^2 (a+b x)^{3/2}}+\frac{A b-2 a B}{b^2 \sqrt{a+b x}}+\frac{B \sqrt{a+b x}}{b^2}\right ) \, dx\\ &=\frac{2 a (A b-a B)}{b^3 \sqrt{a+b x}}+\frac{2 (A b-2 a B) \sqrt{a+b x}}{b^3}+\frac{2 B (a+b x)^{3/2}}{3 b^3}\\ \end{align*}
Mathematica [A] time = 0.0289568, size = 47, normalized size = 0.75 \[ \frac{2 \left (-8 a^2 B+a (6 A b-4 b B x)+b^2 x (3 A+B x)\right )}{3 b^3 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 46, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{2}B{x}^{2}+6\,{b}^{2}Ax-8\,abBx+12\,Aba-16\,B{a}^{2}}{3\,{b}^{3}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08214, size = 82, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}} B - 3 \,{\left (2 \, B a - A b\right )} \sqrt{b x + a}}{b} - \frac{3 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x + a} b}\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38879, size = 123, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (B b^{2} x^{2} - 8 \, B a^{2} + 6 \, A a b -{\left (4 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{b x + a}}{3 \,{\left (b^{4} x + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.2369, size = 60, normalized size = 0.95 \begin{align*} \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3}} - \frac{2 a \left (- A b + B a\right )}{b^{3} \sqrt{a + b x}} + \frac{\sqrt{a + b x} \left (2 A b - 4 B a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14737, size = 93, normalized size = 1.48 \begin{align*} -\frac{2 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x + a} b^{3}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} B b^{6} - 6 \, \sqrt{b x + a} B a b^{6} + 3 \, \sqrt{b x + a} A b^{7}\right )}}{3 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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