Optimal. Leaf size=42 \[ \frac{2 (a+b x)^{3/2} (A b-a B)}{3 b^2}+\frac{2 B (a+b x)^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.0167864, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{2 (a+b x)^{3/2} (A b-a B)}{3 b^2}+\frac{2 B (a+b x)^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \sqrt{a+b x} (A+B x) \, dx &=\int \left (\frac{(A b-a B) \sqrt{a+b x}}{b}+\frac{B (a+b x)^{3/2}}{b}\right ) \, dx\\ &=\frac{2 (A b-a B) (a+b x)^{3/2}}{3 b^2}+\frac{2 B (a+b x)^{5/2}}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0177103, size = 30, normalized size = 0.71 \[ \frac{2 (a+b x)^{3/2} (-2 a B+5 A b+3 b B x)}{15 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 27, normalized size = 0.6 \begin{align*}{\frac{6\,bBx+10\,Ab-4\,Ba}{15\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19657, size = 45, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} B - 5 \,{\left (B a - A b\right )}{\left (b x + a\right )}^{\frac{3}{2}}\right )}}{15 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34161, size = 108, normalized size = 2.57 \begin{align*} \frac{2 \,{\left (3 \, B b^{2} x^{2} - 2 \, B a^{2} + 5 \, A a b +{\left (B a b + 5 \, A b^{2}\right )} x\right )} \sqrt{b x + a}}{15 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.96798, size = 36, normalized size = 0.86 \begin{align*} \frac{2 \left (\frac{B \left (a + b x\right )^{\frac{5}{2}}}{5 b} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14161, size = 55, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{3}{2}} A + \frac{{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} B}{b}\right )}}{15 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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