Optimal. Leaf size=32 \[ \frac{2 (a+b x)^{3/2}}{3 b^2}-\frac{2 a \sqrt{a+b x}}{b^2} \]
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Rubi [A] time = 0.0080337, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2 (a+b x)^{3/2}}{3 b^2}-\frac{2 a \sqrt{a+b x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a+b x}} \, dx &=\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx\\ &=-\frac{2 a \sqrt{a+b x}}{b^2}+\frac{2 (a+b x)^{3/2}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0099597, size = 23, normalized size = 0.72 \[ \frac{2 (b x-2 a) \sqrt{a+b x}}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 0.7 \begin{align*} -{\frac{-2\,bx+4\,a}{3\,{b}^{2}}\sqrt{bx+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.932934, size = 35, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, b^{2}} - \frac{2 \, \sqrt{b x + a} a}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50508, size = 47, normalized size = 1.47 \begin{align*} \frac{2 \, \sqrt{b x + a}{\left (b x - 2 \, a\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.03835, size = 162, normalized size = 5.06 \begin{align*} - \frac{4 a^{\frac{7}{2}} \sqrt{1 + \frac{b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac{4 a^{\frac{7}{2}}}{3 a^{2} b^{2} + 3 a b^{3} x} - \frac{2 a^{\frac{5}{2}} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac{4 a^{\frac{5}{2}} b x}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac{2 a^{\frac{3}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07206, size = 31, normalized size = 0.97 \begin{align*} \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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