Optimal. Leaf size=49 \[ -2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+2 a \sqrt{a+b x}+\frac{2}{3} (a+b x)^{3/2} \]
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Rubi [A] time = 0.0144803, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {50, 63, 208} \[ -2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+2 a \sqrt{a+b x}+\frac{2}{3} (a+b x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{x} \, dx &=\frac{2}{3} (a+b x)^{3/2}+a \int \frac{\sqrt{a+b x}}{x} \, dx\\ &=2 a \sqrt{a+b x}+\frac{2}{3} (a+b x)^{3/2}+a^2 \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=2 a \sqrt{a+b x}+\frac{2}{3} (a+b x)^{3/2}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{b}\\ &=2 a \sqrt{a+b x}+\frac{2}{3} (a+b x)^{3/2}-2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0420943, size = 44, normalized size = 0.9 \[ \frac{2}{3} \sqrt{a+b x} (4 a+b x)-2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 38, normalized size = 0.8 \begin{align*}{\frac{2}{3} \left ( bx+a \right ) ^{{\frac{3}{2}}}}-2\,{a}^{3/2}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) +2\,a\sqrt{bx+a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64594, size = 228, normalized size = 4.65 \begin{align*} \left [a^{\frac{3}{2}} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \frac{2}{3} \,{\left (b x + 4 \, a\right )} \sqrt{b x + a}, 2 \, \sqrt{-a} a \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) + \frac{2}{3} \,{\left (b x + 4 \, a\right )} \sqrt{b x + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.72374, size = 71, normalized size = 1.45 \begin{align*} \frac{8 a^{\frac{3}{2}} \sqrt{1 + \frac{b x}{a}}}{3} + a^{\frac{3}{2}} \log{\left (\frac{b x}{a} \right )} - 2 a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )} + \frac{2 \sqrt{a} b x \sqrt{1 + \frac{b x}{a}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13304, size = 59, normalized size = 1.2 \begin{align*} \frac{2 \, a^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2}{3} \,{\left (b x + a\right )}^{\frac{3}{2}} + 2 \, \sqrt{b x + a} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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