Optimal. Leaf size=54 \[ 2 A \sqrt{a+b x}-2 \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 B (a+b x)^{3/2}}{3 b} \]
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Rubi [A] time = 0.0182262, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {80, 50, 63, 208} \[ 2 A \sqrt{a+b x}-2 \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 B (a+b x)^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x} (A+B x)}{x} \, dx &=\frac{2 B (a+b x)^{3/2}}{3 b}+A \int \frac{\sqrt{a+b x}}{x} \, dx\\ &=2 A \sqrt{a+b x}+\frac{2 B (a+b x)^{3/2}}{3 b}+(a A) \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=2 A \sqrt{a+b x}+\frac{2 B (a+b x)^{3/2}}{3 b}+\frac{(2 a A) \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 B (a+b x)^{3/2}}{3 b}-2 \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0527528, size = 55, normalized size = 1.02 \[ A \left (2 \sqrt{a+b x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\right )+\frac{2 B (a+b x)^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 46, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{b} \left ( 1/3\,B \left ( bx+a \right ) ^{3/2}+Ab\sqrt{bx+a}-A\sqrt{a}b{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) } \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 = 2.47481, size = 279, normalized size = 5.17 \begin{align*} \left [\frac{3 \, A \sqrt{a} b \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (B b x + B a + 3 \, A b\right )} \sqrt{b x + a}}{3 \, b}, \frac{2 \,{\left (3 \, A \sqrt{-a} b \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (B b x + B a + 3 \, A b\right )} \sqrt{b x + a}\right )}}{3 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.3698, size = 54, normalized size = 1. \begin{align*} \frac{2 A a \operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} + 2 A \sqrt{a + b x} + \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17765, size = 74, normalized size = 1.37 \begin{align*} \frac{2 \, A a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} B b^{2} + 3 \, \sqrt{b x + a} A b^{3}\right )}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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