Optimal. Leaf size=32 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b} \]
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Rubi [A] time = 0.0058244, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {609} \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin{align*} \int \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0079605, size = 30, normalized size = 0.94 \[ \frac{x \sqrt{(a+b x)^2} (2 a+b x)}{2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 27, normalized size = 0.8 \begin{align*}{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \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 = 1.85692, size = 23, normalized size = 0.72 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.245859, size = 8, normalized size = 0.25 \begin{align*} a x + \frac{b x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17913, size = 45, normalized size = 1.41 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} \mathrm{sgn}\left (b x + a\right ) + \frac{a^{2} \mathrm{sgn}\left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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