Optimal. Leaf size=25 \[ \frac{(a+b x) \sqrt{c (a+b x)^2}}{2 b} \]
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Rubi [A] time = 0.010434, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac{(a+b x) \sqrt{c (a+b x)^2}}{2 b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \sqrt{c (a+b x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{c x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\sqrt{c (a+b x)^2} \operatorname{Subst}(\int x \, dx,x,a+b x)}{b (a+b x)}\\ &=\frac{(a+b x) \sqrt{c (a+b x)^2}}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0133636, size = 31, normalized size = 1.24 \[ \frac{c x (a+b x) (2 a+b x)}{2 \sqrt{c (a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 29, normalized size = 1.2 \begin{align*}{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a}\sqrt{c \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 = 1.23573, size = 90, normalized size = 3.6 \begin{align*} \frac{\sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}{\left (b x^{2} + 2 \, a x\right )}}{2 \,{\left (b x + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \left (a + b x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11334, size = 49, normalized size = 1.96 \begin{align*} \frac{1}{2} \,{\left ({\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 )}{b}\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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