Optimal. Leaf size=25 \[ \frac{a+b x}{2 b \sqrt{\frac{c}{(a+b x)^2}}} \]
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Rubi [A] time = 0.0073814, 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} \[ \frac{a+b x}{2 b \sqrt{\frac{c}{(a+b x)^2}}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\frac{c}{(a+b x)^2}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{c}{x^2}}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}(\int x \, dx,x,a+b x)}{b \sqrt{\frac{c}{(a+b x)^2}} (a+b x)}\\ &=\frac{a+b x}{2 b \sqrt{\frac{c}{(a+b x)^2}}}\\ \end{align*}
Mathematica [A] time = 0.0128316, size = 32, normalized size = 1.28 \[ \frac{x (2 a+b x)}{2 (a+b x) \sqrt{\frac{c}{(a+b x)^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 29, normalized size = 1.2 \begin{align*}{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a}{\frac{1}{\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14642, size = 20, normalized size = 0.8 \begin{align*} \frac{b x^{2} + 2 \, a x}{2 \, \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.29815, size = 99, normalized size = 3.96 \begin{align*} \frac{{\left (b^{2} x^{3} + 3 \, a b x^{2} + 2 \, a^{2} x\right )} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, c} \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 \frac{1}{\sqrt{\frac{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 = 1.39094, size = 39, normalized size = 1.56 \begin{align*} \frac{b \sqrt{c} x^{2} + 2 \, a \sqrt{c} x}{2 \, c \mathrm{sgn}\left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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