Optimal. Leaf size=41 \[ \frac{2 \sqrt{c+d x}}{b d}-\frac{2 a \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]
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Rubi [A] time = 0.0241772, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {247, 190, 43} \[ \frac{2 \sqrt{c+d x}}{b d}-\frac{2 a \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 247
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{a+b \sqrt{c+d x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \sqrt{x}} \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 \sqrt{c+d x}}{b d}-\frac{2 a \log \left (a+b \sqrt{c+d x}\right )}{b^2 d}\\ \end{align*}
Mathematica [A] time = 0.0188489, size = 39, normalized size = 0.95 \[ \frac{2 \left (\frac{\sqrt{c+d x}}{b}-\frac{a \log \left (a+b \sqrt{c+d x}\right )}{b^2}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 87, normalized size = 2.1 \begin{align*} 2\,{\frac{\sqrt{dx+c}}{bd}}+{\frac{a}{{b}^{2}d}\ln \left ( -a+b\sqrt{dx+c} \right ) }-{\frac{a}{{b}^{2}d}\ln \left ( a+b\sqrt{dx+c} \right ) }-{\frac{a\ln \left ({b}^{2}dx+{b}^{2}c-{a}^{2} \right ) }{{b}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976455, size = 47, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (\frac{a \log \left (\sqrt{d x + c} b + a\right )}{b^{2}} - \frac{\sqrt{d x + c}}{b}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73875, size = 80, normalized size = 1.95 \begin{align*} -\frac{2 \,{\left (a \log \left (\sqrt{d x + c} b + a\right ) - \sqrt{d x + c} b\right )}}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.496423, size = 49, normalized size = 1.2 \begin{align*} \begin{cases} \frac{x}{a} & \text{for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac{x}{a + b \sqrt{c}} & \text{for}\: d = 0 \\- \frac{2 a \log{\left (\frac{a}{b} + \sqrt{c + d x} \right )}}{b^{2} d} + \frac{2 \sqrt{c + d x}}{b d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17932, size = 68, normalized size = 1.66 \begin{align*} -\frac{2 \, a \log \left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{2} d} + \frac{2 \, a \log \left ({\left | a \right |}\right )}{b^{2} d} + \frac{2 \, \sqrt{d x + c}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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