Optimal. Leaf size=62 \[ \frac{2 \sqrt{c+d x}}{b}-\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.0527786, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {50, 63, 208} \[ \frac{2 \sqrt{c+d x}}{b}-\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x}}{a+b x} \, dx &=\frac{2 \sqrt{c+d x}}{b}+\frac{(b c-a d) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{b}\\ &=\frac{2 \sqrt{c+d x}}{b}+\frac{(2 (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{b d}\\ &=\frac{2 \sqrt{c+d x}}{b}-\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0342759, size = 62, normalized size = 1. \[ \frac{2 \sqrt{c+d x}}{b}-\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 92, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{dx+c}}{b}}-2\,{\frac{ad}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{c}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \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 = 2.15195, size = 306, normalized size = 4.94 \begin{align*} \left [\frac{\sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \, \sqrt{d x + c}}{b}, -\frac{2 \,{\left (\sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) - \sqrt{d x + c}\right )}}{b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.06263, size = 61, normalized size = 0.98 \begin{align*} \frac{2 \left (\frac{d \sqrt{c + d x}}{b} - \frac{d \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b^{2} \sqrt{\frac{a d - b c}{b}}}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06447, size = 84, normalized size = 1.35 \begin{align*} \frac{2 \,{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b} + \frac{2 \, \sqrt{d x + c}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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