Optimal. Leaf size=68 \[ \frac{\tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{\sqrt{d} \sqrt{c d-b e}} \]
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Rubi [A] time = 0.0305797, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {724, 206} \[ \frac{\tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{\sqrt{d} \sqrt{c d-b e}} \]
Antiderivative was successfully verified.
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Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )\right )\\ &=\frac{\tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{\sqrt{d} \sqrt{c d-b e}}\\ \end{align*}
Mathematica [A] time = 0.0258039, size = 77, normalized size = 1.13 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{d} \sqrt{x (b+c x)} \sqrt{b e-c d}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.216, size = 132, normalized size = 1.9 \begin{align*} -{\frac{1}{e}\ln \left ({ \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{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 = 2.05929, size = 284, normalized size = 4.18 \begin{align*} \left [\frac{\log \left (\frac{b d +{\left (2 \, c d - b e\right )} x + 2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}}{e x + d}\right )}{\sqrt{c d^{2} - b d e}}, \frac{2 \, \sqrt{-c d^{2} + b d e} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right )}{c d^{2} - b d e}\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 \frac{1}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29285, size = 82, normalized size = 1.21 \begin{align*} \frac{2 \, \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e}}\right )}{\sqrt{-c d^{2} + b d e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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