Optimal. Leaf size=23 \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]
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Rubi [A] time = 0.0133389, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {619, 216} \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\frac{-b^2+4 c}{4 c}+b x-c x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 c}}} \, dx,x,b-2 c x\right )}{2 c}\\ &=-\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0272743, size = 23, normalized size = 1. \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.247, size = 44, normalized size = 1.9 \begin{align*}{\arctan \left ( 2\,{\sqrt{c} \left ( x-1/2\,{\frac{b}{c}} \right ){\frac{1}{\sqrt{-4\,c{x}^{2}+4\,bx-{\frac{{b}^{2}-4\,c}{c}}}}}} \right ){\frac{1}{\sqrt{c}}}} \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 [B] time = 1.97964, size = 323, normalized size = 14.04 \begin{align*} \left [-\frac{\sqrt{-c} \log \left (4 \, c^{2} x^{2} - 4 \, b c x + b^{2} -{\left (2 \, c x - b\right )} \sqrt{-c} \sqrt{-\frac{4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}{c}} - 2 \, c\right )}{2 \, c}, -\frac{\arctan \left (\frac{{\left (2 \, c x - b\right )} \sqrt{c} \sqrt{-\frac{4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}{c}}}{4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}\right )}{\sqrt{c}}\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*} 2 \int \frac{1}{\sqrt{- \frac{b^{2}}{c} + 4 b x - 4 c x^{2} + 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.43435, size = 72, normalized size = 3.13 \begin{align*} -\frac{\log \left ({\left |{\left (2 \, \sqrt{-c} x - \sqrt{-4 \, c x^{2} + 4 \, b x - \frac{b^{2} - 4 \, c}{c}}\right )} \sqrt{-c} + b \right |}\right )}{\sqrt{-c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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