Optimal. Leaf size=65 \[ -\frac{\tan ^{-1}\left (\frac{-a d+b c-2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{x (b c-a d)+a c-b d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]
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Rubi [A] time = 0.0249964, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1981, 621, 204} \[ -\frac{\tan ^{-1}\left (\frac{-a d+b c-2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{x (b c-a d)+a c-b d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 1981
Rule 621
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{(a+b x) (c-d x)}} \, dx &=\int \frac{1}{\sqrt{a c+(b c-a d) x-b d x^2}} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{1}{-4 b d-x^2} \, dx,x,\frac{b c-a d-2 b d x}{\sqrt{a c+(b c-a d) x-b d x^2}}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{b c-a d-2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{a c+(b c-a d) x-b d x^2}}\right )}{\sqrt{b} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.0741378, size = 94, normalized size = 1.45 \[ \frac{2 \sqrt{a+b x} \sqrt{a d+b c} \sqrt{\frac{b (c-d x)}{a d+b c}} \sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d+b c}}\right )}{b \sqrt{d} \sqrt{(a+b x) (c-d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 55, normalized size = 0.9 \begin{align*}{\arctan \left ({\sqrt{bd} \left ( x-{\frac{-ad+bc}{2\,bd}} \right ){\frac{1}{\sqrt{ac+ \left ( -ad+bc \right ) x-bd{x}^{2}}}}} \right ){\frac{1}{\sqrt{bd}}}} \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 = 1.51656, size = 439, normalized size = 6.75 \begin{align*} \left [-\frac{\sqrt{-b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 4 \, \sqrt{-b d x^{2} + a c +{\left (b c - a d\right )} x}{\left (2 \, b d x - b c + a d\right )} \sqrt{-b d} - 8 \,{\left (b^{2} c d - a b d^{2}\right )} x\right )}{2 \, b d}, -\frac{\sqrt{b d} \arctan \left (\frac{\sqrt{-b d x^{2} + a c +{\left (b c - a d\right )} x}{\left (2 \, b d x - b c + a d\right )} \sqrt{b d}}{2 \,{\left (b^{2} d^{2} x^{2} - a b c d -{\left (b^{2} c d - a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23893, size = 80, normalized size = 1.23 \begin{align*} -\frac{\log \left ({\left | b c - a d + 2 \, \sqrt{-b d}{\left (\sqrt{-b d} x - \sqrt{-b d x^{2} + b c x - a d x + a c}\right )} \right |}\right )}{\sqrt{-b d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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