Optimal. Leaf size=42 \[ \frac{2 \sin ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{b (1-c)}{d}+b x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]
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Rubi [A] time = 0.0161989, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {63, 216} \[ \frac{2 \sin ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{b (1-c)}{d}+b x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\frac{b-b c}{d}+b x} \sqrt{c-d x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{b-b c}{b}-\frac{d x^2}{b}}} \, dx,x,\sqrt{\frac{b-b c}{d}+b x}\right )}{b}\\ &=\frac{2 \sin ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{b (1-c)}{d}+b x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.0501009, size = 67, normalized size = 1.6 \[ \frac{2 \sqrt{-d} \sqrt{-c+d x+1} \sinh ^{-1}\left (\frac{\sqrt{-d} \sqrt{c-d x}}{\sqrt{d}}\right )}{d^{3/2} \sqrt{\frac{b (-c+d x+1)}{d}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 118, normalized size = 2.8 \begin{align*}{\sqrt{ \left ({\frac{b \left ( 1-c \right ) }{d}}+bx \right ) \left ( -dx+c \right ) }\arctan \left ({\sqrt{bd} \left ( x-{\frac{-b \left ( 1-c \right ) +bc}{2\,bd}} \right ){\frac{1}{\sqrt{-d{x}^{2}b+ \left ( -b \left ( 1-c \right ) +bc \right ) x+{\frac{b \left ( 1-c \right ) c}{d}}}}}} \right ){\frac{1}{\sqrt{{\frac{b \left ( 1-c \right ) }{d}}+bx}}}{\frac{1}{\sqrt{-dx+c}}}{\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 [B] time = 2.11584, size = 409, normalized size = 9.74 \begin{align*} \left [-\frac{\sqrt{-b d} \log \left (8 \, b d^{2} x^{2} + 8 \, b c^{2} - 8 \,{\left (2 \, b c - b\right )} d x - 4 \, \sqrt{-b d}{\left (2 \, d x - 2 \, c + 1\right )} \sqrt{-d x + c} \sqrt{\frac{b d x - b c + b}{d}} - 8 \, b c + b\right )}{2 \, b d}, -\frac{\sqrt{b d} \arctan \left (\frac{\sqrt{b d}{\left (2 \, d x - 2 \, c + 1\right )} \sqrt{-d x + c} \sqrt{\frac{b d x - b c + b}{d}}}{2 \,{\left (b d^{2} x^{2} + b c^{2} -{\left (2 \, b c - b\right )} d x - b c\right )}}\right )}{b d}\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{b \left (- \frac{c}{d} + x + \frac{1}{d}\right )} \sqrt{c - d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0759, size = 90, normalized size = 2.14 \begin{align*} -\frac{2 \, b \log \left (-\sqrt{-b d} \sqrt{b x - \frac{b c - b}{d}} + \sqrt{-{\left (b x - \frac{b c - b}{d}\right )} b d + b^{2}}\right )}{\sqrt{-b d}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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