Optimal. Leaf size=42 \[ -\frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{-b}}\right )|-\frac{c}{d}\right )}{\sqrt{-b} \sqrt{d}} \]
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Rubi [A] time = 0.0168393, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {110} \[ -\frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{-b}}\right )|-\frac{c}{d}\right )}{\sqrt{-b} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 110
Rubi steps
\begin{align*} \int \frac{\sqrt{1-c x}}{\sqrt{b x} \sqrt{1+d x}} \, dx &=-\frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{-b}}\right )|-\frac{c}{d}\right )}{\sqrt{-b} \sqrt{d}}\\ \end{align*}
Mathematica [B] time = 0.411434, size = 112, normalized size = 2.67 \[ \frac{-2 x^{3/2} \sqrt{1-\frac{1}{c x}} \sqrt{\frac{1}{d x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{1}{c}}}{\sqrt{x}}\right )|-\frac{c}{d}\right )-\frac{2 \sqrt{\frac{1}{c}} (c x-1) (d x+1)}{d}}{\sqrt{\frac{1}{c}} \sqrt{b x} \sqrt{1-c x} \sqrt{d x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 67, normalized size = 1.6 \begin{align*} -2\,{\frac{ \left ( c+d \right ) \sqrt{-dx}\sqrt{-cx+1}}{ \left ( cx-1 \right ) \sqrt{bx}{d}^{2}}{\it EllipticE} \left ( \sqrt{dx+1},\sqrt{{\frac{c}{c+d}}} \right ) \sqrt{-{\frac{ \left ( cx-1 \right ) d}{c+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c x + 1}}{\sqrt{b x} \sqrt{d x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x} \sqrt{-c x + 1} \sqrt{d x + 1}}{b d x^{2} + b x}, x\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{\sqrt{- c x + 1}}{\sqrt{b x} \sqrt{d x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c x + 1}}{\sqrt{b x} \sqrt{d x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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