Optimal. Leaf size=10 \[ \frac{\sin ^{-1}(b x+1)}{b} \]
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Rubi [A] time = 0.009602, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {53, 619, 216} \[ \frac{\sin ^{-1}(b x+1)}{b} \]
Antiderivative was successfully verified.
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Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-b x} \sqrt{2+b x}} \, dx &=\int \frac{1}{\sqrt{-2 b x-b^2 x^2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 b-2 b^2 x\right )}{2 b^2}\\ &=\frac{\sin ^{-1}(1+b x)}{b}\\ \end{align*}
Mathematica [B] time = 0.0121286, size = 51, normalized size = 5.1 \[ \frac{2 \sqrt{x} \sqrt{b x+2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b} \sqrt{-b x (b x+2)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 58, normalized size = 5.8 \begin{align*}{\sqrt{-bx \left ( bx+2 \right ) }\arctan \left ({(x+{b}^{-1})\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,bx}}}} \right ){\frac{1}{\sqrt{-bx}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{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 [B] time = 2.00147, size = 61, normalized size = 6.1 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b x}}{b x}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.41673, size = 24, normalized size = 2.4 \begin{align*} - \frac{2 i \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06147, size = 24, normalized size = 2.4 \begin{align*} \frac{2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{b x + 2}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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