Optimal. Leaf size=19 \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{b x+1}\right )}{b} \]
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Rubi [A] time = 0.0049356, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {63, 215} \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{b x+1}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 215
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+b x} \sqrt{5+b x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{4+x^2}} \, dx,x,\sqrt{1+b x}\right )}{b}\\ &=\frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{1+b x}\right )}{b}\\ \end{align*}
Mathematica [B] time = 0.0113841, size = 39, normalized size = 2.05 \[ \frac{2 \sqrt{b x+1} \sin ^{-1}\left (\frac{1}{2} \sqrt{-b x-1}\right )}{b \sqrt{-b x-1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 66, normalized size = 3.5 \begin{align*}{\sqrt{ \left ( bx+1 \right ) \left ( bx+5 \right ) }\ln \left ({({b}^{2}x+3\,b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+6\,bx+5} \right ){\frac{1}{\sqrt{bx+1}}}{\frac{1}{\sqrt{bx+5}}}{\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 [A] time = 2.13267, size = 65, normalized size = 3.42 \begin{align*} -\frac{\log \left (-b x + \sqrt{b x + 5} \sqrt{b x + 1} - 3\right )}{b} \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 x + 1} \sqrt{b x + 5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14361, size = 32, normalized size = 1.68 \begin{align*} -\frac{2 \, \log \left ({\left | -\sqrt{b x + 5} + \sqrt{b x + 1} \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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