Optimal. Leaf size=16 \[ \sinh ^{-1}\left (\frac{\sqrt{2 x+1}}{\sqrt{2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0052886, antiderivative size = 16, 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 = {54, 215} \[ \sinh ^{-1}\left (\frac{\sqrt{2 x+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+2 x} \sqrt{3+2 x}} \, dx &=\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{4+2 x^2}} \, dx,x,\sqrt{1+2 x}\right )\\ &=\sinh ^{-1}\left (\frac{\sqrt{1+2 x}}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0097572, size = 31, normalized size = 1.94 \[ \frac{\sqrt{2 x+1} \sin ^{-1}\left (\sqrt{-x-\frac{1}{2}}\right )}{\sqrt{-2 x-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.005, size = 57, normalized size = 3.6 \begin{align*}{\frac{\sqrt{4}}{4}\sqrt{ \left ( 1+2\,x \right ) \left ( 3+2\,x \right ) }\ln \left ({\frac{ \left ( 4+4\,x \right ) \sqrt{4}}{4}}+\sqrt{4\,{x}^{2}+8\,x+3} \right ){\frac{1}{\sqrt{1+2\,x}}}{\frac{1}{\sqrt{3+2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.28276, size = 30, normalized size = 1.88 \begin{align*} \frac{1}{2} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} + 8 \, x + 3} + 8\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.47466, size = 66, normalized size = 4.12 \begin{align*} -\frac{1}{2} \, \log \left (\sqrt{2 \, x + 3} \sqrt{2 \, x + 1} - 2 \, x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.16709, size = 27, normalized size = 1.69 \begin{align*} \begin{cases} \operatorname{acosh}{\left (\sqrt{x + \frac{3}{2}} \right )} & \text{for}\: \left |{x + \frac{3}{2}}\right | > 1 \\- i \operatorname{asin}{\left (\sqrt{x + \frac{3}{2}} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.45895, size = 28, normalized size = 1.75 \begin{align*} -\log \left ({\left | -\sqrt{2 \, x + 3} + \sqrt{2 \, x + 1} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]