Optimal. Leaf size=67 \[ \frac{\sqrt{2 x^2+1}}{2 \sqrt{2}}-\frac{1}{8} \sqrt{33} \tanh ^{-1}\left (\frac{\sqrt{\frac{2}{33}} (2-5 x)}{\sqrt{2 x^2+1}}\right )-\frac{5}{8} \sinh ^{-1}\left (\sqrt{2} x\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.139362, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{\sqrt{2 x^2+1}}{2 \sqrt{2}}-\frac{1}{8} \sqrt{33} \tanh ^{-1}\left (\frac{\sqrt{\frac{2}{33}} (2-5 x)}{\sqrt{2 x^2+1}}\right )-\frac{5}{8} \sinh ^{-1}\left (\sqrt{2} x\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 + 4*x^2]/(5 + 4*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.3335, size = 53, normalized size = 0.79 \[ \frac{\sqrt{4 x^{2} + 2}}{4} - \frac{5 \operatorname{asinh}{\left (\sqrt{2} x \right )}}{8} - \frac{\sqrt{33} \operatorname{atanh}{\left (\frac{\sqrt{33} \left (- 20 x + 8\right )}{66 \sqrt{4 x^{2} + 2}} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((4*x**2+2)**(1/2)/(5+4*x),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0914127, size = 71, normalized size = 1.06 \[ \frac{1}{8} \left (2 \sqrt{4 x^2+2}-\sqrt{33} \log \left (2 \sqrt{66} \sqrt{2 x^2+1}-20 x+8\right )+\sqrt{33} \log (4 x+5)-5 \sinh ^{-1}\left (\sqrt{2} x\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 + 4*x^2]/(5 + 4*x),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 56, normalized size = 0.8 \[{\frac{1}{8}\sqrt{16\, \left ( x+5/4 \right ) ^{2}-40\,x-17}}-{\frac{5\,{\it Arcsinh} \left ( x\sqrt{2} \right ) }{8}}-{\frac{\sqrt{33}}{8}{\it Artanh} \left ({\frac{ \left ( -20\,x+8 \right ) \sqrt{33}}{33}{\frac{1}{\sqrt{16\, \left ( x+5/4 \right ) ^{2}-40\,x-17}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((4*x^2+2)^(1/2)/(5+4*x),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.793261, size = 73, normalized size = 1.09 \[ \frac{1}{8} \, \sqrt{33} \operatorname{arsinh}\left (\frac{5 \, \sqrt{2} x}{{\left | 4 \, x + 5 \right |}} - \frac{2 \, \sqrt{2}}{{\left | 4 \, x + 5 \right |}}\right ) + \frac{1}{4} \, \sqrt{4 \, x^{2} + 2} - \frac{5}{8} \, \operatorname{arsinh}\left (\sqrt{2} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 + 2)/(4*x + 5),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.231081, size = 215, normalized size = 3.21 \[ -\frac{8 \, x^{2} - 5 \,{\left (2 \, x - \sqrt{4 \, x^{2} + 2}\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 2}\right ) -{\left (2 \, \sqrt{33} x - \sqrt{33} \sqrt{4 \, x^{2} + 2}\right )} \log \left (\frac{16 \, x^{2} - 2 \, \sqrt{4 \, x^{2} + 2}{\left (4 \, x + \sqrt{33} + 5\right )} + \sqrt{33}{\left (4 \, x + 5\right )} + 20 \, x + 33}{8 \, x^{2} - \sqrt{4 \, x^{2} + 2}{\left (4 \, x + 5\right )} + 10 \, x}\right ) - 4 \, \sqrt{4 \, x^{2} + 2} x + 4}{8 \,{\left (2 \, x - \sqrt{4 \, x^{2} + 2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 + 2)/(4*x + 5),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \sqrt{2} \int \frac{\sqrt{2 x^{2} + 1}}{4 x + 5}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x**2+2)**(1/2)/(5+4*x),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.233618, size = 142, normalized size = 2.12 \[ \frac{1}{16} \, \sqrt{2}{\left (5 \, \sqrt{2}{\rm ln}\left (-\sqrt{2} x + \sqrt{2 \, x^{2} + 1}\right ) + \sqrt{66}{\rm ln}\left (-\frac{{\left | -4 \, \sqrt{2} x - \sqrt{66} - 5 \, \sqrt{2} + 4 \, \sqrt{2 \, x^{2} + 1} \right |}}{4 \, \sqrt{2} x - \sqrt{66} + 5 \, \sqrt{2} - 4 \, \sqrt{2 \, x^{2} + 1}}\right ) + 4 \, \sqrt{2 \, x^{2} + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 + 2)/(4*x + 5),x, algorithm="giac")
[Out]