3.548 \(\int \frac{\sqrt{2+4 x^2}}{5+4 x} \, dx\)

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]

Sqrt[1 + 2*x^2]/(2*Sqrt[2]) - (5*ArcSinh[Sqrt[2]*x])/8 - (Sqrt[33]*ArcTanh[(Sqrt
[2/33]*(2 - 5*x))/Sqrt[1 + 2*x^2]])/8

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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]

Sqrt[1 + 2*x^2]/(2*Sqrt[2]) - (5*ArcSinh[Sqrt[2]*x])/8 - (Sqrt[33]*ArcTanh[(Sqrt
[2/33]*(2 - 5*x))/Sqrt[1 + 2*x^2]])/8

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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]

sqrt(4*x**2 + 2)/4 - 5*asinh(sqrt(2)*x)/8 - sqrt(33)*atanh(sqrt(33)*(-20*x + 8)/
(66*sqrt(4*x**2 + 2)))/8

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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]

(2*Sqrt[2 + 4*x^2] - 5*ArcSinh[Sqrt[2]*x] + Sqrt[33]*Log[5 + 4*x] - Sqrt[33]*Log
[8 - 20*x + 2*Sqrt[66]*Sqrt[1 + 2*x^2]])/8

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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]

1/8*(16*(x+5/4)^2-40*x-17)^(1/2)-5/8*arcsinh(x*2^(1/2))-1/8*33^(1/2)*arctanh(2/3
3*(-10*x+4)*33^(1/2)/(16*(x+5/4)^2-40*x-17)^(1/2))

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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]

1/8*sqrt(33)*arcsinh(5*sqrt(2)*x/abs(4*x + 5) - 2*sqrt(2)/abs(4*x + 5)) + 1/4*sq
rt(4*x^2 + 2) - 5/8*arcsinh(sqrt(2)*x)

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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]

-1/8*(8*x^2 - 5*(2*x - sqrt(4*x^2 + 2))*log(-2*x + sqrt(4*x^2 + 2)) - (2*sqrt(33
)*x - sqrt(33)*sqrt(4*x^2 + 2))*log((16*x^2 - 2*sqrt(4*x^2 + 2)*(4*x + sqrt(33)
+ 5) + sqrt(33)*(4*x + 5) + 20*x + 33)/(8*x^2 - sqrt(4*x^2 + 2)*(4*x + 5) + 10*x
)) - 4*sqrt(4*x^2 + 2)*x + 4)/(2*x - sqrt(4*x^2 + 2))

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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]

sqrt(2)*Integral(sqrt(2*x**2 + 1)/(4*x + 5), x)

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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]

1/16*sqrt(2)*(5*sqrt(2)*ln(-sqrt(2)*x + sqrt(2*x^2 + 1)) + sqrt(66)*ln(-abs(-4*s
qrt(2)*x - sqrt(66) - 5*sqrt(2) + 4*sqrt(2*x^2 + 1))/(4*sqrt(2)*x - sqrt(66) + 5
*sqrt(2) - 4*sqrt(2*x^2 + 1))) + 4*sqrt(2*x^2 + 1))