3.28 \(\int \frac{2+x}{\left (2+4 x-3 x^2\right ) \sqrt{1+3 x+2 x^2}} \, dx\)

Optimal. Leaf size=151 \[ \frac{1}{2} \sqrt{1-\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left (\frac{\left (17+4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )-\frac{1}{2} \sqrt{1+\frac{7 \sqrt{\frac{2}{5}}}{5}} \tanh ^{-1}\left (\frac{\left (17-4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right ) \]

[Out]

-(Sqrt[1 + (7*Sqrt[2/5])/5]*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*
Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/2 + (Sqrt[1 - (7*Sqrt[2/5])/5]*A
rcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1
+ 3*x + 2*x^2])])/2

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Rubi [A]  time = 0.601403, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{10} \sqrt{25-7 \sqrt{10}} \tanh ^{-1}\left (\frac{\left (17+4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )-\frac{1}{10} \sqrt{25+7 \sqrt{10}} \tanh ^{-1}\left (\frac{\left (17-4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x)/((2 + 4*x - 3*x^2)*Sqrt[1 + 3*x + 2*x^2]),x]

[Out]

-(Sqrt[25 + 7*Sqrt[10]]*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*Sqrt
[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/10 + (Sqrt[25 - 7*Sqrt[10]]*ArcTanh[
(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x
+ 2*x^2])])/10

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Rubi in Sympy [A]  time = 40.304, size = 150, normalized size = 0.99 \[ \frac{\sqrt{10} \left (- 2 \sqrt{10} + 16\right ) \operatorname{atanh}{\left (\frac{x \left (-34 + 8 \sqrt{10}\right ) - 24 + 6 \sqrt{10}}{4 \sqrt{- 17 \sqrt{10} + 55} \sqrt{2 x^{2} + 3 x + 1}} \right )}}{40 \sqrt{- 17 \sqrt{10} + 55}} - \frac{\sqrt{10} \left (2 \sqrt{10} + 16\right ) \operatorname{atanh}{\left (\frac{x \left (-34 - 8 \sqrt{10}\right ) - 24 - 6 \sqrt{10}}{4 \sqrt{17 \sqrt{10} + 55} \sqrt{2 x^{2} + 3 x + 1}} \right )}}{40 \sqrt{17 \sqrt{10} + 55}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+x)/(-3*x**2+4*x+2)/(2*x**2+3*x+1)**(1/2),x)

[Out]

sqrt(10)*(-2*sqrt(10) + 16)*atanh((x*(-34 + 8*sqrt(10)) - 24 + 6*sqrt(10))/(4*sq
rt(-17*sqrt(10) + 55)*sqrt(2*x**2 + 3*x + 1)))/(40*sqrt(-17*sqrt(10) + 55)) - sq
rt(10)*(2*sqrt(10) + 16)*atanh((x*(-34 - 8*sqrt(10)) - 24 - 6*sqrt(10))/(4*sqrt(
17*sqrt(10) + 55)*sqrt(2*x**2 + 3*x + 1)))/(40*sqrt(17*sqrt(10) + 55))

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Mathematica [A]  time = 1.253, size = 220, normalized size = 1.46 \[ \frac{\frac{\left (\sqrt{10}-8\right ) \log \left (-2 \sqrt{550-170 \sqrt{10}} \sqrt{2 x^2+3 x+1}-17 \sqrt{10} x+40 x-12 \sqrt{10}+30\right )}{\sqrt{55-17 \sqrt{10}}}+\frac{\left (8+\sqrt{10}\right ) \log \left (2 \sqrt{550+170 \sqrt{10}} \sqrt{2 x^2+3 x+1}+17 \sqrt{10} x+40 x+12 \sqrt{10}+30\right )}{\sqrt{55+17 \sqrt{10}}}-\frac{\left (\sqrt{10}-8\right ) \log \left (-3 x-\sqrt{10}+2\right )}{\sqrt{55-17 \sqrt{10}}}-\frac{\left (8+\sqrt{10}\right ) \log \left (-3 x+\sqrt{10}+2\right )}{\sqrt{55+17 \sqrt{10}}}}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x)/((2 + 4*x - 3*x^2)*Sqrt[1 + 3*x + 2*x^2]),x]

[Out]

(-(((-8 + Sqrt[10])*Log[2 - Sqrt[10] - 3*x])/Sqrt[55 - 17*Sqrt[10]]) - ((8 + Sqr
t[10])*Log[2 + Sqrt[10] - 3*x])/Sqrt[55 + 17*Sqrt[10]] + ((-8 + Sqrt[10])*Log[30
- 12*Sqrt[10] + 40*x - 17*Sqrt[10]*x - 2*Sqrt[550 - 170*Sqrt[10]]*Sqrt[1 + 3*x
+ 2*x^2]])/Sqrt[55 - 17*Sqrt[10]] + ((8 + Sqrt[10])*Log[30 + 12*Sqrt[10] + 40*x
+ 17*Sqrt[10]*x + 2*Sqrt[550 + 170*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2]])/Sqrt[55 + 1
7*Sqrt[10]])/(2*Sqrt[10])

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Maple [A]  time = 0.06, size = 186, normalized size = 1.2 \[{\frac{ \left ( -8+\sqrt{10} \right ) \sqrt{10}}{20\,\sqrt{55-17\,\sqrt{10}}}{\it Artanh} \left ({\frac{9}{2\,\sqrt{55-17\,\sqrt{10}}} \left ({\frac{110}{9}}-{\frac{34\,\sqrt{10}}{9}}+ \left ({\frac{17}{3}}-{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}+{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{18\, \left ( x-2/3+1/3\,\sqrt{10} \right ) ^{2}+9\, \left ({\frac{17}{3}}-4/3\,\sqrt{10} \right ) \left ( x-2/3+1/3\,\sqrt{10} \right ) +55-17\,\sqrt{10}}}}} \right ) }+{\frac{ \left ( 8+\sqrt{10} \right ) \sqrt{10}}{20\,\sqrt{55+17\,\sqrt{10}}}{\it Artanh} \left ({\frac{9}{2\,\sqrt{55+17\,\sqrt{10}}} \left ({\frac{110}{9}}+{\frac{34\,\sqrt{10}}{9}}+ \left ({\frac{17}{3}}+{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}-{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{18\, \left ( x-2/3-1/3\,\sqrt{10} \right ) ^{2}+9\, \left ({\frac{17}{3}}+4/3\,\sqrt{10} \right ) \left ( x-2/3-1/3\,\sqrt{10} \right ) +55+17\,\sqrt{10}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(-3*x^2+4*x+2)/(2*x^2+3*x+1)^(1/2),x)

[Out]

1/20*(-8+10^(1/2))*10^(1/2)/(55-17*10^(1/2))^(1/2)*arctanh(9/2*(110/9-34/9*10^(1
/2)+(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2)))/(55-17*10^(1/2))^(1/2)/(18*(x-2/3+
1/3*10^(1/2))^2+9*(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))+55-17*10^(1/2))^(1/2)
)+1/20*(8+10^(1/2))*10^(1/2)/(55+17*10^(1/2))^(1/2)*arctanh(9/2*(110/9+34/9*10^(
1/2)+(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(55+17*10^(1/2))^(1/2)/(18*(x-2/3
-1/3*10^(1/2))^2+9*(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))+55+17*10^(1/2))^(1/2
))

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Maxima [A]  time = 0.799835, size = 490, normalized size = 3.25 \[ \frac{1}{60} \, \sqrt{10}{\left (\frac{3 \, \sqrt{10} \log \left (\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{2 \, x^{2} + 3 \, x + 1} \sqrt{17 \, \sqrt{10} + 55}}{3 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{34 \, \sqrt{10}}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{110}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{17}{18}\right )}{\sqrt{17 \, \sqrt{10} + 55}} + \frac{\sqrt{10} \log \left (-\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{2 \, x^{2} + 3 \, x + 1} \sqrt{-\frac{17}{9} \, \sqrt{10} + \frac{55}{9}}}{{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} - \frac{34 \, \sqrt{10}}{9 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{110}{9 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{17}{18}\right )}{\sqrt{-\frac{17}{9} \, \sqrt{10} + \frac{55}{9}}} + \frac{24 \, \log \left (\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{2 \, x^{2} + 3 \, x + 1} \sqrt{17 \, \sqrt{10} + 55}}{3 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{34 \, \sqrt{10}}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{110}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{17}{18}\right )}{\sqrt{17 \, \sqrt{10} + 55}} - \frac{8 \, \log \left (-\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{2 \, x^{2} + 3 \, x + 1} \sqrt{-\frac{17}{9} \, \sqrt{10} + \frac{55}{9}}}{{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} - \frac{34 \, \sqrt{10}}{9 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{110}{9 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{17}{18}\right )}{\sqrt{-\frac{17}{9} \, \sqrt{10} + \frac{55}{9}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x + 2)/((3*x^2 - 4*x - 2)*sqrt(2*x^2 + 3*x + 1)),x, algorithm="maxima")

[Out]

1/60*sqrt(10)*(3*sqrt(10)*log(2/9*sqrt(10) + 2/3*sqrt(2*x^2 + 3*x + 1)*sqrt(17*s
qrt(10) + 55)/abs(6*x - 2*sqrt(10) - 4) + 34/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4
) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/sqrt(17*sqrt(10) + 55) + sqrt(10)*l
og(-2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*sqrt(-17/9*sqrt(10) + 55/9)/abs(6*x +
2*sqrt(10) - 4) - 34/9*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 110/9/abs(6*x + 2*s
qrt(10) - 4) + 17/18)/sqrt(-17/9*sqrt(10) + 55/9) + 24*log(2/9*sqrt(10) + 2/3*sq
rt(2*x^2 + 3*x + 1)*sqrt(17*sqrt(10) + 55)/abs(6*x - 2*sqrt(10) - 4) + 34/9*sqrt
(10)/abs(6*x - 2*sqrt(10) - 4) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/sqrt(1
7*sqrt(10) + 55) - 8*log(-2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*sqrt(-17/9*sqrt
(10) + 55/9)/abs(6*x + 2*sqrt(10) - 4) - 34/9*sqrt(10)/abs(6*x + 2*sqrt(10) - 4)
+ 110/9/abs(6*x + 2*sqrt(10) - 4) + 17/18)/sqrt(-17/9*sqrt(10) + 55/9))

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Fricas [A]  time = 0.285419, size = 431, normalized size = 2.85 \[ -\frac{1}{10} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{10}{\left (5 \, \sqrt{10} + 14\right )}} \log \left (-\frac{\sqrt{\frac{1}{2}}{\left (2 \, \sqrt{10} x - 5 \, x\right )} \sqrt{\sqrt{10}{\left (5 \, \sqrt{10} + 14\right )}} + 3 \, \sqrt{10}{\left (x + 1\right )} - 3 \, \sqrt{10} \sqrt{2 \, x^{2} + 3 \, x + 1} + 15 \, x}{x}\right ) + \frac{1}{10} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{10}{\left (5 \, \sqrt{10} + 14\right )}} \log \left (\frac{\sqrt{\frac{1}{2}}{\left (2 \, \sqrt{10} x - 5 \, x\right )} \sqrt{\sqrt{10}{\left (5 \, \sqrt{10} + 14\right )}} - 3 \, \sqrt{10}{\left (x + 1\right )} + 3 \, \sqrt{10} \sqrt{2 \, x^{2} + 3 \, x + 1} - 15 \, x}{x}\right ) - \frac{1}{10} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{10}{\left (5 \, \sqrt{10} - 14\right )}} \log \left (-\frac{\sqrt{\frac{1}{2}}{\left (2 \, \sqrt{10} x + 5 \, x\right )} \sqrt{\sqrt{10}{\left (5 \, \sqrt{10} - 14\right )}} + 3 \, \sqrt{10}{\left (x + 1\right )} - 3 \, \sqrt{10} \sqrt{2 \, x^{2} + 3 \, x + 1} - 15 \, x}{x}\right ) + \frac{1}{10} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{10}{\left (5 \, \sqrt{10} - 14\right )}} \log \left (\frac{\sqrt{\frac{1}{2}}{\left (2 \, \sqrt{10} x + 5 \, x\right )} \sqrt{\sqrt{10}{\left (5 \, \sqrt{10} - 14\right )}} - 3 \, \sqrt{10}{\left (x + 1\right )} + 3 \, \sqrt{10} \sqrt{2 \, x^{2} + 3 \, x + 1} + 15 \, x}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x + 2)/((3*x^2 - 4*x - 2)*sqrt(2*x^2 + 3*x + 1)),x, algorithm="fricas")

[Out]

-1/10*sqrt(1/2)*sqrt(sqrt(10)*(5*sqrt(10) + 14))*log(-(sqrt(1/2)*(2*sqrt(10)*x -
5*x)*sqrt(sqrt(10)*(5*sqrt(10) + 14)) + 3*sqrt(10)*(x + 1) - 3*sqrt(10)*sqrt(2*
x^2 + 3*x + 1) + 15*x)/x) + 1/10*sqrt(1/2)*sqrt(sqrt(10)*(5*sqrt(10) + 14))*log(
(sqrt(1/2)*(2*sqrt(10)*x - 5*x)*sqrt(sqrt(10)*(5*sqrt(10) + 14)) - 3*sqrt(10)*(x
+ 1) + 3*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 15*x)/x) - 1/10*sqrt(1/2)*sqrt(sqrt(1
0)*(5*sqrt(10) - 14))*log(-(sqrt(1/2)*(2*sqrt(10)*x + 5*x)*sqrt(sqrt(10)*(5*sqrt
(10) - 14)) + 3*sqrt(10)*(x + 1) - 3*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 15*x)/x) +
1/10*sqrt(1/2)*sqrt(sqrt(10)*(5*sqrt(10) - 14))*log((sqrt(1/2)*(2*sqrt(10)*x +
5*x)*sqrt(sqrt(10)*(5*sqrt(10) - 14)) - 3*sqrt(10)*(x + 1) + 3*sqrt(10)*sqrt(2*x
^2 + 3*x + 1) + 15*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{3 x^{2} \sqrt{2 x^{2} + 3 x + 1} - 4 x \sqrt{2 x^{2} + 3 x + 1} - 2 \sqrt{2 x^{2} + 3 x + 1}}\, dx - \int \frac{2}{3 x^{2} \sqrt{2 x^{2} + 3 x + 1} - 4 x \sqrt{2 x^{2} + 3 x + 1} - 2 \sqrt{2 x^{2} + 3 x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+x)/(-3*x**2+4*x+2)/(2*x**2+3*x+1)**(1/2),x)

[Out]

-Integral(x/(3*x**2*sqrt(2*x**2 + 3*x + 1) - 4*x*sqrt(2*x**2 + 3*x + 1) - 2*sqrt
(2*x**2 + 3*x + 1)), x) - Integral(2/(3*x**2*sqrt(2*x**2 + 3*x + 1) - 4*x*sqrt(2
*x**2 + 3*x + 1) - 2*sqrt(2*x**2 + 3*x + 1)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x + 2)/((3*x^2 - 4*x - 2)*sqrt(2*x^2 + 3*x + 1)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError