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

Optimal. Leaf size=139 \[ \frac{1}{2} \sqrt{\sqrt{10}-\frac{13}{5}} \tan ^{-1}\left (\frac{\left (1+4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )+\frac{1}{2} \sqrt{\frac{13}{5}+\sqrt{10}} \tanh ^{-1}\left (\frac{\left (1-4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right ) \]

[Out]

(Sqrt[-13/5 + Sqrt[10]]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1
+ Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2 + (Sqrt[13/5 + Sqrt[10]]*ArcTanh[(3*(4 +
Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])
/2

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Rubi [A]  time = 0.484639, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{2} \sqrt{\frac{1}{5} \left (5 \sqrt{10}-13\right )} \tan ^{-1}\left (\frac{\left (1+4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )+\frac{1}{2} \sqrt{\frac{1}{5} \left (13+5 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (1-4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(-13 + 5*Sqrt[10])/5]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sq
rt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/2 + (Sqrt[(13 + 5*Sqrt[10])/5]*ArcTanh
[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqrt[1 + 3*x - 2
*x^2])])/2

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Rubi in Sympy [A]  time = 39.8429, size = 144, normalized size = 1.04 \[ - \frac{\sqrt{10} \left (- 2 \sqrt{10} + 16\right ) \operatorname{atan}{\left (\frac{x \left (- 8 \sqrt{10} - 2\right ) - 24 + 6 \sqrt{10}}{4 \sqrt{1 + \sqrt{10}} \sqrt{- 2 x^{2} + 3 x + 1}} \right )}}{40 \sqrt{1 + \sqrt{10}}} - \frac{\sqrt{10} \left (2 \sqrt{10} + 16\right ) \operatorname{atanh}{\left (\frac{x \left (-2 + 8 \sqrt{10}\right ) - 24 - 6 \sqrt{10}}{4 \sqrt{-1 + \sqrt{10}} \sqrt{- 2 x^{2} + 3 x + 1}} \right )}}{40 \sqrt{-1 + \sqrt{10}}} \]

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)*atan((x*(-8*sqrt(10) - 2) - 24 + 6*sqrt(10))/(4*sqr
t(1 + sqrt(10))*sqrt(-2*x**2 + 3*x + 1)))/(40*sqrt(1 + sqrt(10))) - sqrt(10)*(2*
sqrt(10) + 16)*atanh((x*(-2 + 8*sqrt(10)) - 24 - 6*sqrt(10))/(4*sqrt(-1 + sqrt(1
0))*sqrt(-2*x**2 + 3*x + 1)))/(40*sqrt(-1 + sqrt(10)))

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Mathematica [A]  time = 0.674782, size = 161, normalized size = 1.16 \[ \frac{-\sqrt{1+\sqrt{10}} \left (8+\sqrt{10}\right ) \left (\log \left (-3 x+\sqrt{10}+2\right )-\log \left (2 \sqrt{10 \left (\sqrt{10}-1\right )} \sqrt{-2 x^2+3 x+1}+\sqrt{10} x-40 x+12 \sqrt{10}+30\right )\right )-\left (\sqrt{10}-8\right ) \sqrt{\sqrt{10}-1} \tan ^{-1}\left (\frac{4 \sqrt{10} x+x-3 \sqrt{10}+12}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )}{6 \sqrt{10}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-((-8 + Sqrt[10])*Sqrt[-1 + Sqrt[10]]*ArcTan[(12 - 3*Sqrt[10] + x + 4*Sqrt[10]*
x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])]) - Sqrt[1 + Sqrt[10]]*(8 + Sqrt
[10])*(Log[2 + Sqrt[10] - 3*x] - Log[30 + 12*Sqrt[10] - 40*x + Sqrt[10]*x + 2*Sq
rt[10*(-1 + Sqrt[10])]*Sqrt[1 + 3*x - 2*x^2]]))/(6*Sqrt[10])

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Maple [B]  time = 0.079, size = 324, normalized size = 2.3 \[{\frac{2\,\sqrt{10}}{5\,\sqrt{1+\sqrt{10}}}\arctan \left ({\frac{9}{2\,\sqrt{1+\sqrt{10}}} \left ( -{\frac{2}{9}}-{\frac{2\,\sqrt{10}}{9}}+ \left ({\frac{1}{3}}+{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}+{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{-18\, \left ( x-2/3+1/3\,\sqrt{10} \right ) ^{2}+9\, \left ( 1/3+4/3\,\sqrt{10} \right ) \left ( x-2/3+1/3\,\sqrt{10} \right ) -1-\sqrt{10}}}}} \right ) }-{\frac{1}{2\,\sqrt{1+\sqrt{10}}}\arctan \left ({\frac{9}{2\,\sqrt{1+\sqrt{10}}} \left ( -{\frac{2}{9}}-{\frac{2\,\sqrt{10}}{9}}+ \left ({\frac{1}{3}}+{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}+{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{-18\, \left ( x-2/3+1/3\,\sqrt{10} \right ) ^{2}+9\, \left ( 1/3+4/3\,\sqrt{10} \right ) \left ( x-2/3+1/3\,\sqrt{10} \right ) -1-\sqrt{10}}}}} \right ) }+{\frac{2\,\sqrt{10}}{5\,\sqrt{-1+\sqrt{10}}}{\it Artanh} \left ({\frac{9}{2\,\sqrt{-1+\sqrt{10}}} \left ( -{\frac{2}{9}}+{\frac{2\,\sqrt{10}}{9}}+ \left ({\frac{1}{3}}-{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}-{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{-18\, \left ( x-2/3-1/3\,\sqrt{10} \right ) ^{2}+9\, \left ( 1/3-4/3\,\sqrt{10} \right ) \left ( x-2/3-1/3\,\sqrt{10} \right ) -1+\sqrt{10}}}}} \right ) }+{\frac{1}{2\,\sqrt{-1+\sqrt{10}}}{\it Artanh} \left ({\frac{9}{2\,\sqrt{-1+\sqrt{10}}} \left ( -{\frac{2}{9}}+{\frac{2\,\sqrt{10}}{9}}+ \left ({\frac{1}{3}}-{\frac{4\,\sqrt{10}}{3}} \right ) \left ( x-{\frac{2}{3}}-{\frac{\sqrt{10}}{3}} \right ) \right ){\frac{1}{\sqrt{-18\, \left ( x-2/3-1/3\,\sqrt{10} \right ) ^{2}+9\, \left ( 1/3-4/3\,\sqrt{10} \right ) \left ( x-2/3-1/3\,\sqrt{10} \right ) -1+\sqrt{10}}}}} \right ) } \]

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]

2/5*10^(1/2)/(1+10^(1/2))^(1/2)*arctan(9/2*(-2/9-2/9*10^(1/2)+(1/3+4/3*10^(1/2))
*(x-2/3+1/3*10^(1/2)))/(1+10^(1/2))^(1/2)/(-18*(x-2/3+1/3*10^(1/2))^2+9*(1/3+4/3
*10^(1/2))*(x-2/3+1/3*10^(1/2))-1-10^(1/2))^(1/2))-1/2/(1+10^(1/2))^(1/2)*arctan
(9/2*(-2/9-2/9*10^(1/2)+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2)))/(1+10^(1/2))^(1
/2)/(-18*(x-2/3+1/3*10^(1/2))^2+9*(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1-10^(
1/2))^(1/2))+2/5*10^(1/2)/(-1+10^(1/2))^(1/2)*arctanh(9/2*(-2/9+2/9*10^(1/2)+(1/
3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(-1+10^(1/2))^(1/2)/(-18*(x-2/3-1/3*10^(1/
2))^2+9*(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1+10^(1/2))^(1/2))+1/2/(-1+10^(1
/2))^(1/2)*arctanh(9/2*(-2/9+2/9*10^(1/2)+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)
))/(-1+10^(1/2))^(1/2)/(-18*(x-2/3-1/3*10^(1/2))^2+9*(1/3-4/3*10^(1/2))*(x-2/3-1
/3*10^(1/2))-1+10^(1/2))^(1/2))

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Maxima [A]  time = 0.800441, size = 487, normalized size = 3.5 \[ -\frac{1}{20} \, \sqrt{10}{\left (\frac{\sqrt{10} \arcsin \left (\frac{8 \, \sqrt{17} \sqrt{10} x}{17 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{2 \, \sqrt{17} x}{17 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} - \frac{6 \, \sqrt{17} \sqrt{10}}{17 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{24 \, \sqrt{17}}{17 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}}\right )}{\sqrt{\sqrt{10} + 1}} - \frac{\sqrt{10} \log \left (-\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{-2 \, x^{2} + 3 \, x + 1} \sqrt{\sqrt{10} - 1}}{3 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{2 \, \sqrt{10}}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} - \frac{2}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{1}{18}\right )}{\sqrt{\sqrt{10} - 1}} - \frac{8 \, \arcsin \left (\frac{8 \, \sqrt{17} \sqrt{10} x}{17 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{2 \, \sqrt{17} x}{17 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} - \frac{6 \, \sqrt{17} \sqrt{10}}{17 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}} + \frac{24 \, \sqrt{17}}{17 \,{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}}\right )}{\sqrt{\sqrt{10} + 1}} - \frac{8 \, \log \left (-\frac{2}{9} \, \sqrt{10} + \frac{2 \, \sqrt{-2 \, x^{2} + 3 \, x + 1} \sqrt{\sqrt{10} - 1}}{3 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{2 \, \sqrt{10}}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} - \frac{2}{9 \,{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}} + \frac{1}{18}\right )}{\sqrt{\sqrt{10} - 1}}\right )} \]

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]

-1/20*sqrt(10)*(sqrt(10)*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) -
4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(17)*sqrt(10)/abs(6*x
+ 2*sqrt(10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/sqrt(sqrt(10) + 1)
- sqrt(10)*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10) - 1)/ab
s(6*x - 2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x -
2*sqrt(10) - 4) + 1/18)/sqrt(sqrt(10) - 1) - 8*arcsin(8/17*sqrt(17)*sqrt(10)*x/
abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqr
t(17)*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) -
4))/sqrt(sqrt(10) + 1) - 8*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(
sqrt(10) - 1)/abs(6*x - 2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4)
- 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/18)/sqrt(sqrt(10) - 1))

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Fricas [A]  time = 0.300685, size = 466, normalized size = 3.35 \[ 2 \, \sqrt{\frac{1}{10}} \sqrt{\sqrt{2}{\left (10 \, \sqrt{5} - 13 \, \sqrt{2}\right )}} \arctan \left (-\frac{\sqrt{\frac{1}{10}}{\left (7 \, \sqrt{5} x + 10 \, \sqrt{2} x\right )} \sqrt{\sqrt{2}{\left (10 \, \sqrt{5} - 13 \, \sqrt{2}\right )}}}{9 \,{\left (\sqrt{2}{\left (x + 1\right )} + \sqrt{5} x - x \sqrt{\frac{\sqrt{2}{\left (\sqrt{2}{\left (3 \, x^{2} + 5 \, x + 2\right )} + \sqrt{5}{\left (3 \, x^{2} + 2 \, x\right )} - 2 \, \sqrt{-2 \, x^{2} + 3 \, x + 1}{\left (\sqrt{2}{\left (x + 1\right )} + \sqrt{5} x\right )}\right )}}{x^{2}}} - \sqrt{2} \sqrt{-2 \, x^{2} + 3 \, x + 1}\right )}}\right ) - \frac{1}{2} \, \sqrt{\frac{1}{10}} \sqrt{\sqrt{2}{\left (10 \, \sqrt{5} + 13 \, \sqrt{2}\right )}} \log \left (-\frac{\sqrt{\frac{1}{10}}{\left (7 \, \sqrt{5} x - 10 \, \sqrt{2} x\right )} \sqrt{\sqrt{2}{\left (10 \, \sqrt{5} + 13 \, \sqrt{2}\right )}} + 9 \, \sqrt{2}{\left (x + 1\right )} - 9 \, \sqrt{5} x - 9 \, \sqrt{2} \sqrt{-2 \, x^{2} + 3 \, x + 1}}{x}\right ) + \frac{1}{2} \, \sqrt{\frac{1}{10}} \sqrt{\sqrt{2}{\left (10 \, \sqrt{5} + 13 \, \sqrt{2}\right )}} \log \left (\frac{\sqrt{\frac{1}{10}}{\left (7 \, \sqrt{5} x - 10 \, \sqrt{2} x\right )} \sqrt{\sqrt{2}{\left (10 \, \sqrt{5} + 13 \, \sqrt{2}\right )}} - 9 \, \sqrt{2}{\left (x + 1\right )} + 9 \, \sqrt{5} x + 9 \, \sqrt{2} \sqrt{-2 \, x^{2} + 3 \, x + 1}}{x}\right ) \]

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]

2*sqrt(1/10)*sqrt(sqrt(2)*(10*sqrt(5) - 13*sqrt(2)))*arctan(-1/9*sqrt(1/10)*(7*s
qrt(5)*x + 10*sqrt(2)*x)*sqrt(sqrt(2)*(10*sqrt(5) - 13*sqrt(2)))/(sqrt(2)*(x + 1
) + sqrt(5)*x - x*sqrt(sqrt(2)*(sqrt(2)*(3*x^2 + 5*x + 2) + sqrt(5)*(3*x^2 + 2*x
) - 2*sqrt(-2*x^2 + 3*x + 1)*(sqrt(2)*(x + 1) + sqrt(5)*x))/x^2) - sqrt(2)*sqrt(
-2*x^2 + 3*x + 1))) - 1/2*sqrt(1/10)*sqrt(sqrt(2)*(10*sqrt(5) + 13*sqrt(2)))*log
(-(sqrt(1/10)*(7*sqrt(5)*x - 10*sqrt(2)*x)*sqrt(sqrt(2)*(10*sqrt(5) + 13*sqrt(2)
)) + 9*sqrt(2)*(x + 1) - 9*sqrt(5)*x - 9*sqrt(2)*sqrt(-2*x^2 + 3*x + 1))/x) + 1/
2*sqrt(1/10)*sqrt(sqrt(2)*(10*sqrt(5) + 13*sqrt(2)))*log((sqrt(1/10)*(7*sqrt(5)*
x - 10*sqrt(2)*x)*sqrt(sqrt(2)*(10*sqrt(5) + 13*sqrt(2))) - 9*sqrt(2)*(x + 1) +
9*sqrt(5)*x + 9*sqrt(2)*sqrt(-2*x^2 + 3*x + 1))/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*sq
rt(-2*x**2 + 3*x + 1)), x) - Integral(2/(3*x**2*sqrt(-2*x**2 + 3*x + 1) - 4*x*sq
rt(-2*x**2 + 3*x + 1) - 2*sqrt(-2*x**2 + 3*x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x + 2}{{\left (3 \, x^{2} - 4 \, x - 2\right )} \sqrt{-2 \, x^{2} + 3 \, x + 1}}\,{d x} \]

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]

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