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

Optimal. Leaf size=166 \[ -\frac{2 (14 x+15)}{17 \sqrt{-2 x^2+3 x+1}}-\frac{9}{2} \sqrt{\frac{1}{5} \left (\sqrt{10}-3\right )} \tan ^{-1}\left (\frac{\left (1+4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )+\frac{9}{2} \sqrt{\frac{1}{5} \left (3+\sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (1-4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right ) \]

[Out]

(-2*(15 + 14*x))/(17*Sqrt[1 + 3*x - 2*x^2]) - (9*Sqrt[(-3 + Sqrt[10])/5]*ArcTan[
(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x
^2])])/2 + (9*Sqrt[(3 + 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

_______________________________________________________________________________________

Rubi [A]  time = 0.581049, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 (14 x+15)}{17 \sqrt{-2 x^2+3 x+1}}-\frac{9}{2} \sqrt{\frac{1}{5} \left (\sqrt{10}-3\right )} \tan ^{-1}\left (\frac{\left (1+4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )+\frac{9}{2} \sqrt{\frac{1}{5} \left (3+\sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (1-4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(15 + 14*x))/(17*Sqrt[1 + 3*x - 2*x^2]) - (9*Sqrt[(-3 + Sqrt[10])/5]*ArcTan[
(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x
^2])])/2 + (9*Sqrt[(3 + 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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 50.5332, size = 168, normalized size = 1.01 \[ - \frac{2 \left (14 x + 15\right )}{17 \sqrt{- 2 x^{2} + 3 x + 1}} - \frac{9 \sqrt{10} \left (- 2 \sqrt{10} + 4\right ) \operatorname{atan}{\left (\frac{x \left (- 8 \sqrt{10} - 2\right ) - 24 + 6 \sqrt{10}}{4 \sqrt{1 + \sqrt{10}} \sqrt{- 2 x^{2} + 3 x + 1}} \right )}}{40 \sqrt{1 + \sqrt{10}}} - \frac{9 \sqrt{10} \left (4 + 2 \sqrt{10}\right ) \operatorname{atanh}{\left (\frac{x \left (-2 + 8 \sqrt{10}\right ) - 24 - 6 \sqrt{10}}{4 \sqrt{-1 + \sqrt{10}} \sqrt{- 2 x^{2} + 3 x + 1}} \right )}}{40 \sqrt{-1 + \sqrt{10}}} \]

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)**(3/2),x)

[Out]

-2*(14*x + 15)/(17*sqrt(-2*x**2 + 3*x + 1)) - 9*sqrt(10)*(-2*sqrt(10) + 4)*atan(
(x*(-8*sqrt(10) - 2) - 24 + 6*sqrt(10))/(4*sqrt(1 + sqrt(10))*sqrt(-2*x**2 + 3*x
+ 1)))/(40*sqrt(1 + sqrt(10))) - 9*sqrt(10)*(4 + 2*sqrt(10))*atanh((x*(-2 + 8*s
qrt(10)) - 24 - 6*sqrt(10))/(4*sqrt(-1 + sqrt(10))*sqrt(-2*x**2 + 3*x + 1)))/(40
*sqrt(-1 + sqrt(10)))

_______________________________________________________________________________________

Mathematica [A]  time = 1.52075, size = 263, normalized size = 1.58 \[ \frac{1}{170} \left (-\frac{280 x}{\sqrt{-2 x^2+3 x+1}}-\frac{300}{\sqrt{-2 x^2+3 x+1}}+51 \sqrt{10 \left (1+\sqrt{10}\right )} \log \left (2 \sqrt{10 \left (\sqrt{10}-1\right )} \sqrt{-2 x^2+3 x+1}+\sqrt{10} x-40 x+12 \sqrt{10}+30\right )+255 \sqrt{1+\sqrt{10}} \log \left (2 \sqrt{10 \left (\sqrt{10}-1\right )} \sqrt{-2 x^2+3 x+1}+\sqrt{10} x-40 x+12 \sqrt{10}+30\right )+\frac{153 \left (\sqrt{10}-5\right ) \tan ^{-1}\left (\frac{4 \sqrt{10} x+x-3 \sqrt{10}+12}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )}{\sqrt{1+\sqrt{10}}}-51 \sqrt{1+\sqrt{10}} \left (5+\sqrt{10}\right ) \log \left (-3 x+\sqrt{10}+2\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(-300/Sqrt[1 + 3*x - 2*x^2] - (280*x)/Sqrt[1 + 3*x - 2*x^2] + (153*(-5 + 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]] - 51*Sqrt[1 + Sqrt[10]]*(5 + Sqrt[10])*Log[2 +
Sqrt[10] - 3*x] + 255*Sqrt[1 + Sqrt[10]]*Log[30 + 12*Sqrt[10] - 40*x + Sqrt[10]*
x + 2*Sqrt[10*(-1 + Sqrt[10])]*Sqrt[1 + 3*x - 2*x^2]] + 51*Sqrt[10*(1 + Sqrt[10]
)]*Log[30 + 12*Sqrt[10] - 40*x + Sqrt[10]*x + 2*Sqrt[10*(-1 + Sqrt[10])]*Sqrt[1
+ 3*x - 2*x^2]])/170

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 760, normalized size = 4.6 \[ \text{result too large to display} \]

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)^(3/2),x)

[Out]

26/255*10^(1/2)/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2)
)*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)+32/765/(-1/9-1/9*10^(1/2))/(-2*(x
-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(
1/2)*10^(1/2)*x-62/153/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*1
0^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)*x+7/51/(-1/9-1/9*10^(1/2))
/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(
1/2))^(1/2)+2/5*10^(1/2)/(-1/9-1/9*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/(-1/9-1/9*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))-26/255*10^(1/2)
/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10
^(1/2))-1/9+1/9*10^(1/2))^(1/2)-32/765/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/
2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*10^(1/2)*x
-62/153/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3
-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*x+7/51/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3
*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)+2/5
*10^(1/2)/(-1/9+1/9*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/(-1/9
+1/9*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))

_______________________________________________________________________________________

Maxima [A]  time = 0.806922, size = 915, normalized size = 5.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/340*sqrt(10)*(124*sqrt(10)*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 +
3*x + 1)) - 124*sqrt(10)*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x
+ 1)) + 153*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(10)*sqrt(sqrt(10
) + 1) + sqrt(sqrt(10) + 1)) - 128*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*
x^2 + 3*x + 1)) - 128*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1
)) - 1224*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(10)*sqrt(sqrt(10) + 1) + sq
rt(sqrt(10) + 1)) + 153*sqrt(10)*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(10) - 1)^(3/2) - 42*sqrt(10)
/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) + 42*sqrt(10)/(sqrt(
10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 1224*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*sq
rt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/18)/(sqrt(1
0) - 1)^(3/2) - 312/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) -
312/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)))

_______________________________________________________________________________________

Fricas [A]  time = 0.29416, size = 586, normalized size = 3.53 \[ -\frac{36 \, \sqrt{\frac{1}{2}}{\left (4 \, x^{2} + \sqrt{-2 \, x^{2} + 3 \, x + 1}{\left (3 \, x + 2\right )} - 6 \, x - 2\right )} \sqrt{-\sqrt{10}{\left (3 \, \sqrt{10} - 10\right )}} \arctan \left (-\frac{\sqrt{\frac{1}{2}}{\left (\sqrt{10} x + 3 \, x\right )} \sqrt{-\sqrt{10}{\left (3 \, \sqrt{10} - 10\right )}}}{\sqrt{10}{\left (x + 1\right )} - x \sqrt{\frac{\sqrt{10}{\left (15 \, x^{2} + \sqrt{10}{\left (3 \, x^{2} + 5 \, x + 2\right )} - 2 \, \sqrt{-2 \, x^{2} + 3 \, x + 1}{\left (\sqrt{10}{\left (x + 1\right )} + 5 \, x\right )} + 10 \, x\right )}}{x^{2}}} - \sqrt{10} \sqrt{-2 \, x^{2} + 3 \, x + 1} + 5 \, x}\right ) + 9 \, \sqrt{\frac{1}{2}}{\left (4 \, x^{2} + \sqrt{-2 \, x^{2} + 3 \, x + 1}{\left (3 \, x + 2\right )} - 6 \, x - 2\right )} \sqrt{\sqrt{10}{\left (3 \, \sqrt{10} + 10\right )}} \log \left (-\frac{9 \,{\left (\sqrt{\frac{1}{2}}{\left (\sqrt{10} x - 3 \, x\right )} \sqrt{\sqrt{10}{\left (3 \, \sqrt{10} + 10\right )}} + \sqrt{10}{\left (x + 1\right )} - \sqrt{10} \sqrt{-2 \, x^{2} + 3 \, x + 1} - 5 \, x\right )}}{x}\right ) - 9 \, \sqrt{\frac{1}{2}}{\left (4 \, x^{2} + \sqrt{-2 \, x^{2} + 3 \, x + 1}{\left (3 \, x + 2\right )} - 6 \, x - 2\right )} \sqrt{\sqrt{10}{\left (3 \, \sqrt{10} + 10\right )}} \log \left (\frac{9 \,{\left (\sqrt{\frac{1}{2}}{\left (\sqrt{10} x - 3 \, x\right )} \sqrt{\sqrt{10}{\left (3 \, \sqrt{10} + 10\right )}} - \sqrt{10}{\left (x + 1\right )} + \sqrt{10} \sqrt{-2 \, x^{2} + 3 \, x + 1} + 5 \, x\right )}}{x}\right ) + 120 \, x^{2} + 20 \, \sqrt{-2 \, x^{2} + 3 \, x + 1} x - 20 \, x}{10 \,{\left (4 \, x^{2} + \sqrt{-2 \, x^{2} + 3 \, x + 1}{\left (3 \, x + 2\right )} - 6 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(36*sqrt(1/2)*(4*x^2 + sqrt(-2*x^2 + 3*x + 1)*(3*x + 2) - 6*x - 2)*sqrt(-s
qrt(10)*(3*sqrt(10) - 10))*arctan(-sqrt(1/2)*(sqrt(10)*x + 3*x)*sqrt(-sqrt(10)*(
3*sqrt(10) - 10))/(sqrt(10)*(x + 1) - x*sqrt(sqrt(10)*(15*x^2 + sqrt(10)*(3*x^2
+ 5*x + 2) - 2*sqrt(-2*x^2 + 3*x + 1)*(sqrt(10)*(x + 1) + 5*x) + 10*x)/x^2) - sq
rt(10)*sqrt(-2*x^2 + 3*x + 1) + 5*x)) + 9*sqrt(1/2)*(4*x^2 + sqrt(-2*x^2 + 3*x +
1)*(3*x + 2) - 6*x - 2)*sqrt(sqrt(10)*(3*sqrt(10) + 10))*log(-9*(sqrt(1/2)*(sqr
t(10)*x - 3*x)*sqrt(sqrt(10)*(3*sqrt(10) + 10)) + sqrt(10)*(x + 1) - sqrt(10)*sq
rt(-2*x^2 + 3*x + 1) - 5*x)/x) - 9*sqrt(1/2)*(4*x^2 + sqrt(-2*x^2 + 3*x + 1)*(3*
x + 2) - 6*x - 2)*sqrt(sqrt(10)*(3*sqrt(10) + 10))*log(9*(sqrt(1/2)*(sqrt(10)*x
- 3*x)*sqrt(sqrt(10)*(3*sqrt(10) + 10)) - sqrt(10)*(x + 1) + sqrt(10)*sqrt(-2*x^
2 + 3*x + 1) + 5*x)/x) + 120*x^2 + 20*sqrt(-2*x^2 + 3*x + 1)*x - 20*x)/(4*x^2 +
sqrt(-2*x^2 + 3*x + 1)*(3*x + 2) - 6*x - 2)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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