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

Optimal. Leaf size=174 \[ \frac{2 (22 x+21)}{5 \sqrt{2 x^2+3 x+1}}-\frac{1}{10} \sqrt{\frac{3}{5} \left (2065+653 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (17-4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )+\frac{1}{10} \sqrt{\frac{3}{5} \left (2065-653 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (17+4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right ) \]

[Out]

(2*(21 + 22*x))/(5*Sqrt[1 + 3*x + 2*x^2]) - (Sqrt[(3*(2065 + 653*Sqrt[10]))/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])])/10 + (Sqrt[(3*(2065 - 653*Sqrt[10]))/5]*ArcTanh[(3*(4 + Sqrt[
10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/1
0

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Rubi [A]  time = 0.724863, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 (22 x+21)}{5 \sqrt{2 x^2+3 x+1}}-\frac{1}{10} \sqrt{\frac{3}{5} \left (2065+653 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (17-4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )+\frac{1}{10} \sqrt{\frac{3}{5} \left (2065-653 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (17+4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(21 + 22*x))/(5*Sqrt[1 + 3*x + 2*x^2]) - (Sqrt[(3*(2065 + 653*Sqrt[10]))/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])])/10 + (Sqrt[(3*(2065 - 653*Sqrt[10]))/5]*ArcTanh[(3*(4 + Sqrt[
10]) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/1
0

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Rubi in Sympy [A]  time = 53.0049, size = 170, normalized size = 0.98 \[ \frac{2 \left (66 x + 63\right )}{15 \sqrt{2 x^{2} + 3 x + 1}} + \frac{\sqrt{10} \left (81 \sqrt{10} + 270\right ) \operatorname{atanh}{\left (\frac{x \left (-34 + 8 \sqrt{10}\right ) - 24 + 6 \sqrt{10}}{4 \sqrt{- 17 \sqrt{10} + 55} \sqrt{2 x^{2} + 3 x + 1}} \right )}}{300 \sqrt{- 17 \sqrt{10} + 55}} - \frac{\sqrt{10} \left (- 81 \sqrt{10} + 270\right ) \operatorname{atanh}{\left (\frac{x \left (-34 - 8 \sqrt{10}\right ) - 24 - 6 \sqrt{10}}{4 \sqrt{17 \sqrt{10} + 55} \sqrt{2 x^{2} + 3 x + 1}} \right )}}{300 \sqrt{17 \sqrt{10} + 55}} \]

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)**(3/2),x)

[Out]

2*(66*x + 63)/(15*sqrt(2*x**2 + 3*x + 1)) + sqrt(10)*(81*sqrt(10) + 270)*atanh((
x*(-34 + 8*sqrt(10)) - 24 + 6*sqrt(10))/(4*sqrt(-17*sqrt(10) + 55)*sqrt(2*x**2 +
3*x + 1)))/(300*sqrt(-17*sqrt(10) + 55)) - sqrt(10)*(-81*sqrt(10) + 270)*atanh(
(x*(-34 - 8*sqrt(10)) - 24 - 6*sqrt(10))/(4*sqrt(17*sqrt(10) + 55)*sqrt(2*x**2 +
3*x + 1)))/(300*sqrt(17*sqrt(10) + 55))

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Mathematica [A]  time = 1.7923, size = 252, normalized size = 1.45 \[ \frac{2 (22 x+21)}{5 \sqrt{2 x^2+3 x+1}}-\frac{9 \left (10+3 \sqrt{10}\right ) \log \left (-2 \sqrt{550-170 \sqrt{10}} \sqrt{2 x^2+3 x+1}-17 \sqrt{10} x+40 x-12 \sqrt{10}+30\right )}{10 \sqrt{550-170 \sqrt{10}}}-\frac{9 \left (3 \sqrt{10}-10\right ) \log \left (2 \sqrt{550+170 \sqrt{10}} \sqrt{2 x^2+3 x+1}+17 \sqrt{10} x+40 x+12 \sqrt{10}+30\right )}{10 \sqrt{550+170 \sqrt{10}}}+\frac{9 \left (10+3 \sqrt{10}\right ) \log \left (-3 x-\sqrt{10}+2\right )}{10 \sqrt{550-170 \sqrt{10}}}+\frac{9 \left (3 \sqrt{10}-10\right ) \log \left (-3 x+\sqrt{10}+2\right )}{10 \sqrt{550+170 \sqrt{10}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(21 + 22*x))/(5*Sqrt[1 + 3*x + 2*x^2]) + (9*(10 + 3*Sqrt[10])*Log[2 - Sqrt[10
] - 3*x])/(10*Sqrt[550 - 170*Sqrt[10]]) + (9*(-10 + 3*Sqrt[10])*Log[2 + Sqrt[10]
- 3*x])/(10*Sqrt[550 + 170*Sqrt[10]]) - (9*(10 + 3*Sqrt[10])*Log[30 - 12*Sqrt[1
0] + 40*x - 17*Sqrt[10]*x - 2*Sqrt[550 - 170*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2]])/(
10*Sqrt[550 - 170*Sqrt[10]]) - (9*(-10 + 3*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]])/(10*Sqrt[5
50 + 170*Sqrt[10]])

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Maple [B]  time = 0.026, size = 466, normalized size = 2.7 \[ \text{result too large to display} \]

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]

-1/20*(-8+10^(1/2))*10^(1/2)*(1/3/(55/9-17/9*10^(1/2))/(2*(x-2/3+1/3*10^(1/2))^2
+(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))+55/9-17/9*10^(1/2))^(1/2)-1/3*(17/3-4/
3*10^(1/2))/(55/9-17/9*10^(1/2))*(3+4*x)/(440/9-136/9*10^(1/2)-(17/3-4/3*10^(1/2
))^2)/(2*(x-2/3+1/3*10^(1/2))^2+(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))+55/9-17
/9*10^(1/2))^(1/2)-1/(55/9-17/9*10^(1/2))/(55-17*10^(1/2))^(1/2)*arctanh(9/2*(11
0/9-34/9*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*1
0^(1/2))^(1/2)))-1/20*(8+10^(1/2))*10^(1/2)*(1/3/(55/9+17/9*10^(1/2))/(2*(x-2/3-
1/3*10^(1/2))^2+(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))+55/9+17/9*10^(1/2))^(1/
2)-1/3*(17/3+4/3*10^(1/2))/(55/9+17/9*10^(1/2))*(3+4*x)/(440/9+136/9*10^(1/2)-(1
7/3+4/3*10^(1/2))^2)/(2*(x-2/3-1/3*10^(1/2))^2+(17/3+4/3*10^(1/2))*(x-2/3-1/3*10
^(1/2))+55/9+17/9*10^(1/2))^(1/2)-1/(55/9+17/9*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+1
7*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 = 0.804822, size = 902, normalized size = 5.18 \[ \text{result too large to display} \]

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]

-1/60*sqrt(10)*(588*sqrt(10)*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^
2 + 3*x + 1)) - 588*sqrt(10)*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^
2 + 3*x + 1)) + 2112*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x
+ 1)) + 2112*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) -
27*sqrt(10)*log(2/9*sqrt(10) + 2/3*sqrt(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)/(17*sqrt(10) + 55)^(3/2) - sqrt(10)*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)/(-17/9*sqrt(10) + 55/9)^(3/2) + 450*sqrt(10)/(17*sqrt(10)*sqrt(2*x^2 +
3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) - 450*sqrt(10)/(17*sqrt(10)*sqrt(2*x^2 + 3
*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) - 216*log(2/9*sqrt(10) + 2/3*sqrt(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)/(17*sqrt(10) + 55)^
(3/2) + 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/ab
s(6*x + 2*sqrt(10) - 4) + 17/18)/(-17/9*sqrt(10) + 55/9)^(3/2) + 1656/(17*sqrt(1
0)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) + 1656/(17*sqrt(10)*sqrt(2*
x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)))

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Fricas [A]  time = 0.288493, size = 763, normalized size = 4.39 \[ \text{result too large to display} \]

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="fricas")

[Out]

1/10*(72*x^2 + (sqrt(3/10)*sqrt(2*x^2 + 3*x + 1)*sqrt(sqrt(10)*(413*sqrt(10) + 1
306))*(3*x + 2) - 2*sqrt(3/10)*(2*x^2 + 3*x + 1)*sqrt(sqrt(10)*(413*sqrt(10) + 1
306)))*log(-(5*sqrt(3/10)*(13*sqrt(10)*x - 41*x)*sqrt(sqrt(10)*(413*sqrt(10) + 1
306)) + 9*sqrt(10)*(x + 1) - 9*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 45*x)/x) - (sqrt
(3/10)*sqrt(2*x^2 + 3*x + 1)*sqrt(sqrt(10)*(413*sqrt(10) + 1306))*(3*x + 2) - 2*
sqrt(3/10)*(2*x^2 + 3*x + 1)*sqrt(sqrt(10)*(413*sqrt(10) + 1306)))*log((5*sqrt(3
/10)*(13*sqrt(10)*x - 41*x)*sqrt(sqrt(10)*(413*sqrt(10) + 1306)) - 9*sqrt(10)*(x
+ 1) + 9*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 45*x)/x) + (sqrt(3/10)*sqrt(2*x^2 + 3
*x + 1)*sqrt(sqrt(10)*(413*sqrt(10) - 1306))*(3*x + 2) - 2*sqrt(3/10)*(2*x^2 + 3
*x + 1)*sqrt(sqrt(10)*(413*sqrt(10) - 1306)))*log(-(5*sqrt(3/10)*(13*sqrt(10)*x
+ 41*x)*sqrt(sqrt(10)*(413*sqrt(10) - 1306)) + 9*sqrt(10)*(x + 1) - 9*sqrt(10)*s
qrt(2*x^2 + 3*x + 1) - 45*x)/x) - (sqrt(3/10)*sqrt(2*x^2 + 3*x + 1)*sqrt(sqrt(10
)*(413*sqrt(10) - 1306))*(3*x + 2) - 2*sqrt(3/10)*(2*x^2 + 3*x + 1)*sqrt(sqrt(10
)*(413*sqrt(10) - 1306)))*log((5*sqrt(3/10)*(13*sqrt(10)*x + 41*x)*sqrt(sqrt(10)
*(413*sqrt(10) - 1306)) - 9*sqrt(10)*(x + 1) + 9*sqrt(10)*sqrt(2*x^2 + 3*x + 1)
+ 45*x)/x) - 76*sqrt(2*x^2 + 3*x + 1)*x + 76*x)/(4*x^2 - sqrt(2*x^2 + 3*x + 1)*(
3*x + 2) + 6*x + 2)

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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

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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)*(2*x^2 + 3*x + 1)^(3/2)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError