3.2568 \(\int \frac{(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{\sqrt{2 x+3} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}+\frac{\sqrt{2 x+3} (2529 x+2090)}{6 \left (3 x^2+5 x+2\right )}+966 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-1247 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

-(Sqrt[3 + 2*x]*(121 + 139*x))/(6*(2 + 5*x + 3*x^2)^2) + (Sqrt[3 + 2*x]*(2090 +
2529*x))/(6*(2 + 5*x + 3*x^2)) + 966*ArcTanh[Sqrt[3 + 2*x]] - 1247*Sqrt[3/5]*Arc
Tanh[Sqrt[3/5]*Sqrt[3 + 2*x]]

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Rubi [A]  time = 0.188675, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{\sqrt{2 x+3} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}+\frac{\sqrt{2 x+3} (2529 x+2090)}{6 \left (3 x^2+5 x+2\right )}+966 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-1247 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(3 + 2*x)^(3/2))/(2 + 5*x + 3*x^2)^3,x]

[Out]

-(Sqrt[3 + 2*x]*(121 + 139*x))/(6*(2 + 5*x + 3*x^2)^2) + (Sqrt[3 + 2*x]*(2090 +
2529*x))/(6*(2 + 5*x + 3*x^2)) + 966*ArcTanh[Sqrt[3 + 2*x]] - 1247*Sqrt[3/5]*Arc
Tanh[Sqrt[3/5]*Sqrt[3 + 2*x]]

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Rubi in Sympy [A]  time = 35.1458, size = 87, normalized size = 0.87 \[ - \frac{\sqrt{2 x + 3} \left (139 x + 121\right )}{6 \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{\sqrt{2 x + 3} \left (12645 x + 10450\right )}{30 \left (3 x^{2} + 5 x + 2\right )} - \frac{1247 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{5} + 966 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**(3/2)/(3*x**2+5*x+2)**3,x)

[Out]

-sqrt(2*x + 3)*(139*x + 121)/(6*(3*x**2 + 5*x + 2)**2) + sqrt(2*x + 3)*(12645*x
+ 10450)/(30*(3*x**2 + 5*x + 2)) - 1247*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)
/5 + 966*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.355018, size = 98, normalized size = 0.98 \[ \frac{\sqrt{2 x+3} \left (2529 x^3+6305 x^2+5123 x+1353\right )}{2 \left (3 x^2+5 x+2\right )^2}-483 \log \left (1-\sqrt{2 x+3}\right )+483 \log \left (\sqrt{2 x+3}+1\right )-1247 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(3 + 2*x)^(3/2))/(2 + 5*x + 3*x^2)^3,x]

[Out]

(Sqrt[3 + 2*x]*(1353 + 5123*x + 6305*x^2 + 2529*x^3))/(2*(2 + 5*x + 3*x^2)^2) -
1247*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]] - 483*Log[1 - Sqrt[3 + 2*x]] + 4
83*Log[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.028, size = 124, normalized size = 1.2 \[ 3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+68\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-483\,\ln \left ( -1+\sqrt{3+2\,x} \right ) +18\,{\frac{1}{ \left ( 4+6\,x \right ) ^{2}} \left ({\frac{145\, \left ( 3+2\,x \right ) ^{3/2}}{2}}-{\frac{2345\,\sqrt{3+2\,x}}{18}} \right ) }-{\frac{1247\,\sqrt{15}}{5}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+68\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+483\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^(3/2)/(3*x^2+5*x+2)^3,x)

[Out]

3/(-1+(3+2*x)^(1/2))^2+68/(-1+(3+2*x)^(1/2))-483*ln(-1+(3+2*x)^(1/2))+18*(145/2*
(3+2*x)^(3/2)-2345/18*(3+2*x)^(1/2))/(4+6*x)^2-1247/5*arctanh(1/5*15^(1/2)*(3+2*
x)^(1/2))*15^(1/2)-3/(1+(3+2*x)^(1/2))^2+68/(1+(3+2*x)^(1/2))+483*ln(1+(3+2*x)^(
1/2))

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Maxima [A]  time = 0.79496, size = 180, normalized size = 1.8 \[ \frac{1247}{10} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{2529 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 10151 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 13115 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 5445 \, \sqrt{2 \, x + 3}}{9 \,{\left (2 \, x + 3\right )}^{4} - 48 \,{\left (2 \, x + 3\right )}^{3} + 94 \,{\left (2 \, x + 3\right )}^{2} - 160 \, x - 215} + 483 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 483 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^(3/2)*(x - 5)/(3*x^2 + 5*x + 2)^3,x, algorithm="maxima")

[Out]

1247/10*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)))
+ (2529*(2*x + 3)^(7/2) - 10151*(2*x + 3)^(5/2) + 13115*(2*x + 3)^(3/2) - 5445*
sqrt(2*x + 3))/(9*(2*x + 3)^4 - 48*(2*x + 3)^3 + 94*(2*x + 3)^2 - 160*x - 215) +
483*log(sqrt(2*x + 3) + 1) - 483*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.296042, size = 243, normalized size = 2.43 \[ \frac{\sqrt{5}{\left (966 \, \sqrt{5}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 966 \, \sqrt{5}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 1247 \, \sqrt{3}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} - 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + \sqrt{5}{\left (2529 \, x^{3} + 6305 \, x^{2} + 5123 \, x + 1353\right )} \sqrt{2 \, x + 3}\right )}}{10 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^(3/2)*(x - 5)/(3*x^2 + 5*x + 2)^3,x, algorithm="fricas")

[Out]

1/10*sqrt(5)*(966*sqrt(5)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(2*x + 3)
+ 1) - 966*sqrt(5)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(2*x + 3) - 1)
+ 1247*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log((sqrt(5)*(3*x + 7) - 5*s
qrt(3)*sqrt(2*x + 3))/(3*x + 2)) + sqrt(5)*(2529*x^3 + 6305*x^2 + 5123*x + 1353)
*sqrt(2*x + 3))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

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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((5-x)*(3+2*x)**(3/2)/(3*x**2+5*x+2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272978, size = 161, normalized size = 1.61 \[ \frac{1247}{10} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + \frac{2529 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 10151 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 13115 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 5445 \, \sqrt{2 \, x + 3}}{{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 483 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 483 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^(3/2)*(x - 5)/(3*x^2 + 5*x + 2)^3,x, algorithm="giac")

[Out]

1247/10*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*
x + 3))) + (2529*(2*x + 3)^(7/2) - 10151*(2*x + 3)^(5/2) + 13115*(2*x + 3)^(3/2)
- 5445*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19)^2 + 483*ln(sqrt(2*x + 3) + 1)
- 483*ln(abs(sqrt(2*x + 3) - 1))