3.2407 \(\int (5-x) (3+2 x)^2 \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=110 \[ -\frac{1}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac{(3006 x+7969) \left (3 x^2+5 x+2\right )^{3/2}}{1620}+\frac{2267 (6 x+5) \sqrt{3 x^2+5 x+2}}{2592}-\frac{2267 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{5184 \sqrt{3}} \]

[Out]

(2267*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/2592 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/
2))/15 + ((7969 + 3006*x)*(2 + 5*x + 3*x^2)^(3/2))/1620 - (2267*ArcTanh[(5 + 6*x
)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(5184*Sqrt[3])

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Rubi [A]  time = 0.137828, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{1}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac{(3006 x+7969) \left (3 x^2+5 x+2\right )^{3/2}}{1620}+\frac{2267 (6 x+5) \sqrt{3 x^2+5 x+2}}{2592}-\frac{2267 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{5184 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(2267*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/2592 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/
2))/15 + ((7969 + 3006*x)*(2 + 5*x + 3*x^2)^(3/2))/1620 - (2267*ArcTanh[(5 + 6*x
)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(5184*Sqrt[3])

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Rubi in Sympy [A]  time = 18.0037, size = 99, normalized size = 0.9 \[ - \frac{\left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{15} + \frac{2267 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{2592} + \frac{\left (3006 x + 7969\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{1620} - \frac{2267 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{15552} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**2*(3*x**2+5*x+2)**(1/2),x)

[Out]

-(2*x + 3)**2*(3*x**2 + 5*x + 2)**(3/2)/15 + 2267*(6*x + 5)*sqrt(3*x**2 + 5*x +
2)/2592 + (3006*x + 7969)*(3*x**2 + 5*x + 2)**(3/2)/1620 - 2267*sqrt(3)*atanh(sq
rt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/15552

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Mathematica [A]  time = 0.0776135, size = 70, normalized size = 0.64 \[ \frac{-11335 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (10368 x^4-23760 x^3-229416 x^2-375250 x-168627\right )}{77760} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-168627 - 375250*x - 229416*x^2 - 23760*x^3 + 10368*x
^4) - 11335*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/77760

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Maple [A]  time = 0.01, size = 96, normalized size = 0.9 \[{\frac{11335+13602\,x}{2592}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{2267\,\sqrt{3}}{15552}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{6997}{1620} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{19\,x}{18} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{4\,{x}^{2}}{15} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^2*(3*x^2+5*x+2)^(1/2),x)

[Out]

2267/2592*(5+6*x)*(3*x^2+5*x+2)^(1/2)-2267/15552*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2
+5*x+2)^(1/2))*3^(1/2)+6997/1620*(3*x^2+5*x+2)^(3/2)+19/18*x*(3*x^2+5*x+2)^(3/2)
-4/15*x^2*(3*x^2+5*x+2)^(3/2)

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Maxima [A]  time = 0.770675, size = 140, normalized size = 1.27 \[ -\frac{4}{15} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + \frac{19}{18} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{6997}{1620} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{2267}{432} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{2267}{15552} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{11335}{2592} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/15*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 19/18*(3*x^2 + 5*x + 2)^(3/2)*x + 6997/1620*
(3*x^2 + 5*x + 2)^(3/2) + 2267/432*sqrt(3*x^2 + 5*x + 2)*x - 2267/15552*sqrt(3)*
log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 11335/2592*sqrt(3*x^2 + 5*x + 2
)

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Fricas [A]  time = 0.274705, size = 108, normalized size = 0.98 \[ -\frac{1}{155520} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (10368 \, x^{4} - 23760 \, x^{3} - 229416 \, x^{2} - 375250 \, x - 168627\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 11335 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/155520*sqrt(3)*(4*sqrt(3)*(10368*x^4 - 23760*x^3 - 229416*x^2 - 375250*x - 16
8627)*sqrt(3*x^2 + 5*x + 2) - 11335*log(sqrt(3)*(72*x^2 + 120*x + 49) - 12*sqrt(
3*x^2 + 5*x + 2)*(6*x + 5)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 51 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 8 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 4 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 45 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**2*(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(-51*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-8*x**2*sqrt(3*x**2 + 5*x
+ 2), x) - Integral(4*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-45*sqrt(3*x**2
+ 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.327197, size = 93, normalized size = 0.85 \[ -\frac{1}{12960} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (24 \, x - 55\right )} x - 9559\right )} x - 187625\right )} x - 168627\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{2267}{15552} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12960*(2*(12*(18*(24*x - 55)*x - 9559)*x - 187625)*x - 168627)*sqrt(3*x^2 + 5
*x + 2) + 2267/15552*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2
)) - 5))