3.2496 \(\int \frac{(5-x) (3+2 x)^2}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{1}{9} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{1}{54} (194 x+699) \sqrt{3 x^2+5 x+2}+\frac{1147 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{108 \sqrt{3}} \]

[Out]

-((3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/9 + ((699 + 194*x)*Sqrt[2 + 5*x + 3*x^2])/5
4 + (1147*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(108*Sqrt[3])

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Rubi [A]  time = 0.138218, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{1}{9} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{1}{54} (194 x+699) \sqrt{3 x^2+5 x+2}+\frac{1147 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{108 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-((3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/9 + ((699 + 194*x)*Sqrt[2 + 5*x + 3*x^2])/5
4 + (1147*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(108*Sqrt[3])

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Rubi in Sympy [A]  time = 17.696, size = 76, normalized size = 0.87 \[ - \frac{\left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 5 x + 2}}{9} + \frac{\left (582 x + 2097\right ) \sqrt{3 x^{2} + 5 x + 2}}{162} + \frac{1147 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{324} \]

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*sqrt(3*x**2 + 5*x + 2)/9 + (582*x + 2097)*sqrt(3*x**2 + 5*x + 2)/1
62 + 1147*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/324

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Mathematica [A]  time = 0.0593156, size = 60, normalized size = 0.69 \[ \frac{1}{324} \left (1147 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (24 x^2-122 x-645\right )\right ) \]

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]*(-645 - 122*x + 24*x^2) + 1147*Sqrt[3]*Log[-5 - 6*x -
2*Sqrt[6 + 15*x + 9*x^2]])/324

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Maple [A]  time = 0.009, size = 77, normalized size = 0.9 \[{\frac{1147\,\sqrt{3}}{324}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{215}{18}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{61\,x}{27}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{4\,{x}^{2}}{9}\sqrt{3\,{x}^{2}+5\,x+2}} \]

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]

1147/324*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+215/18*(3*x^2+5*x
+2)^(1/2)+61/27*x*(3*x^2+5*x+2)^(1/2)-4/9*x^2*(3*x^2+5*x+2)^(1/2)

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Maxima [A]  time = 0.797513, size = 101, normalized size = 1.16 \[ -\frac{4}{9} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{2} + \frac{61}{27} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{1147}{324} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{215}{18} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/9*sqrt(3*x^2 + 5*x + 2)*x^2 + 61/27*sqrt(3*x^2 + 5*x + 2)*x + 1147/324*sqrt(3
)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 215/18*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.273955, size = 95, normalized size = 1.09 \[ -\frac{1}{648} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (24 \, x^{2} - 122 \, x - 645\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 1147 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/648*sqrt(3)*(4*sqrt(3)*(24*x^2 - 122*x - 645)*sqrt(3*x^2 + 5*x + 2) - 1147*lo
g(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. \[ - \int \left (- \frac{51 x}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{8 x^{2}}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{4 x^{3}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{45}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

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 = 0.270374, size = 80, normalized size = 0.92 \[ -\frac{1}{54} \,{\left (2 \,{\left (12 \, x - 61\right )} x - 645\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1147}{324} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*(2*(12*x - 61)*x - 645)*sqrt(3*x^2 + 5*x + 2) - 1147/324*sqrt(3)*ln(abs(-2
*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))