3.2408 \(\int (5-x) (3+2 x) \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=85 \[ \frac{1}{108} (109-18 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac{559}{864} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{559 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1728 \sqrt{3}} \]

[Out]

(559*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/864 + ((109 - 18*x)*(2 + 5*x + 3*x^2)^(3/2
))/108 - (559*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1728*Sqrt[3
])

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Rubi [A]  time = 0.0808971, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{1}{108} (109-18 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac{559}{864} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{559 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1728 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(559*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/864 + ((109 - 18*x)*(2 + 5*x + 3*x^2)^(3/2
))/108 - (559*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1728*Sqrt[3
])

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Rubi in Sympy [A]  time = 9.57009, size = 76, normalized size = 0.89 \[ \frac{\left (- 18 x + 109\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{108} + \frac{559 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{864} - \frac{559 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{5184} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-18*x + 109)*(3*x**2 + 5*x + 2)**(3/2)/108 + 559*(6*x + 5)*sqrt(3*x**2 + 5*x +
2)/864 - 559*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/5184

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Mathematica [A]  time = 0.0694037, size = 65, normalized size = 0.76 \[ \frac{-559 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )-6 \sqrt{3 x^2+5 x+2} \left (432 x^3-1896 x^2-7426 x-4539\right )}{5184} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-4539 - 7426*x - 1896*x^2 + 432*x^3) - 559*Sqrt[3]*Lo
g[5 + 6*x + 2*Sqrt[6 + 15*x + 9*x^2]])/5184

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Maple [A]  time = 0.026, size = 79, normalized size = 0.9 \[{\frac{2795+3354\,x}{864}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{559\,\sqrt{3}}{5184}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{109}{108} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{x}{6} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

559/864*(5+6*x)*(3*x^2+5*x+2)^(1/2)-559/5184*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x
+2)^(1/2))*3^(1/2)+109/108*(3*x^2+5*x+2)^(3/2)-1/6*x*(3*x^2+5*x+2)^(3/2)

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Maxima [A]  time = 0.769338, size = 117, normalized size = 1.38 \[ -\frac{1}{6} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{109}{108} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{559}{144} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{559}{5184} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{2795}{864} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*x^2 + 5*x + 2)^(3/2)*x + 109/108*(3*x^2 + 5*x + 2)^(3/2) + 559/144*sqrt(
3*x^2 + 5*x + 2)*x - 559/5184*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x
+ 5) + 2795/864*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.272552, size = 101, normalized size = 1.19 \[ -\frac{1}{10368} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (432 \, x^{3} - 1896 \, x^{2} - 7426 \, x - 4539\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 559 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10368*sqrt(3)*(4*sqrt(3)*(432*x^3 - 1896*x^2 - 7426*x - 4539)*sqrt(3*x^2 + 5*
x + 2) - 559*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. \[ - \int \left (- 7 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 2 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 15 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-7*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(2*x**2*sqrt(3*x**2 + 5*x +
2), x) - Integral(-15*sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.313731, size = 86, normalized size = 1.01 \[ -\frac{1}{864} \,{\left (2 \,{\left (12 \,{\left (18 \, x - 79\right )} x - 3713\right )} x - 4539\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{559}{5184} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/864*(2*(12*(18*x - 79)*x - 3713)*x - 4539)*sqrt(3*x^2 + 5*x + 2) + 559/5184*s
qrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))