3.2409 \(\int (5-x) \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=80 \[ -\frac{1}{9} \left (3 x^2+5 x+2\right )^{3/2}+\frac{35}{72} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{144 \sqrt{3}} \]

[Out]

(35*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/72 - (2 + 5*x + 3*x^2)^(3/2)/9 - (35*ArcTan
h[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(144*Sqrt[3])

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Rubi [A]  time = 0.0552115, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{1}{9} \left (3 x^2+5 x+2\right )^{3/2}+\frac{35}{72} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{144 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(35*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/72 - (2 + 5*x + 3*x^2)^(3/2)/9 - (35*ArcTan
h[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(144*Sqrt[3])

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Rubi in Sympy [A]  time = 6.20666, size = 71, normalized size = 0.89 \[ \frac{35 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{72} - \frac{\left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{9} - \frac{35 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{432} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)/72 - (3*x**2 + 5*x + 2)**(3/2)/9 - 35*sqrt(3
)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/432

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Mathematica [A]  time = 0.039914, size = 60, normalized size = 0.75 \[ \frac{1}{432} \left (-6 \sqrt{3 x^2+5 x+2} \left (24 x^2-170 x-159\right )-35 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-159 - 170*x + 24*x^2) - 35*Sqrt[3]*Log[5 + 6*x + 2*S
qrt[6 + 15*x + 9*x^2]])/432

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Maple [A]  time = 0.007, size = 64, normalized size = 0.8 \[ -{\frac{1}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{175+210\,x}{72}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{35\,\sqrt{3}}{432}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(3*x^2+5*x+2)^(3/2)+35/72*(5+6*x)*(3*x^2+5*x+2)^(1/2)-35/432*ln(1/3*(5/2+3*
x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)

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Maxima [A]  time = 0.769649, size = 97, normalized size = 1.21 \[ -\frac{1}{9} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{35}{12} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{35}{432} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{175}{72} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(3*x^2 + 5*x + 2)^(3/2) + 35/12*sqrt(3*x^2 + 5*x + 2)*x - 35/432*sqrt(3)*lo
g(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 175/72*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.273507, size = 95, normalized size = 1.19 \[ -\frac{1}{864} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (24 \, x^{2} - 170 \, x - 159\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 35 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/864*sqrt(3)*(4*sqrt(3)*(24*x^2 - 170*x - 159)*sqrt(3*x^2 + 5*x + 2) - 35*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. \[ - \int x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 5 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.266799, size = 80, normalized size = 1. \[ -\frac{1}{72} \,{\left (2 \,{\left (12 \, x - 85\right )} x - 159\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{35}{432} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*(2*(12*x - 85)*x - 159)*sqrt(3*x^2 + 5*x + 2) + 35/432*sqrt(3)*ln(abs(-2*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))