3.2410 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx\)

Optimal. Leaf size=100 \[ \frac{1}{24} \sqrt{3 x^2+5 x+2} (73-6 x)-\frac{311 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{48 \sqrt{3}}+\frac{13}{8} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

((73 - 6*x)*Sqrt[2 + 5*x + 3*x^2])/24 - (311*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2
+ 5*x + 3*x^2])])/(48*Sqrt[3]) + (13*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[
2 + 5*x + 3*x^2])])/8

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Rubi [A]  time = 0.178318, 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 \[ \frac{1}{24} \sqrt{3 x^2+5 x+2} (73-6 x)-\frac{311 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{48 \sqrt{3}}+\frac{13}{8} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((73 - 6*x)*Sqrt[2 + 5*x + 3*x^2])/24 - (311*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2
+ 5*x + 3*x^2])])/(48*Sqrt[3]) + (13*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[
2 + 5*x + 3*x^2])])/8

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Rubi in Sympy [A]  time = 24.7595, size = 92, normalized size = 0.92 \[ \frac{\left (- 6 x + 73\right ) \sqrt{3 x^{2} + 5 x + 2}}{24} - \frac{311 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{144} - \frac{13 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-6*x + 73)*sqrt(3*x**2 + 5*x + 2)/24 - 311*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*s
qrt(3*x**2 + 5*x + 2)))/144 - 13*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**
2 + 5*x + 2)))/8

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Mathematica [A]  time = 0.0791446, size = 114, normalized size = 1.14 \[ \frac{1}{144} \left (-36 \sqrt{3 x^2+5 x+2} x+438 \sqrt{3 x^2+5 x+2}-234 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-311 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )+234 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(438*Sqrt[2 + 5*x + 3*x^2] - 36*x*Sqrt[2 + 5*x + 3*x^2] + 234*Sqrt[5]*Log[3 + 2*
x] - 234*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] - 311*Sqrt[3]*L
og[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/144

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Maple [A]  time = 0.026, size = 127, normalized size = 1.3 \[ -{\frac{5+6\,x}{24}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{\sqrt{3}}{144}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{13}{8}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13\,\sqrt{3}}{6}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }-{\frac{13\,\sqrt{5}}{8}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(5+6*x)*(3*x^2+5*x+2)^(1/2)+1/144*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(
1/2))*3^(1/2)+13/8*(12*(x+3/2)^2-16*x-19)^(1/2)-13/6*ln(1/3*(5/2+3*x)*3^(1/2)+(3
*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-13/8*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/
(12*(x+3/2)^2-16*x-19)^(1/2))

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Maxima [A]  time = 0.776889, size = 134, normalized size = 1.34 \[ -\frac{1}{4} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{311}{144} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{13}{8} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{73}{24} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(3*x^2 + 5*x + 2)*x - 311/144*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2)
+ 3*x + 5/2) - 13/8*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/
2/abs(2*x + 3) - 2) + 73/24*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.286586, size = 161, normalized size = 1.61 \[ -\frac{1}{288} \, \sqrt{3}{\left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 73\right )} - 78 \, \sqrt{5} \sqrt{3} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 311 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/288*sqrt(3)*(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x - 73) - 78*sqrt(5)*sqrt(3)*
log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 +
12*x + 9)) - 311*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. \[ - \int \left (- \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.30223, size = 170, normalized size = 1.7 \[ -\frac{1}{24} \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 73\right )} + \frac{13}{8} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{311}{144} \, \sqrt{3}{\rm ln}\left ({\left | -6 \, \sqrt{3} x - 5 \, \sqrt{3} + 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*sqrt(3*x^2 + 5*x + 2)*(6*x - 73) + 13/8*sqrt(5)*ln(abs(-4*sqrt(3)*x - 2*sq
rt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sq
rt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 311/144*sqrt(3)*ln(abs(-6*sqrt(3)*x - 5*sqrt
(3) + 6*sqrt(3*x^2 + 5*x + 2)))