3.2505 \(\int \frac{(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{2 (139 x+121) (2 x+3)^3}{3 \sqrt{3 x^2+5 x+2}}+\frac{1664}{27} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{10}{81} (1438 x+3369) \sqrt{3 x^2+5 x+2}+\frac{6265 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{81 \sqrt{3}} \]

[Out]

(-2*(3 + 2*x)^3*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (1664*(3 + 2*x)^2*Sqr
t[2 + 5*x + 3*x^2])/27 + (10*(3369 + 1438*x)*Sqrt[2 + 5*x + 3*x^2])/81 + (6265*A
rcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(81*Sqrt[3])

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Rubi [A]  time = 0.216318, antiderivative size = 117, 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 \[ -\frac{2 (139 x+121) (2 x+3)^3}{3 \sqrt{3 x^2+5 x+2}}+\frac{1664}{27} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{10}{81} (1438 x+3369) \sqrt{3 x^2+5 x+2}+\frac{6265 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{81 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2)^(3/2),x]

[Out]

(-2*(3 + 2*x)^3*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (1664*(3 + 2*x)^2*Sqr
t[2 + 5*x + 3*x^2])/27 + (10*(3369 + 1438*x)*Sqrt[2 + 5*x + 3*x^2])/81 + (6265*A
rcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(81*Sqrt[3])

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Rubi in Sympy [A]  time = 26.0737, size = 107, normalized size = 0.91 \[ - \frac{2 \left (2 x + 3\right )^{3} \left (139 x + 121\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} + \frac{1664 \left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 5 x + 2}}{27} + \frac{\left (43140 x + 101070\right ) \sqrt{3 x^{2} + 5 x + 2}}{243} + \frac{6265 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**4/(3*x**2+5*x+2)**(3/2),x)

[Out]

-2*(2*x + 3)**3*(139*x + 121)/(3*sqrt(3*x**2 + 5*x + 2)) + 1664*(2*x + 3)**2*sqr
t(3*x**2 + 5*x + 2)/27 + (43140*x + 101070)*sqrt(3*x**2 + 5*x + 2)/243 + 6265*sq
rt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/243

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Mathematica [A]  time = 0.127331, size = 70, normalized size = 0.6 \[ \frac{1}{243} \left (6265 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{6 \left (72 x^4-102 x^3-3331 x^2+6920 x+9591\right )}{\sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2)^(3/2),x]

[Out]

((-6*(9591 + 6920*x - 3331*x^2 - 102*x^3 + 72*x^4))/Sqrt[2 + 5*x + 3*x^2] + 6265
*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/243

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Maple [A]  time = 0.019, size = 130, normalized size = 1.1 \[ -{\frac{12625+15150\,x}{162}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{25739}{162}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{6265\,x}{81}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{6265\,\sqrt{3}}{243}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{6662\,{x}^{2}}{81}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{68\,{x}^{3}}{27}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{16\,{x}^{4}}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^4/(3*x^2+5*x+2)^(3/2),x)

[Out]

-2525/162*(5+6*x)/(3*x^2+5*x+2)^(1/2)-25739/162/(3*x^2+5*x+2)^(1/2)-6265/81*x/(3
*x^2+5*x+2)^(1/2)+6265/243*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
+6662/81*x^2/(3*x^2+5*x+2)^(1/2)+68/27*x^3/(3*x^2+5*x+2)^(1/2)-16/9*x^4/(3*x^2+5
*x+2)^(1/2)

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Maxima [A]  time = 0.798701, size = 147, normalized size = 1.26 \[ -\frac{16 \, x^{4}}{9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{68 \, x^{3}}{27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{6662 \, x^{2}}{81 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{6265}{243} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{13840 \, x}{81 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{6394}{27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 5*x + 2)^(3/2),x, algorithm="maxima")

[Out]

-16/9*x^4/sqrt(3*x^2 + 5*x + 2) + 68/27*x^3/sqrt(3*x^2 + 5*x + 2) + 6662/81*x^2/
sqrt(3*x^2 + 5*x + 2) + 6265/243*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6
*x + 5) - 13840/81*x/sqrt(3*x^2 + 5*x + 2) - 6394/27/sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.292953, size = 138, normalized size = 1.18 \[ -\frac{\sqrt{3}{\left (4 \, \sqrt{3}{\left (72 \, x^{4} - 102 \, x^{3} - 3331 \, x^{2} + 6920 \, x + 9591\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 6265 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{486 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 5*x + 2)^(3/2),x, algorithm="fricas")

[Out]

-1/486*sqrt(3)*(4*sqrt(3)*(72*x^4 - 102*x^3 - 3331*x^2 + 6920*x + 9591)*sqrt(3*x
^2 + 5*x + 2) - 6265*(3*x^2 + 5*x + 2)*log(sqrt(3)*(72*x^2 + 120*x + 49) + 12*sq
rt(3*x^2 + 5*x + 2)*(6*x + 5)))/(3*x^2 + 5*x + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{999 x}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{864 x^{2}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{264 x^{3}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{16 x^{4}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{16 x^{5}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{405}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**4/(3*x**2+5*x+2)**(3/2),x)

[Out]

-Integral(-999*x/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2
*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-864*x**2/(3*x**2*sqrt(3*x**2 + 5*x + 2)
+ 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-264*x*
*3/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 +
5*x + 2)), x) - Integral(16*x**4/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x*
*2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - Integral(16*x**5/(3*x**2*sqrt(3*
x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - I
ntegral(-405/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqr
t(3*x**2 + 5*x + 2)), x)

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GIAC/XCAS [A]  time = 0.281682, size = 90, normalized size = 0.77 \[ -\frac{6265}{243} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{2 \,{\left ({\left ({\left (6 \,{\left (12 \, x - 17\right )} x - 3331\right )} x + 6920\right )} x + 9591\right )}}{81 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 5*x + 2)^(3/2),x, algorithm="giac")

[Out]

-6265/243*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) -
2/81*(((6*(12*x - 17)*x - 3331)*x + 6920)*x + 9591)/sqrt(3*x^2 + 5*x + 2)