3.2515 \(\int \frac{(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (7976 x+6809) (2 x+3)}{27 \sqrt{3 x^2+5 x+2}}-\frac{6848}{9} \sqrt{3 x^2+5 x+2}+\frac{152 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{27 \sqrt{3}} \]

[Out]

(-2*(3 + 2*x)^3*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(3 + 2*x)*(6809
+ 7976*x))/(27*Sqrt[2 + 5*x + 3*x^2]) - (6848*Sqrt[2 + 5*x + 3*x^2])/9 + (152*Ar
cTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(27*Sqrt[3])

_______________________________________________________________________________________

Rubi [A]  time = 0.18884, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (7976 x+6809) (2 x+3)}{27 \sqrt{3 x^2+5 x+2}}-\frac{6848}{9} \sqrt{3 x^2+5 x+2}+\frac{152 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{27 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3 + 2*x)^3*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(3 + 2*x)*(6809
+ 7976*x))/(27*Sqrt[2 + 5*x + 3*x^2]) - (6848*Sqrt[2 + 5*x + 3*x^2])/9 + (152*Ar
cTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(27*Sqrt[3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.7056, size = 107, normalized size = 0.93 \[ - \frac{2 \left (2 x + 3\right )^{3} \left (139 x + 121\right )}{9 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \left (2 x + 3\right ) \left (7976 x + 6809\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} - \frac{6848 \sqrt{3 x^{2} + 5 x + 2}}{9} + \frac{152 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{81} \]

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)**(5/2),x)

[Out]

-2*(2*x + 3)**3*(139*x + 121)/(9*(3*x**2 + 5*x + 2)**(3/2)) + 4*(2*x + 3)*(7976*
x + 6809)/(27*sqrt(3*x**2 + 5*x + 2)) - 6848*sqrt(3*x**2 + 5*x + 2)/9 + 152*sqrt
(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/81

_______________________________________________________________________________________

Mathematica [A]  time = 0.122294, size = 70, normalized size = 0.61 \[ -\frac{2}{81} \left (\frac{3 \left (72 x^4-58720 x^3-146180 x^2-118153 x-30819\right )}{\left (3 x^2+5 x+2\right )^{3/2}}-76 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*((3*(-30819 - 118153*x - 146180*x^2 - 58720*x^3 + 72*x^4))/(2 + 5*x + 3*x^2)
^(3/2) - 76*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]]))/81

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 178, normalized size = 1.6 \[ -{\frac{80905+97086\,x}{1458} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{295120+354144\,x}{243}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{145763}{1458} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{14639\,x}{81} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{2380\,{x}^{2}}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{152\,{x}^{3}}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{152\,x}{27}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{380}{81}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{152\,\sqrt{3}}{81}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{16\,{x}^{4}}{3} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16181/1458*(5+6*x)/(3*x^2+5*x+2)^(3/2)+59024/243*(5+6*x)/(3*x^2+5*x+2)^(1/2)-14
5763/1458/(3*x^2+5*x+2)^(3/2)-14639/81*x/(3*x^2+5*x+2)^(3/2)-2380/27*x^2/(3*x^2+
5*x+2)^(3/2)-152/27*x^3/(3*x^2+5*x+2)^(3/2)-152/27*x/(3*x^2+5*x+2)^(1/2)+380/81/
(3*x^2+5*x+2)^(1/2)+152/81*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
-16/3*x^4/(3*x^2+5*x+2)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.795651, size = 289, normalized size = 2.51 \[ -\frac{16 \, x^{4}}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{152}{81} \, x{\left (\frac{1410 \, x}{\sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{9 \, x^{2}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} + \frac{1175}{\sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{55 \, x}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{46}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}\right )} + \frac{152}{81} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{71440}{81} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{60704 \, x}{81 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{920 \, x^{2}}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{15680}{27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{13066 \, x}{81 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{6766}{81 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/3*x^4/(3*x^2 + 5*x + 2)^(3/2) - 152/81*x*(1410*x/sqrt(3*x^2 + 5*x + 2) + 9*x
^2/(3*x^2 + 5*x + 2)^(3/2) + 1175/sqrt(3*x^2 + 5*x + 2) - 55*x/(3*x^2 + 5*x + 2)
^(3/2) - 46/(3*x^2 + 5*x + 2)^(3/2)) + 152/81*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 +
5*x + 2) + 6*x + 5) + 71440/81*sqrt(3*x^2 + 5*x + 2) - 60704/81*x/sqrt(3*x^2 +
5*x + 2) - 920/9*x^2/(3*x^2 + 5*x + 2)^(3/2) - 15680/27/sqrt(3*x^2 + 5*x + 2) -
13066/81*x/(3*x^2 + 5*x + 2)^(3/2) - 6766/81/(3*x^2 + 5*x + 2)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.284432, size = 163, normalized size = 1.42 \[ -\frac{2 \, \sqrt{3}{\left (\sqrt{3}{\left (72 \, x^{4} - 58720 \, x^{3} - 146180 \, x^{2} - 118153 \, x - 30819\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 38 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{81 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/81*sqrt(3)*(sqrt(3)*(72*x^4 - 58720*x^3 - 146180*x^2 - 118153*x - 30819)*sqrt
(3*x^2 + 5*x + 2) - 38*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(3)*(72*x^2
+ 120*x + 49) + 12*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)))/(9*x^4 + 30*x^3 + 37*x^2 +
20*x + 4)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{999 x}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{864 x^{2}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{264 x^{3}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{16 x^{4}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{16 x^{5}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{405}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-999*x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2)
+ 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2
+ 5*x + 2)), x) - Integral(-864*x**2/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sq
rt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x +
2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-264*x**3/(9*x**4*sqrt(3*x**2 + 5
*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x
*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(16*x**4/(9*x*
*4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2
+ 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Inte
gral(16*x**5/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 3
7*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*
x + 2)), x) - Integral(-405/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2
+ 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*s
qrt(3*x**2 + 5*x + 2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.290185, size = 92, normalized size = 0.8 \[ -\frac{152}{81} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{2 \,{\left ({\left (4 \,{\left (2 \,{\left (9 \, x - 7340\right )} x - 36545\right )} x - 118153\right )} x - 30819\right )}}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-152/81*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/
27*((4*(2*(9*x - 7340)*x - 36545)*x - 118153)*x - 30819)/(3*x^2 + 5*x + 2)^(3/2)