3.2419 \(\int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=133 \[ -\frac{1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}+\frac{(2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}}{1890}+\frac{1129 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac{1129 (6 x+5) \sqrt{3 x^2+5 x+2}}{20736}+\frac{1129 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{41472 \sqrt{3}} \]

[Out]

(-1129*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/20736 + (1129*(5 + 6*x)*(2 + 5*x + 3*x^2
)^(3/2))/2592 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2))/21 + ((5827 + 2370*x)*(2 +
5*x + 3*x^2)^(5/2))/1890 + (1129*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*
x^2])])/(41472*Sqrt[3])

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Rubi [A]  time = 0.17128, antiderivative size = 133, 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 \[ -\frac{1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}+\frac{(2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}}{1890}+\frac{1129 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac{1129 (6 x+5) \sqrt{3 x^2+5 x+2}}{20736}+\frac{1129 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{41472 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-1129*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/20736 + (1129*(5 + 6*x)*(2 + 5*x + 3*x^2
)^(3/2))/2592 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2))/21 + ((5827 + 2370*x)*(2 +
5*x + 3*x^2)^(5/2))/1890 + (1129*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*
x^2])])/(41472*Sqrt[3])

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Rubi in Sympy [A]  time = 18.2073, size = 121, normalized size = 0.91 \[ - \frac{\left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{21} + \frac{1129 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{2592} - \frac{1129 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{20736} + \frac{\left (7110 x + 17481\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{5670} + \frac{1129 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{124416} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**2*(3*x**2 + 5*x + 2)**(5/2)/21 + 1129*(6*x + 5)*(3*x**2 + 5*x + 2)**
(3/2)/2592 - 1129*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)/20736 + (7110*x + 17481)*(3*x
**2 + 5*x + 2)**(5/2)/5670 + 1129*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2
+ 5*x + 2)))/124416

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Mathematica [A]  time = 0.0946945, size = 80, normalized size = 0.6 \[ \frac{39515 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (1244160 x^6-311040 x^5-27084672 x^4-79049520 x^3-94861176 x^2-51971350 x-10669737\right )}{4354560} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-10669737 - 51971350*x - 94861176*x^2 - 79049520*x^3
- 27084672*x^4 - 311040*x^5 + 1244160*x^6) + 39515*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt
[6 + 15*x + 9*x^2]])/4354560

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Maple [A]  time = 0.009, size = 115, normalized size = 0.9 \[{\frac{5645+6774\,x}{2592} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{5645+6774\,x}{20736}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{1129\,\sqrt{3}}{124416}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{5017}{1890} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{43\,x}{63} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{4\,{x}^{2}}{21} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1129/2592*(5+6*x)*(3*x^2+5*x+2)^(3/2)-1129/20736*(5+6*x)*(3*x^2+5*x+2)^(1/2)+112
9/124416*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+5017/1890*(3*x^2+
5*x+2)^(5/2)+43/63*x*(3*x^2+5*x+2)^(5/2)-4/21*x^2*(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 0.775952, size = 180, normalized size = 1.35 \[ -\frac{4}{21} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{43}{63} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{5017}{1890} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{1129}{432} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{5645}{2592} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{1129}{3456} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{1129}{124416} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{5645}{20736} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/21*(3*x^2 + 5*x + 2)^(5/2)*x^2 + 43/63*(3*x^2 + 5*x + 2)^(5/2)*x + 5017/1890*
(3*x^2 + 5*x + 2)^(5/2) + 1129/432*(3*x^2 + 5*x + 2)^(3/2)*x + 5645/2592*(3*x^2
+ 5*x + 2)^(3/2) - 1129/3456*sqrt(3*x^2 + 5*x + 2)*x + 1129/124416*sqrt(3)*log(2
*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 5645/20736*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.280373, size = 122, normalized size = 0.92 \[ -\frac{1}{8709120} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (1244160 \, x^{6} - 311040 \, x^{5} - 27084672 \, x^{4} - 79049520 \, x^{3} - 94861176 \, x^{2} - 51971350 \, x - 10669737\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 39515 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8709120*sqrt(3)*(4*sqrt(3)*(1244160*x^6 - 311040*x^5 - 27084672*x^4 - 7904952
0*x^3 - 94861176*x^2 - 51971350*x - 10669737)*sqrt(3*x^2 + 5*x + 2) - 39515*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. \[ - \int \left (- 327 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 406 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 185 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 4 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 12 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 90 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-327*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-406*x**2*sqrt(3*x**2 + 5
*x + 2), x) - Integral(-185*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-4*x**4*s
qrt(3*x**2 + 5*x + 2), x) - Integral(12*x**5*sqrt(3*x**2 + 5*x + 2), x) - Integr
al(-90*sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.266641, size = 107, normalized size = 0.8 \[ -\frac{1}{725760} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (90 \,{\left (4 \, x - 1\right )} x - 7837\right )} x - 182985\right )} x - 3952549\right )} x - 25985675\right )} x - 10669737\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1129}{124416} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/725760*(2*(12*(18*(8*(90*(4*x - 1)*x - 7837)*x - 182985)*x - 3952549)*x - 259
85675)*x - 10669737)*sqrt(3*x^2 + 5*x + 2) - 1129/124416*sqrt(3)*ln(abs(-2*sqrt(
3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))