3.2437 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx\)

Optimal. Leaf size=151 \[ -\frac{(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac{1}{192} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac{1}{512} (3865-8082 x) \sqrt{3 x^2+5 x+2}+\frac{41053 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024 \sqrt{3}}-\frac{1325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

-((3865 - 8082*x)*Sqrt[2 + 5*x + 3*x^2])/512 - ((65 - 1194*x)*(2 + 5*x + 3*x^2)^
(3/2))/192 - ((34 + x)*(2 + 5*x + 3*x^2)^(5/2))/(10*(3 + 2*x)) + (41053*ArcTanh[
(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1024*Sqrt[3]) - (1325*Sqrt[5]*Arc
Tanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/128

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Rubi [A]  time = 0.292817, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac{1}{192} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac{1}{512} (3865-8082 x) \sqrt{3 x^2+5 x+2}+\frac{41053 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024 \sqrt{3}}-\frac{1325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((3865 - 8082*x)*Sqrt[2 + 5*x + 3*x^2])/512 - ((65 - 1194*x)*(2 + 5*x + 3*x^2)^
(3/2))/192 - ((34 + x)*(2 + 5*x + 3*x^2)^(5/2))/(10*(3 + 2*x)) + (41053*ArcTanh[
(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1024*Sqrt[3]) - (1325*Sqrt[5]*Arc
Tanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/128

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Rubi in Sympy [A]  time = 41.4221, size = 136, normalized size = 0.9 \[ - \frac{\left (- 290952 x + 139140\right ) \sqrt{3 x^{2} + 5 x + 2}}{18432} - \frac{\left (- 7164 x + 390\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{1152} + \frac{41053 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{3072} + \frac{1325 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{128} - \frac{\left (2 x + 68\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{20 \left (2 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-290952*x + 139140)*sqrt(3*x**2 + 5*x + 2)/18432 - (-7164*x + 390)*(3*x**2 + 5
*x + 2)**(3/2)/1152 + 41053*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x
+ 2)))/3072 + 1325*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2))
)/128 - (2*x + 68)*(3*x**2 + 5*x + 2)**(5/2)/(20*(2*x + 3))

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Mathematica [A]  time = 0.315982, size = 131, normalized size = 0.87 \[ \frac{31800 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+41053 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{2 \sqrt{3 x^2+5 x+2} \left (6912 x^5-28512 x^4-80064 x^3-118996 x^2+40412 x+293973\right )}{5 (2 x+3)}-31800 \sqrt{5} \log (2 x+3)}{3072} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[2 + 5*x + 3*x^2]*(293973 + 40412*x - 118996*x^2 - 80064*x^3 - 28512*x^
4 + 6912*x^5))/(5*(3 + 2*x)) - 31800*Sqrt[5]*Log[3 + 2*x] + 31800*Sqrt[5]*Log[-7
- 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] + 41053*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[
6 + 15*x + 9*x^2]])/3072

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Maple [A]  time = 0.017, size = 195, normalized size = 1.3 \[ -{\frac{13}{10} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{53}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{995+1194\,x}{192} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{6735+8082\,x}{512}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{41053\,\sqrt{3}}{3072}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }-{\frac{265}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{1325}{128}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{1325\,\sqrt{5}}{128}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{65+78\,x}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/10/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-53/20*(3*(x+3/2)^2-4*x-19/4)^(5/2)+1
99/192*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)+1347/512*(5+6*x)*(3*(x+3/2)^2-4*x-19
/4)^(1/2)+41053/3072*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1
/2)-265/48*(3*(x+3/2)^2-4*x-19/4)^(3/2)-1325/128*(12*(x+3/2)^2-16*x-19)^(1/2)+13
25/128*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+13/2
0*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)

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Maxima [A]  time = 0.802485, size = 220, normalized size = 1.46 \[ -\frac{1}{20} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{199}{32} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{65}{192} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{4 \,{\left (2 \, x + 3\right )}} + \frac{4041}{256} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{41053}{3072} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{1325}{128} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{3865}{512} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(3*x^2 + 5*x + 2)^(5/2) + 199/32*(3*x^2 + 5*x + 2)^(3/2)*x - 65/192*(3*x^2
+ 5*x + 2)^(3/2) - 13/4*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 4041/256*sqrt(3*x^2
+ 5*x + 2)*x + 41053/3072*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2
) + 1325/128*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*
x + 3) - 2) - 3865/512*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.28943, size = 212, normalized size = 1.4 \[ \frac{\sqrt{3}{\left (159000 \, \sqrt{5} \sqrt{3}{\left (2 \, x + 3\right )} \log \left (-\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, \sqrt{3}{\left (6912 \, x^{5} - 28512 \, x^{4} - 80064 \, x^{3} - 118996 \, x^{2} + 40412 \, x + 293973\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 615795 \,{\left (2 \, x + 3\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{92160 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/92160*sqrt(3)*(159000*sqrt(5)*sqrt(3)*(2*x + 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)) - 4*sqrt(3)*(6912*
x^5 - 28512*x^4 - 80064*x^3 - 118996*x^2 + 40412*x + 293973)*sqrt(3*x^2 + 5*x +
2) + 615795*(2*x + 3)*log(sqrt(3)*(72*x^2 + 120*x + 49) + 12*sqrt(3*x^2 + 5*x +
2)*(6*x + 5)))/(2*x + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-96*x*sq
rt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-165*x**2*sqrt(3*x**2 +
5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(4*
x**2 + 12*x + 9), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x +
9), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x)

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GIAC/XCAS [A]  time = 0.644244, size = 906, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-41053/3072*sqrt(3)*ln(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)
+ 2*sqrt(5)/(2*x + 3))/abs(2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)
+ 2*sqrt(5)/(2*x + 3)))*sign(1/(2*x + 3)) + 1325/128*sqrt(5)*ln(abs(sqrt(5)*(sq
rt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sign(1/(2*x + 3)
) - 325/128*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)*sign(1/(2*x + 3)) + 1/7680*(1
304805*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^9*sign(1/(2*
x + 3)) - 2064120*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x
+ 3))^8*sign(1/(2*x + 3)) - 4382950*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + s
qrt(5)/(2*x + 3))^7*sign(1/(2*x + 3)) + 10490640*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/
(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^6*sign(1/(2*x + 3)) + 19083456*(sqrt(-8/(2
*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^5*sign(1/(2*x + 3)) - 33372000
*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sign(1/(
2*x + 3)) - 42760170*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)
)^3*sign(1/(2*x + 3)) + 60102000*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)
+ sqrt(5)/(2*x + 3))^2*sign(1/(2*x + 3)) + 21448395*(sqrt(-8/(2*x + 3) + 5/(2*x
+ 3)^2 + 3) + sqrt(5)/(2*x + 3))*sign(1/(2*x + 3)) - 36498600*sqrt(5)*sign(1/(2
*x + 3)))/((sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2 - 3)^5