3.2430 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx\)

Optimal. Leaf size=199 \[ -\frac{3879 \left (3 x^2+5 x+2\right )^{5/2}}{12500 (2 x+3)^5}-\frac{717 \left (3 x^2+5 x+2\right )^{5/2}}{2000 (2 x+3)^6}-\frac{19 \left (3 x^2+5 x+2\right )^{5/2}}{50 (2 x+3)^7}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}+\frac{51309 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{800000 (2 x+3)^4}-\frac{153927 (8 x+7) \sqrt{3 x^2+5 x+2}}{32000000 (2 x+3)^2}+\frac{153927 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{64000000 \sqrt{5}} \]

[Out]

(-153927*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(32000000*(3 + 2*x)^2) + (51309*(7 + 8
*x)*(2 + 5*x + 3*x^2)^(3/2))/(800000*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))
/(40*(3 + 2*x)^8) - (19*(2 + 5*x + 3*x^2)^(5/2))/(50*(3 + 2*x)^7) - (717*(2 + 5*
x + 3*x^2)^(5/2))/(2000*(3 + 2*x)^6) - (3879*(2 + 5*x + 3*x^2)^(5/2))/(12500*(3
+ 2*x)^5) + (153927*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(64000
000*Sqrt[5])

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Rubi [A]  time = 0.376224, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{3879 \left (3 x^2+5 x+2\right )^{5/2}}{12500 (2 x+3)^5}-\frac{717 \left (3 x^2+5 x+2\right )^{5/2}}{2000 (2 x+3)^6}-\frac{19 \left (3 x^2+5 x+2\right )^{5/2}}{50 (2 x+3)^7}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}+\frac{51309 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{800000 (2 x+3)^4}-\frac{153927 (8 x+7) \sqrt{3 x^2+5 x+2}}{32000000 (2 x+3)^2}+\frac{153927 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{64000000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^9,x]

[Out]

(-153927*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(32000000*(3 + 2*x)^2) + (51309*(7 + 8
*x)*(2 + 5*x + 3*x^2)^(3/2))/(800000*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))
/(40*(3 + 2*x)^8) - (19*(2 + 5*x + 3*x^2)^(5/2))/(50*(3 + 2*x)^7) - (717*(2 + 5*
x + 3*x^2)^(5/2))/(2000*(3 + 2*x)^6) - (3879*(2 + 5*x + 3*x^2)^(5/2))/(12500*(3
+ 2*x)^5) + (153927*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(64000
000*Sqrt[5])

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Rubi in Sympy [A]  time = 53.1668, size = 189, normalized size = 0.95 \[ - \frac{153927 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{320000000} - \frac{153927 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{32000000 \left (2 x + 3\right )^{2}} + \frac{51309 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{800000 \left (2 x + 3\right )^{4}} - \frac{3879 \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{12500 \left (2 x + 3\right )^{5}} - \frac{717 \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{2000 \left (2 x + 3\right )^{6}} - \frac{19 \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{50 \left (2 x + 3\right )^{7}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{40 \left (2 x + 3\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**9,x)

[Out]

-153927*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/320000000
- 153927*(8*x + 7)*sqrt(3*x**2 + 5*x + 2)/(32000000*(2*x + 3)**2) + 51309*(8*x +
7)*(3*x**2 + 5*x + 2)**(3/2)/(800000*(2*x + 3)**4) - 3879*(3*x**2 + 5*x + 2)**(
5/2)/(12500*(2*x + 3)**5) - 717*(3*x**2 + 5*x + 2)**(5/2)/(2000*(2*x + 3)**6) -
19*(3*x**2 + 5*x + 2)**(5/2)/(50*(2*x + 3)**7) - 13*(3*x**2 + 5*x + 2)**(5/2)/(4
0*(2*x + 3)**8)

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Mathematica [A]  time = 0.121528, size = 124, normalized size = 0.62 \[ -\frac{153927 \sqrt{5} (2 x+3)^8 \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+10 \sqrt{3 x^2+5 x+2} \left (5681664 x^7+60161472 x^6+272314944 x^5+682163760 x^4+1007243840 x^3+924451956 x^2+512781828 x+131091161\right )-153927 \sqrt{5} (2 x+3)^8 \log (2 x+3)}{320000000 (2 x+3)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^9,x]

[Out]

-(10*Sqrt[2 + 5*x + 3*x^2]*(131091161 + 512781828*x + 924451956*x^2 + 1007243840
*x^3 + 682163760*x^4 + 272314944*x^5 + 60161472*x^6 + 5681664*x^7) - 153927*Sqrt
[5]*(3 + 2*x)^8*Log[3 + 2*x] + 153927*Sqrt[5]*(3 + 2*x)^8*Log[-7 - 8*x + 2*Sqrt[
5]*Sqrt[2 + 5*x + 3*x^2]])/(320000000*(3 + 2*x)^8)

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Maple [A]  time = 0.03, size = 274, normalized size = 1.4 \[{\frac{51309}{200000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{153927}{320000000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13}{10240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-8}}-{\frac{19}{6400} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}}-{\frac{717}{128000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{3879}{400000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{51309}{3200000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{51309}{2000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1590579}{40000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{1487961}{25000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{7439805+8927766\,x}{50000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{769635+923562\,x}{40000000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{153927\,\sqrt{5}}{320000000}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^9,x)

[Out]

51309/200000000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+153927/320000000*(12*(x+3/2)^2-16*x
-19)^(1/2)-13/10240/(x+3/2)^8*(3*(x+3/2)^2-4*x-19/4)^(5/2)-19/6400/(x+3/2)^7*(3*
(x+3/2)^2-4*x-19/4)^(5/2)-717/128000/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(5/2)-3879
/400000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(5/2)-51309/3200000/(x+3/2)^4*(3*(x+3/2
)^2-4*x-19/4)^(5/2)-51309/2000000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1590579
/40000000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1487961/25000000/(x+3/2)*(3*(x+
3/2)^2-4*x-19/4)^(5/2)+1487961/50000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-153
927/40000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-153927/320000000*5^(1/2)*arcta
nh(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 = 0.788676, size = 532, normalized size = 2.67 \[ \frac{4771737}{40000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{40 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac{19 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{50 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{717 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{2000 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{3879 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{12500 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{51309 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{200000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{51309 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{250000 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1590579 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{10000000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{461781}{20000000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{153927}{320000000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{2924613}{160000000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1487961 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{10000000 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^9,x, algorithm="maxima")

[Out]

4771737/40000000*(3*x^2 + 5*x + 2)^(3/2) - 13/40*(3*x^2 + 5*x + 2)^(5/2)/(256*x^
8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 3499
2*x + 6561) - 19/50*(3*x^2 + 5*x + 2)^(5/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 151
20*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 717/2000*(3*x^2 + 5*x + 2)^(5
/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 3879/12
500*(3*x^2 + 5*x + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243
) - 51309/200000*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81
) - 51309/250000*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1590579/
10000000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) - 461781/20000000*sqrt(3*x^2
+ 5*x + 2)*x - 153927/320000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2
*x + 3) + 5/2/abs(2*x + 3) - 2) - 2924613/160000000*sqrt(3*x^2 + 5*x + 2) - 1487
961/10000000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 0.286125, size = 257, normalized size = 1.29 \[ -\frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (5681664 \, x^{7} + 60161472 \, x^{6} + 272314944 \, x^{5} + 682163760 \, x^{4} + 1007243840 \, x^{3} + 924451956 \, x^{2} + 512781828 \, x + 131091161\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 153927 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{640000000 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^9,x, algorithm="fricas")

[Out]

-1/640000000*sqrt(5)*(4*sqrt(5)*(5681664*x^7 + 60161472*x^6 + 272314944*x^5 + 68
2163760*x^4 + 1007243840*x^3 + 924451956*x^2 + 512781828*x + 131091161)*sqrt(3*x
^2 + 5*x + 2) - 153927*(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 +
108864*x^3 + 81648*x^2 + 34992*x + 6561)*log((sqrt(5)*(124*x^2 + 212*x + 89) +
20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x^2 + 12*x + 9)))/(256*x^8 + 3072*x^7 + 1
6128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**9,x)

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 145152
*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683
), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7
+ 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098*
x + 19683), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8
+ 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x*
*2 + 118098*x + 19683), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(512*x**9 +
6912*x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 +
314928*x**2 + 118098*x + 19683), x)

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GIAC/XCAS [A]  time = 0.31191, size = 691, normalized size = 3.47 \[ \frac{153927}{320000000} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{19702656 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 443309760 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 13775440320 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 88813739520 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 1135723030560 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 3326100961968 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 20795205897360 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 31719485197440 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 108381222834920 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 93303707056820 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 182905948708404 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 90199904722080 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 98616726439110 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 25302796273485 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 12323187970155 \, \sqrt{3} x + 954490882968 \, \sqrt{3} - 12323187970155 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{32000000 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^9,x, algorithm="giac")

[Out]

153927/320000000*sqrt(5)*ln(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*
x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x +
2))) - 1/32000000*(19702656*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 + 443309760*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 13775440320*(sqrt(3)*x - sqrt(3*
x^2 + 5*x + 2))^13 + 88813739520*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12
+ 1135723030560*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 3326100961968*sqrt(3)*(
sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 20795205897360*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^9 + 31719485197440*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 10
8381222834920*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 93303707056820*sqrt(3)*(sq
rt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 182905948708404*(sqrt(3)*x - sqrt(3*x^2 + 5
*x + 2))^5 + 90199904722080*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 9861
6726439110*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 25302796273485*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 12323187970155*sqrt(3)*x + 954490882968*sqrt(3
) - 12323187970155*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^8