3.2459 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx\)

Optimal. Leaf size=197 \[ \frac{(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac{(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac{(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac{3 (434104 x+559841) \sqrt{3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac{27}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{1673211 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{8192000 \sqrt{5}} \]

[Out]

(3*(559841 + 434104*x)*Sqrt[2 + 5*x + 3*x^2])/(4096000*(3 + 2*x)^2) + ((20959 +
17096*x)*(2 + 5*x + 3*x^2)^(3/2))/(102400*(3 + 2*x)^4) + ((881 + 664*x)*(2 + 5*x
+ 3*x^2)^(5/2))/(6400*(3 + 2*x)^6) + ((757 + 808*x)*(2 + 5*x + 3*x^2)^(7/2))/(1
120*(3 + 2*x)^8) - (27*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2
])])/512 + (1673211*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(81920
00*Sqrt[5])

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Rubi [A]  time = 0.392961, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac{(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac{(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac{3 (434104 x+559841) \sqrt{3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac{27}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{1673211 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{8192000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^9,x]

[Out]

(3*(559841 + 434104*x)*Sqrt[2 + 5*x + 3*x^2])/(4096000*(3 + 2*x)^2) + ((20959 +
17096*x)*(2 + 5*x + 3*x^2)^(3/2))/(102400*(3 + 2*x)^4) + ((881 + 664*x)*(2 + 5*x
+ 3*x^2)^(5/2))/(6400*(3 + 2*x)^6) + ((757 + 808*x)*(2 + 5*x + 3*x^2)^(7/2))/(1
120*(3 + 2*x)^8) - (27*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2
])])/512 + (1673211*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(81920
00*Sqrt[5])

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Rubi in Sympy [A]  time = 50.8971, size = 180, normalized size = 0.91 \[ - \frac{27 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{512} - \frac{1673211 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{40960000} + \frac{\left (156277440 x + 201542760\right ) \sqrt{3 x^{2} + 5 x + 2}}{491520000 \left (2 x + 3\right )^{2}} + \frac{\left (3077280 x + 3772620\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{18432000 \left (2 x + 3\right )^{4}} + \frac{\left (19920 x + 26430\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{192000 \left (2 x + 3\right )^{6}} + \frac{\left (808 x + 757\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{1120 \left (2 x + 3\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**9,x)

[Out]

-27*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/512 - 1673211*sq
rt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/40960000 + (15627744
0*x + 201542760)*sqrt(3*x**2 + 5*x + 2)/(491520000*(2*x + 3)**2) + (3077280*x +
3772620)*(3*x**2 + 5*x + 2)**(3/2)/(18432000*(2*x + 3)**4) + (19920*x + 26430)*(
3*x**2 + 5*x + 2)**(5/2)/(192000*(2*x + 3)**6) + (808*x + 757)*(3*x**2 + 5*x + 2
)**(7/2)/(1120*(2*x + 3)**8)

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Mathematica [A]  time = 0.266222, size = 139, normalized size = 0.71 \[ \frac{-11712477 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-15120000 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )+\frac{10 \sqrt{3 x^2+5 x+2} \left (1478785536 x^7+12182619328 x^6+45214440256 x^5+97176896240 x^4+129405924160 x^3+105874603844 x^2+48950756372 x+9818427389\right )}{(2 x+3)^8}[phone] \sqrt{5} \log (2 x+3)}{286720000} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^9,x]

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(9818427389 + 48950756372*x + 105874603844*x^2 + 1294
05924160*x^3 + 97176896240*x^4 + 45214440256*x^5 + 12182619328*x^6 + 1478785536*
x^7))/(3 + 2*x)^8 + 11712477*Sqrt[5]*Log[3 + 2*x] - 11712477*Sqrt[5]*Log[-7 - 8*
x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] - 15120000*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6
+ 15*x + 9*x^2]])/286720000

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Maple [B]  time = 0.041, size = 379, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^9,x)

[Out]

1673211/280000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+1673211/160000000*(3*(x+3/2)^2-4*
x-19/4)^(5/2)+557737/25600000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+1673211/40960000*(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)^(9/2)-81/4480
0/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(9/2)-523/179200/(x+3/2)^6*(3*(x+3/2)^2-4*x-1
9/4)^(9/2)-363/80000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-158331/22400000/(x+3
/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-150503/14000000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19
/4)^(9/2)-664383/40000000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)-767427/35000000
/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+767427/70000000*(5+6*x)*(3*(x+3/2)^2-4*x-1
9/4)^(7/2)-135591/40000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-25627/6400000*(5
+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-53211/5120000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^
(1/2)-27/512*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-1673
211/40960000*5^(1/2)*arctanh(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 = 0.825794, size = 647, normalized size = 3.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^9,x, algorithm="maxima")

[Out]

1993149/40000000*(3*x^2 + 5*x + 2)^(7/2) - 13/40*(3*x^2 + 5*x + 2)^(9/2)/(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) - 81/350*(3*x^2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15
120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 523/2800*(3*x^2 + 5*x + 2)^(
9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 363/25
00*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
- 158331/1400000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 8
1) - 150503/1750000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 66438
3/10000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 406773/20000000*(3*x^2 +
5*x + 2)^(5/2)*x - 1038609/160000000*(3*x^2 + 5*x + 2)^(5/2) - 767427/14000000*
(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) - 76881/3200000*(3*x^2 + 5*x + 2)^(3/2)*x + 45
197/25600000*(3*x^2 + 5*x + 2)^(3/2) - 159633/2560000*sqrt(3*x^2 + 5*x + 2)*x -
27/512*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 1673211/40960000
*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2)
+ 608991/20480000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.299465, size = 366, normalized size = 1.86 \[ \frac{\sqrt{5}{\left (3024000 \, \sqrt{5} \sqrt{3}{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 4 \, \sqrt{5}{\left (1478785536 \, x^{7} + 12182619328 \, x^{6} + 45214440256 \, x^{5} + 97176896240 \, x^{4} + 129405924160 \, x^{3} + 105874603844 \, x^{2} + 48950756372 \, x + 9818427389\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 11712477 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{573440000 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^9,x, algorithm="fricas")

[Out]

1/573440000*sqrt(5)*(3024000*sqrt(5)*sqrt(3)*(256*x^8 + 3072*x^7 + 16128*x^6 + 4
8384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)*log(-4*sqrt(3)*s
qrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 4*sqrt(5)*(1478785536*x^
7 + 12182619328*x^6 + 45214440256*x^5 + 97176896240*x^4 + 129405924160*x^3 + 105
874603844*x^2 + 48950756372*x + 9818427389)*sqrt(3*x^2 + 5*x + 2) + 11712477*(25
6*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 + 16128*x^6 + 48384*x^5 + 907
20*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**9,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^9,x, algorithm="giac")

[Out]

undef