3.2445 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx\)

Optimal. Leaf size=204 \[ -\frac{1321 \left (3 x^2+5 x+2\right )^{7/2}}{5250 (2 x+3)^7}-\frac{527 \left (3 x^2+5 x+2\right )^{7/2}}{1800 (2 x+3)^8}-\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}+\frac{6167 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{120000 (2 x+3)^6}-\frac{6167 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1920000 (2 x+3)^4}+\frac{6167 (8 x+7) \sqrt{3 x^2+5 x+2}}{25600000 (2 x+3)^2}-\frac{6167 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{51200000 \sqrt{5}} \]

[Out]

(6167*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(25600000*(3 + 2*x)^2) - (6167*(7 + 8*x)*
(2 + 5*x + 3*x^2)^(3/2))/(1920000*(3 + 2*x)^4) + (6167*(7 + 8*x)*(2 + 5*x + 3*x^
2)^(5/2))/(120000*(3 + 2*x)^6) - (13*(2 + 5*x + 3*x^2)^(7/2))/(45*(3 + 2*x)^9) -
(527*(2 + 5*x + 3*x^2)^(7/2))/(1800*(3 + 2*x)^8) - (1321*(2 + 5*x + 3*x^2)^(7/2
))/(5250*(3 + 2*x)^7) - (6167*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
)])/(51200000*Sqrt[5])

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Rubi [A]  time = 0.35419, antiderivative size = 204, 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 \[ -\frac{1321 \left (3 x^2+5 x+2\right )^{7/2}}{5250 (2 x+3)^7}-\frac{527 \left (3 x^2+5 x+2\right )^{7/2}}{1800 (2 x+3)^8}-\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}+\frac{6167 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{120000 (2 x+3)^6}-\frac{6167 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1920000 (2 x+3)^4}+\frac{6167 (8 x+7) \sqrt{3 x^2+5 x+2}}{25600000 (2 x+3)^2}-\frac{6167 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{51200000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^10,x]

[Out]

(6167*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(25600000*(3 + 2*x)^2) - (6167*(7 + 8*x)*
(2 + 5*x + 3*x^2)^(3/2))/(1920000*(3 + 2*x)^4) + (6167*(7 + 8*x)*(2 + 5*x + 3*x^
2)^(5/2))/(120000*(3 + 2*x)^6) - (13*(2 + 5*x + 3*x^2)^(7/2))/(45*(3 + 2*x)^9) -
(527*(2 + 5*x + 3*x^2)^(7/2))/(1800*(3 + 2*x)^8) - (1321*(2 + 5*x + 3*x^2)^(7/2
))/(5250*(3 + 2*x)^7) - (6167*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
)])/(51200000*Sqrt[5])

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Rubi in Sympy [A]  time = 55.5231, size = 194, normalized size = 0.95 \[ \frac{6167 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{256000000} + \frac{6167 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{25600000 \left (2 x + 3\right )^{2}} - \frac{6167 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{1920000 \left (2 x + 3\right )^{4}} + \frac{6167 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{120000 \left (2 x + 3\right )^{6}} - \frac{1321 \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{5250 \left (2 x + 3\right )^{7}} - \frac{527 \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{1800 \left (2 x + 3\right )^{8}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{45 \left (2 x + 3\right )^{9}} \]

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)**10,x)

[Out]

6167*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/256000000 + 6
167*(8*x + 7)*sqrt(3*x**2 + 5*x + 2)/(25600000*(2*x + 3)**2) - 6167*(8*x + 7)*(3
*x**2 + 5*x + 2)**(3/2)/(1920000*(2*x + 3)**4) + 6167*(8*x + 7)*(3*x**2 + 5*x +
2)**(5/2)/(120000*(2*x + 3)**6) - 1321*(3*x**2 + 5*x + 2)**(7/2)/(5250*(2*x + 3)
**7) - 527*(3*x**2 + 5*x + 2)**(7/2)/(1800*(2*x + 3)**8) - 13*(3*x**2 + 5*x + 2)
**(7/2)/(45*(2*x + 3)**9)

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Mathematica [A]  time = 0.208976, size = 115, normalized size = 0.56 \[ \frac{388521 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{10 \sqrt{3 x^2+5 x+2} \left (333241344 x^8+4204480128 x^7+23288995392 x^6+76435267296 x^5+149661252080 x^4+173974546136 x^3+117870367452 x^2+43246799138 x+6706847909\right )}{(2 x+3)^9}-388521 \sqrt{5} \log (2 x+3)}{16128000000} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^10,x]

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(6706847909 + 43246799138*x + 117870367452*x^2 + 1739
74546136*x^3 + 149661252080*x^4 + 76435267296*x^5 + 23288995392*x^6 + 4204480128
*x^7 + 333241344*x^8))/(3 + 2*x)^9 - 388521*Sqrt[5]*Log[3 + 2*x] + 388521*Sqrt[5
]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/16128000000

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Maple [A]  time = 0.047, size = 332, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^10,x)

[Out]

-6167/1000000000*(3*(x+3/2)^2-4*x-19/4)^(5/2)-6167/480000000*(3*(x+3/2)^2-4*x-19
/4)^(3/2)-6167/256000000*(12*(x+3/2)^2-16*x-19)^(1/2)-13/23040/(x+3/2)^9*(3*(x+3
/2)^2-4*x-19/4)^(7/2)-527/460800/(x+3/2)^8*(3*(x+3/2)^2-4*x-19/4)^(7/2)-1321/672
000/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(7/2)-6167/1920000/(x+3/2)^6*(3*(x+3/2)^2-4
*x-19/4)^(7/2)-6167/1200000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(7/2)-129507/160000
00/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(7/2)-376187/30000000/(x+3/2)^3*(3*(x+3/2)^2
-4*x-19/4)^(7/2)-11464453/600000000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(7/2)+35830
27/250000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-3583027/125000000/(x+3/2)*(3*(
x+3/2)^2-4*x-19/4)^(7/2)-178843/120000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)+6
167/32000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+6167/256000000*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 = 0.815316, size = 653, normalized size = 3.2 \[ \text{result too large to display} \]

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)^10,x, algorithm="maxima")

[Out]

11464453/200000000*(3*x^2 + 5*x + 2)^(5/2) - 13/45*(3*x^2 + 5*x + 2)^(7/2)/(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) - 527/1800*(3*x^2 + 5*x + 2)^(7/2)/(256*x^8 + 30
72*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x +
6561) - 1321/5250*(3*x^2 + 5*x + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120
*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 6167/30000*(3*x^2 + 5*x + 2)^(7
/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 6167/37
500*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243
) - 129507/1000000*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x +
81) - 376187/3750000*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1146
4453/150000000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 178843/20000000*(3*x
^2 + 5*x + 2)^(3/2)*x - 3583027/480000000*(3*x^2 + 5*x + 2)^(3/2) - 3583027/5000
0000*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 18501/16000000*sqrt(3*x^2 + 5*x + 2)*x
+ 6167/256000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/ab
s(2*x + 3) - 2) + 117173/128000000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.292992, size = 277, normalized size = 1.36 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (333241344 \, x^{8} + 4204480128 \, x^{7} + 23288995392 \, x^{6} + 76435267296 \, x^{5} + 149661252080 \, x^{4} + 173974546136 \, x^{3} + 117870367452 \, x^{2} + 43246799138 \, x + 6706847909\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 388521 \,{\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} - 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{32256000000 \,{\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} \]

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)^10,x, algorithm="fricas")

[Out]

1/32256000000*sqrt(5)*(4*sqrt(5)*(333241344*x^8 + 4204480128*x^7 + 23288995392*x
^6 + 76435267296*x^5 + 149661252080*x^4 + 173974546136*x^3 + 117870367452*x^2 +
43246799138*x + 6706847909)*sqrt(3*x^2 + 5*x + 2) + 388521*(512*x^9 + 6912*x^8 +
41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 11
8098*x + 19683)*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)))/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 32
6592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683)

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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)**(5/2)/(3+2*x)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.308663, size = 760, normalized size = 3.73 \[ \text{result too large to display} \]

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)^10,x, algorithm="giac")

[Out]

-6167/256000000*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
))) + 1/1612800000*(99461376*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 + 2536265088
*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 83954355072*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^15 - 341000936640*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^
14 + 17778066768000*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 177356386111968*sqr
t(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 2399974462831392*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2))^11 + 6844601123556624*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^10 + 41172892580130560*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 609368726885
85000*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 204498063708405624*(sqrt(3
)*x - sqrt(3*x^2 + 5*x + 2))^7 + 174436297943297292*sqrt(3)*(sqrt(3)*x - sqrt(3*
x^2 + 5*x + 2))^6 + 339439601929212792*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 1
64994557892929730*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 17493677251469
4750*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 42504221165006223*sqrt(3)*(sqrt(3)*
x - sqrt(3*x^2 + 5*x + 2))^2 + 19065836258759367*sqrt(3)*x + 1323473153587704*sq
rt(3) - 19065836258759367*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)^9