3.2462 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx\)
Optimal. Leaf size=234 \[ -\frac{3904 \left (3 x^2+5 x+2\right )^{9/2}}{20625 (2 x+3)^9}-\frac{621 \left (3 x^2+5 x+2\right )^{9/2}}{2750 (2 x+3)^{10}}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}+\frac{7671 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{200000 (2 x+3)^8}-\frac{17899 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{8000000 (2 x+3)^6}+\frac{17899 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{128000000 (2 x+3)^4}-\frac{53697 (8 x+7) \sqrt{3 x^2+5 x+2}}{5120000000 (2 x+3)^2}+\frac{53697 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{10240000000 \sqrt{5}} \]
[Out]
(-53697*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(5120000000*(3 + 2*x)^2) + (17899*(7 +
8*x)*(2 + 5*x + 3*x^2)^(3/2))/(128000000*(3 + 2*x)^4) - (17899*(7 + 8*x)*(2 + 5*
x + 3*x^2)^(5/2))/(8000000*(3 + 2*x)^6) + (7671*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2
))/(200000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) - (621*
(2 + 5*x + 3*x^2)^(9/2))/(2750*(3 + 2*x)^10) - (3904*(2 + 5*x + 3*x^2)^(9/2))/(2
0625*(3 + 2*x)^9) + (53697*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])
/(10240000000*Sqrt[5])
_______________________________________________________________________________________
Rubi [A] time = 0.40248, antiderivative size = 234, 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
\[ -\frac{3904 \left (3 x^2+5 x+2\right )^{9/2}}{20625 (2 x+3)^9}-\frac{621 \left (3 x^2+5 x+2\right )^{9/2}}{2750 (2 x+3)^{10}}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}+\frac{7671 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{200000 (2 x+3)^8}-\frac{17899 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{8000000 (2 x+3)^6}+\frac{17899 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{128000000 (2 x+3)^4}-\frac{53697 (8 x+7) \sqrt{3 x^2+5 x+2}}{5120000000 (2 x+3)^2}+\frac{53697 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{10240000000 \sqrt{5}} \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^12,x]
[Out]
(-53697*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(5120000000*(3 + 2*x)^2) + (17899*(7 +
8*x)*(2 + 5*x + 3*x^2)^(3/2))/(128000000*(3 + 2*x)^4) - (17899*(7 + 8*x)*(2 + 5*
x + 3*x^2)^(5/2))/(8000000*(3 + 2*x)^6) + (7671*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2
))/(200000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) - (621*
(2 + 5*x + 3*x^2)^(9/2))/(2750*(3 + 2*x)^10) - (3904*(2 + 5*x + 3*x^2)^(9/2))/(2
0625*(3 + 2*x)^9) + (53697*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])
/(10240000000*Sqrt[5])
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Rubi in Sympy [A] time = 62.201, size = 223, normalized size = 0.95 \[ - \frac{53697 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{51200000000} - \frac{53697 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{5120000000 \left (2 x + 3\right )^{2}} + \frac{17899 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{128000000 \left (2 x + 3\right )^{4}} - \frac{17899 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{8000000 \left (2 x + 3\right )^{6}} + \frac{7671 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{200000 \left (2 x + 3\right )^{8}} - \frac{3904 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{20625 \left (2 x + 3\right )^{9}} - \frac{621 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{2750 \left (2 x + 3\right )^{10}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{55 \left (2 x + 3\right )^{11}} \]
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)**12,x)
[Out]
-53697*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/51200000000
- 53697*(8*x + 7)*sqrt(3*x**2 + 5*x + 2)/(5120000000*(2*x + 3)**2) + 17899*(8*x
+ 7)*(3*x**2 + 5*x + 2)**(3/2)/(128000000*(2*x + 3)**4) - 17899*(8*x + 7)*(3*x*
*2 + 5*x + 2)**(5/2)/(8000000*(2*x + 3)**6) + 7671*(8*x + 7)*(3*x**2 + 5*x + 2)*
*(7/2)/(200000*(2*x + 3)**8) - 3904*(3*x**2 + 5*x + 2)**(9/2)/(20625*(2*x + 3)**
9) - 621*(3*x**2 + 5*x + 2)**(9/2)/(2750*(2*x + 3)**10) - 13*(3*x**2 + 5*x + 2)*
*(9/2)/(55*(2*x + 3)**11)
_______________________________________________________________________________________
Mathematica [A] time = 0.146058, size = 127, normalized size = 0.54 \[ \frac{-53697 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{10 \sqrt{3 x^2+5 x+2} \left (30557343744 x^{10}+479034140160 x^9+3387337708800 x^8+14992486229760 x^7+41485308553600 x^6+72251114756992 x^5+80329740407040 x^4+56898923222800 x^3+24817198954840 x^2+6058472990850 x+629890144539\right )}{33 (2 x+3)^{11}}+53697 \sqrt{5} \log (2 x+3)}{51200000000} \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^12,x]
[Out]
((10*Sqrt[2 + 5*x + 3*x^2]*(629890144539 + 6058472990850*x + 24817198954840*x^2
+ 56898923222800*x^3 + 80329740407040*x^4 + 72251114756992*x^5 + 41485308553600*
x^6 + 14992486229760*x^7 + 3387337708800*x^8 + 479034140160*x^9 + 30557343744*x^
10))/(33*(3 + 2*x)^11) + 53697*Sqrt[5]*Log[3 + 2*x] - 53697*Sqrt[5]*Log[-7 - 8*x
+ 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/51200000000
_______________________________________________________________________________________
Maple [B] time = 0.117, size = 411, normalized size = 1.8 \[ \text{result too large to display} \]
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)^12,x)
[Out]
7671/50000000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+53697/200000000000*(3*(x+3/2)^2-4*
x-19/4)^(5/2)+17899/32000000000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+53697/51200000000*(
12*(x+3/2)^2-16*x-19)^(1/2)-13/112640/(x+3/2)^11*(3*(x+3/2)^2-4*x-19/4)^(9/2)-62
1/2816000/(x+3/2)^10*(3*(x+3/2)^2-4*x-19/4)^(9/2)-61/165000/(x+3/2)^9*(3*(x+3/2)
^2-4*x-19/4)^(9/2)-7671/12800000/(x+3/2)^8*(3*(x+3/2)^2-4*x-19/4)^(9/2)-7671/800
0000/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(9/2)-48583/32000000/(x+3/2)^6*(3*(x+3/2)^
2-4*x-19/4)^(9/2)-237801/100000000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-147359
91/4000000000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-14112083/2500000000/(x+3/2)
^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-427819341/50000000000/(x+3/2)^2*(3*(x+3/2)^2-4*x
-19/4)^(9/2)-80215647/6250000000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+80215647/1
2500000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-31197957/50000000000*(5+6*x)*(3*
(x+3/2)^2-4*x-19/4)^(5/2)+519071/8000000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)
-53697/6400000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-53697/51200000000*5^(1/2)
*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
_______________________________________________________________________________________
Maxima [A] time = 0.827189, size = 878, normalized size = 3.75 \[ \text{result too large to display} \]
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)^12,x, algorithm="maxima")
[Out]
1283458023/50000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/55*(3*x^2 + 5*x + 2)^(9/2)/(
2048*x^11 + 33792*x^10 + 253440*x^9 + 1140480*x^8 + 3421440*x^7 + 7185024*x^6 +
10777536*x^5 + 11547360*x^4 + 8660520*x^3 + 4330260*x^2 + 1299078*x + 177147) -
621/2750*(3*x^2 + 5*x + 2)^(9/2)/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^
7 + 1088640*x^6 + 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660
*x + 59049) - 3904/20625*(3*x^2 + 5*x + 2)^(9/2)/(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 + 1
9683) - 7671/50000*(3*x^2 + 5*x + 2)^(9/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 483
84*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 7671/62500*(3*x^
2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 2041
2*x^2 + 10206*x + 2187) - 48583/500000*(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) - 237801/3125000*(3*x^2 + 5*x
+ 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 14735991/2500
00000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1411208
3/312500000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 427819341/125
00000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 93593871/25000000000*(3*x^
2 + 5*x + 2)^(5/2)*x - 623905443/200000000000*(3*x^2 + 5*x + 2)^(5/2) - 80215647
/2500000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 1557213/4000000000*(3*x^2 + 5*x
+ 2)^(3/2)*x + 10399319/32000000000*(3*x^2 + 5*x + 2)^(3/2) - 161091/3200000000*
sqrt(3*x^2 + 5*x + 2)*x - 53697/51200000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x
+ 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 1020243/25600000000*sqrt(3*x^2 + 5*
x + 2)
_______________________________________________________________________________________
Fricas [A] time = 0.295049, size = 317, normalized size = 1.35 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (30557343744 \, x^{10} + 479034140160 \, x^{9} + 3387337708800 \, x^{8} + 14992486229760 \, x^{7} + 41485308553600 \, x^{6} + 72251114756992 \, x^{5} + 80329740407040 \, x^{4} + 56898923222800 \, x^{3} + 24817198954840 \, x^{2} + 6058472990850 \, x + 629890144539\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 1772001 \,{\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{3379200000000 \,{\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\right )}} \]
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)^12,x, algorithm="fricas")
[Out]
1/3379200000000*sqrt(5)*(4*sqrt(5)*(30557343744*x^10 + 479034140160*x^9 + 338733
7708800*x^8 + 14992486229760*x^7 + 41485308553600*x^6 + 72251114756992*x^5 + 803
29740407040*x^4 + 56898923222800*x^3 + 24817198954840*x^2 + 6058472990850*x + 62
9890144539)*sqrt(3*x^2 + 5*x + 2) + 1772001*(2048*x^11 + 33792*x^10 + 253440*x^9
+ 1140480*x^8 + 3421440*x^7 + 7185024*x^6 + 10777536*x^5 + 11547360*x^4 + 86605
20*x^3 + 4330260*x^2 + 1299078*x + 177147)*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)))/(2048*x^11 + 33792*x^1
0 + 253440*x^9 + 1140480*x^8 + 3421440*x^7 + 7185024*x^6 + 10777536*x^5 + 115473
60*x^4 + 8660520*x^3 + 4330260*x^2 + 1299078*x + 177147)
_______________________________________________________________________________________
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)**12,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.34654, size = 898, normalized size = 3.84 \[ \text{result too large to display} \]
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)^12,x, algorithm="giac")
[Out]
53697/51200000000*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))) - 1/168960000000*(1814529024*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^21 + 5715
7664256*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^20 + 57290941171200*(sqrt(3)
*x - sqrt(3*x^2 + 5*x + 2))^19 + 557490020440320*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^18 + 3116590396465920*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 - 4057
1342658595584*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 10986534193921313
28*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 - 4929229513296950400*sqrt(3)*(sqrt(3)
*x - sqrt(3*x^2 + 5*x + 2))^14 - 44860439685628251520*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^13 - 101067124429527527040*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
^12 - 530008429621517017088*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 - 73594491188
4403670592*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 24658078943595848872
00*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 2226326899649908579920*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 4870616002552398497520*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^7 - 2849658548882889760632*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^6 - 3959763769847021107884*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 142016354
1040959876150*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 114153742472719985
6070*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 215130617786249721765*sqrt(3)*(sqrt
(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 76323347715579462729*sqrt(3)*x - 426152045940
2725896*sqrt(3) + 76323347715579462729*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sq
rt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^11