3.2463 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx\)
Optimal. Leaf size=259 \[ -\frac{6379 \left (3 x^2+5 x+2\right )^{9/2}}{41250 (2 x+3)^9}-\frac{2067 \left (3 x^2+5 x+2\right )^{9/2}}{11000 (2 x+3)^{10}}-\frac{12 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}+\frac{25017 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800000 (2 x+3)^8}-\frac{58373 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{32000000 (2 x+3)^6}+\frac{58373 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{512000000 (2 x+3)^4}-\frac{175119 (8 x+7) \sqrt{3 x^2+5 x+2}}{20480000000 (2 x+3)^2}+\frac{175119 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40960000000 \sqrt{5}} \]
[Out]
(-175119*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000000*(3 + 2*x)^2) + (58373*(7
+ 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(512000000*(3 + 2*x)^4) - (58373*(7 + 8*x)*(2 +
5*x + 3*x^2)^(5/2))/(32000000*(3 + 2*x)^6) + (25017*(7 + 8*x)*(2 + 5*x + 3*x^2)^
(7/2))/(800000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(60*(3 + 2*x)^12) - (
12*(2 + 5*x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) - (2067*(2 + 5*x + 3*x^2)^(9/2))/(
11000*(3 + 2*x)^10) - (6379*(2 + 5*x + 3*x^2)^(9/2))/(41250*(3 + 2*x)^9) + (1751
19*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40960000000*Sqrt[5])
_______________________________________________________________________________________
Rubi [A] time = 0.484801, antiderivative size = 259, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185
\[ -\frac{6379 \left (3 x^2+5 x+2\right )^{9/2}}{41250 (2 x+3)^9}-\frac{2067 \left (3 x^2+5 x+2\right )^{9/2}}{11000 (2 x+3)^{10}}-\frac{12 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}+\frac{25017 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800000 (2 x+3)^8}-\frac{58373 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{32000000 (2 x+3)^6}+\frac{58373 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{512000000 (2 x+3)^4}-\frac{175119 (8 x+7) \sqrt{3 x^2+5 x+2}}{20480000000 (2 x+3)^2}+\frac{175119 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40960000000 \sqrt{5}} \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^13,x]
[Out]
(-175119*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000000*(3 + 2*x)^2) + (58373*(7
+ 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(512000000*(3 + 2*x)^4) - (58373*(7 + 8*x)*(2 +
5*x + 3*x^2)^(5/2))/(32000000*(3 + 2*x)^6) + (25017*(7 + 8*x)*(2 + 5*x + 3*x^2)^
(7/2))/(800000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(60*(3 + 2*x)^12) - (
12*(2 + 5*x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) - (2067*(2 + 5*x + 3*x^2)^(9/2))/(
11000*(3 + 2*x)^10) - (6379*(2 + 5*x + 3*x^2)^(9/2))/(41250*(3 + 2*x)^9) + (1751
19*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40960000000*Sqrt[5])
_______________________________________________________________________________________
Rubi in Sympy [A] time = 70.7525, size = 246, normalized size = 0.95 \[ - \frac{175119 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{204800000000} - \frac{175119 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{20480000000 \left (2 x + 3\right )^{2}} + \frac{58373 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{512000000 \left (2 x + 3\right )^{4}} - \frac{58373 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{32000000 \left (2 x + 3\right )^{6}} + \frac{25017 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{800000 \left (2 x + 3\right )^{8}} - \frac{6379 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{41250 \left (2 x + 3\right )^{9}} - \frac{2067 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{11000 \left (2 x + 3\right )^{10}} - \frac{12 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{55 \left (2 x + 3\right )^{11}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{60 \left (2 x + 3\right )^{12}} \]
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)**13,x)
[Out]
-175119*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/2048000000
00 - 175119*(8*x + 7)*sqrt(3*x**2 + 5*x + 2)/(20480000000*(2*x + 3)**2) + 58373*
(8*x + 7)*(3*x**2 + 5*x + 2)**(3/2)/(512000000*(2*x + 3)**4) - 58373*(8*x + 7)*(
3*x**2 + 5*x + 2)**(5/2)/(32000000*(2*x + 3)**6) + 25017*(8*x + 7)*(3*x**2 + 5*x
+ 2)**(7/2)/(800000*(2*x + 3)**8) - 6379*(3*x**2 + 5*x + 2)**(9/2)/(41250*(2*x
+ 3)**9) - 2067*(3*x**2 + 5*x + 2)**(9/2)/(11000*(2*x + 3)**10) - 12*(3*x**2 + 5
*x + 2)**(9/2)/(55*(2*x + 3)**11) - 13*(3*x**2 + 5*x + 2)**(9/2)/(60*(2*x + 3)**
12)
_______________________________________________________________________________________
Mathematica [A] time = 0.165293, size = 144, normalized size = 0.56 \[ -\frac{5778927 \sqrt{5} (2 x+3)^{12} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-5778927 \sqrt{5} (2 x+3)^{12} \log (2 x+3)-10 \sqrt{3 x^2+5 x+2} \left (60734693376 x^{11}+1044584776704 x^{10}+8182662620160 x^9+38544695427840 x^8+123629135656960 x^7+273282692080768 x^6+410468875350912 x^5+412855931529440 x^4+271870111600160 x^3+111795175925940 x^2+25843081681156 x+2531527640959\right )}{6758400000000 (2 x+3)^{12}} \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^13,x]
[Out]
-(-10*Sqrt[2 + 5*x + 3*x^2]*(2531527640959 + 25843081681156*x + 111795175925940*
x^2 + 271870111600160*x^3 + 412855931529440*x^4 + 410468875350912*x^5 + 27328269
2080768*x^6 + 123629135656960*x^7 + 38544695427840*x^8 + 8182662620160*x^9 + 104
4584776704*x^10 + 60734693376*x^11) - 5778927*Sqrt[5]*(3 + 2*x)^12*Log[3 + 2*x]
+ 5778927*Sqrt[5]*(3 + 2*x)^12*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/
(6758400000000*(3 + 2*x)^12)
_______________________________________________________________________________________
Maple [A] time = 0.183, size = 432, normalized size = 1.7 \[ \text{result too large to display} \]
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)^13,x)
[Out]
25017/200000000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+175119/800000000000*(3*(x+3/2)^2
-4*x-19/4)^(5/2)+58373/128000000000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+175119/20480000
0000*(12*(x+3/2)^2-16*x-19)^(1/2)-13/245760/(x+3/2)^12*(3*(x+3/2)^2-4*x-19/4)^(9
/2)-3/28160/(x+3/2)^11*(3*(x+3/2)^2-4*x-19/4)^(9/2)-2067/11264000/(x+3/2)^10*(3*
(x+3/2)^2-4*x-19/4)^(9/2)-6379/21120000/(x+3/2)^9*(3*(x+3/2)^2-4*x-19/4)^(9/2)-2
5017/51200000/(x+3/2)^8*(3*(x+3/2)^2-4*x-19/4)^(9/2)-25017/32000000/(x+3/2)^7*(3
*(x+3/2)^2-4*x-19/4)^(9/2)-158441/128000000/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(9/
2)-775527/400000000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-48057657/16000000000/
(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-46022941/10000000000/(x+3/2)^3*(3*(x+3/2)
^2-4*x-19/4)^(9/2)-1395223107/200000000000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2
)-261602769/25000000000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+261602769/500000000
00*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-101744139/200000000000*(5+6*x)*(3*(x+3/2
)^2-4*x-19/4)^(5/2)+1692817/32000000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-175
119/25600000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-175119/204800000000*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 = 0.83529, size = 980, normalized size = 3.78 \[ \text{result too large to display} \]
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)^13,x, algorithm="maxima")
[Out]
4185669321/200000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/60*(3*x^2 + 5*x + 2)^(9/2)/
(4096*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^9 + 10264320*x^8 + 24634368*x^
7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^3 + 15588936*x^2 + 4
251528*x + 531441) - 12/55*(3*x^2 + 5*x + 2)^(9/2)/(2048*x^11 + 33792*x^10 + 253
440*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) - 2067/11000*(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) - 6379/41250*(
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 + 19683) - 25017/200000*(3*x^
2 + 5*x + 2)^(9/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108
864*x^3 + 81648*x^2 + 34992*x + 6561) - 25017/250000*(3*x^2 + 5*x + 2)^(9/2)/(12
8*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187
) - 158441/2000000*(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) - 775527/12500000*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5
+ 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 48057657/1000000000*(3*x^2 + 5*x
+ 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 46022941/1250000000*(3*x^
2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1395223107/50000000000*(3*x^2
+ 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 305232417/100000000000*(3*x^2 + 5*x + 2)^(
5/2)*x - 2034707661/800000000000*(3*x^2 + 5*x + 2)^(5/2) - 261602769/10000000000
*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 5078451/16000000000*(3*x^2 + 5*x + 2)^(3/2)
*x + 33914713/128000000000*(3*x^2 + 5*x + 2)^(3/2) - 525357/12800000000*sqrt(3*x
^2 + 5*x + 2)*x - 175119/204800000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/
abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 3327261/102400000000*sqrt(3*x^2 + 5*x + 2
)
_______________________________________________________________________________________
Fricas [A] time = 0.304723, size = 338, normalized size = 1.31 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (60734693376 \, x^{11} + 1044584776704 \, x^{10} + 8182662620160 \, x^{9} + 38544695427840 \, x^{8} + 123629135656960 \, x^{7} + 273282692080768 \, x^{6} + 410468875350912 \, x^{5} + 412855931529440 \, x^{4} + 271870111600160 \, x^{3} + 111795175925940 \, x^{2} + 25843081681156 \, x + 2531527640959\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 5778927 \,{\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{13516800000000 \,{\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )}} \]
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)^13,x, algorithm="fricas")
[Out]
1/13516800000000*sqrt(5)*(4*sqrt(5)*(60734693376*x^11 + 1044584776704*x^10 + 818
2662620160*x^9 + 38544695427840*x^8 + 123629135656960*x^7 + 273282692080768*x^6
+ 410468875350912*x^5 + 412855931529440*x^4 + 271870111600160*x^3 + 111795175925
940*x^2 + 25843081681156*x + 2531527640959)*sqrt(3*x^2 + 5*x + 2) + 5778927*(409
6*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^9 + 10264320*x^8 + 24634368*x^7 +
43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^3 + 15588936*x^2 + 42515
28*x + 531441)*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)))/(4096*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^
9 + 10264320*x^8 + 24634368*x^7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 3
4642080*x^3 + 15588936*x^2 + 4251528*x + 531441)
_______________________________________________________________________________________
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)**13,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.361313, size = 967, 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)^(7/2)*(x - 5)/(2*x + 3)^13,x, algorithm="giac")
[Out]
175119/204800000000*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))) - 1/675840000000*(11835242496*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^23 + 4
08315866112*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^22 + 20039038086144*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^21 + 535243596890112*sqrt(3)*(sqrt(3)*x - sqrt(3
*x^2 + 5*x + 2))^20 + 13859706456921600*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^19 +
31535346744025344*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 - 789031961976
842496*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 - 7977976824329385984*sqrt(3)*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 113078650509677476096*(sqrt(3)*x - sqrt(3*x
^2 + 5*x + 2))^15 - 358779889050339715200*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^14 - 2538162771649151164032*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 - 46602
43350382625915904*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 - 2049912252415
5108829248*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 - 24347916060701730772704*sqrt
(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 70788415443572756925600*(sqrt(3)*x
- sqrt(3*x^2 + 5*x + 2))^9 - 56076083911431114398208*sqrt(3)*(sqrt(3)*x - sqrt(3
*x^2 + 5*x + 2))^8 - 108598043564223524909928*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)
)^7 - 56663550021725424101412*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 70
668287639831997261828*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 228760370849032471
15200*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 16680770211437743348146*(s
qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 2864949797863813201587*sqrt(3)*(sqrt(3)*x
- sqrt(3*x^2 + 5*x + 2))^2 - 930278306769206446269*sqrt(3)*x - 47729262032858665
512*sqrt(3) + 930278306769206446269*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)^12