3.2461 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx\)
Optimal. Leaf size=209 \[ -\frac{29 \left (3 x^2+5 x+2\right )^{9/2}}{125 (2 x+3)^9}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}+\frac{1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{40000 (2 x+3)^8}-\frac{4417 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{1600000 (2 x+3)^6}+\frac{4417 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{25600000 (2 x+3)^4}-\frac{13251 (8 x+7) \sqrt{3 x^2+5 x+2}}{1024000000 (2 x+3)^2}+\frac{13251 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2048000000 \sqrt{5}} \]
[Out]
(-13251*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(1024000000*(3 + 2*x)^2) + (4417*(7 + 8
*x)*(2 + 5*x + 3*x^2)^(3/2))/(25600000*(3 + 2*x)^4) - (4417*(7 + 8*x)*(2 + 5*x +
3*x^2)^(5/2))/(1600000*(3 + 2*x)^6) + (1893*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/
(40000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(50*(3 + 2*x)^10) - (29*(2 +
5*x + 3*x^2)^(9/2))/(125*(3 + 2*x)^9) + (13251*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt
[2 + 5*x + 3*x^2])])/(2048000000*Sqrt[5])
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Rubi [A] time = 0.329533, antiderivative size = 209, 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
\[ -\frac{29 \left (3 x^2+5 x+2\right )^{9/2}}{125 (2 x+3)^9}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}+\frac{1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{40000 (2 x+3)^8}-\frac{4417 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{1600000 (2 x+3)^6}+\frac{4417 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{25600000 (2 x+3)^4}-\frac{13251 (8 x+7) \sqrt{3 x^2+5 x+2}}{1024000000 (2 x+3)^2}+\frac{13251 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2048000000 \sqrt{5}} \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^11,x]
[Out]
(-13251*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(1024000000*(3 + 2*x)^2) + (4417*(7 + 8
*x)*(2 + 5*x + 3*x^2)^(3/2))/(25600000*(3 + 2*x)^4) - (4417*(7 + 8*x)*(2 + 5*x +
3*x^2)^(5/2))/(1600000*(3 + 2*x)^6) + (1893*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/
(40000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(50*(3 + 2*x)^10) - (29*(2 +
5*x + 3*x^2)^(9/2))/(125*(3 + 2*x)^9) + (13251*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt
[2 + 5*x + 3*x^2])])/(2048000000*Sqrt[5])
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Rubi in Sympy [A] time = 53.0878, size = 199, normalized size = 0.95 \[ - \frac{13251 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{10240000000} - \frac{13251 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{1024000000 \left (2 x + 3\right )^{2}} + \frac{4417 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{25600000 \left (2 x + 3\right )^{4}} - \frac{4417 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{1600000 \left (2 x + 3\right )^{6}} + \frac{1893 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{40000 \left (2 x + 3\right )^{8}} - \frac{29 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{125 \left (2 x + 3\right )^{9}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{50 \left (2 x + 3\right )^{10}} \]
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)**11,x)
[Out]
-13251*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/10240000000
- 13251*(8*x + 7)*sqrt(3*x**2 + 5*x + 2)/(1024000000*(2*x + 3)**2) + 4417*(8*x
+ 7)*(3*x**2 + 5*x + 2)**(3/2)/(25600000*(2*x + 3)**4) - 4417*(8*x + 7)*(3*x**2
+ 5*x + 2)**(5/2)/(1600000*(2*x + 3)**6) + 1893*(8*x + 7)*(3*x**2 + 5*x + 2)**(7
/2)/(40000*(2*x + 3)**8) - 29*(3*x**2 + 5*x + 2)**(9/2)/(125*(2*x + 3)**9) - 13*
(3*x**2 + 5*x + 2)**(9/2)/(50*(2*x + 3)**10)
_______________________________________________________________________________________
Mathematica [A] time = 0.146252, size = 134, normalized size = 0.64 \[ -\frac{13251 \sqrt{5} (2 x+3)^{10} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-10 \sqrt{3 x^2+5 x+2} \left (371791872 x^9+5268182272 x^8+40186580992 x^7+148740043392 x^6+304078211712 x^5+372602220928 x^4+281702072128 x^3+128970753208 x^2+32786922608 x+3544392763\right )-13251 \sqrt{5} (2 x+3)^{10} \log (2 x+3)}{10240000000 (2 x+3)^{10}} \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^11,x]
[Out]
-(-10*Sqrt[2 + 5*x + 3*x^2]*(3544392763 + 32786922608*x + 128970753208*x^2 + 281
702072128*x^3 + 372602220928*x^4 + 304078211712*x^5 + 148740043392*x^6 + 4018658
0992*x^7 + 5268182272*x^8 + 371791872*x^9) - 13251*Sqrt[5]*(3 + 2*x)^10*Log[3 +
2*x] + 13251*Sqrt[5]*(3 + 2*x)^10*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
])/(10240000000*(3 + 2*x)^10)
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Maple [B] time = 0.08, size = 390, 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)^11,x)
[Out]
1893/10000000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+13251/40000000000*(3*(x+3/2)^2-4*x
-19/4)^(5/2)+4417/6400000000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+13251/10240000000*(12*
(x+3/2)^2-16*x-19)^(1/2)-13/51200/(x+3/2)^10*(3*(x+3/2)^2-4*x-19/4)^(9/2)-29/640
00/(x+3/2)^9*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1893/2560000/(x+3/2)^8*(3*(x+3/2)^2-4*
x-19/4)^(9/2)-1893/1600000/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(9/2)-11989/6400000/
(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(9/2)-58683/20000000/(x+3/2)^5*(3*(x+3/2)^2-4*x
-19/4)^(9/2)-3636453/800000000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-3482489/50
0000000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-105574503/10000000000/(x+3/2)^2*(
3*(x+3/2)^2-4*x-19/4)^(9/2)-19795101/1250000000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(
9/2)+19795101/2500000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-7698831/1000000000
0*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+128093/1600000000*(5+6*x)*(3*(x+3/2)^2-4*
x-19/4)^(3/2)-13251/1280000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-13251/102400
00000*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 = 0.826755, size = 782, normalized size = 3.74 \[ \text{result too large to display} \]
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)^11,x, algorithm="maxima")
[Out]
316723509/10000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/50*(3*x^2 + 5*x + 2)^(9/2)/(1
024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 + 244
9440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049) - 29/125*(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) - 1893/10000*(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 + 34992*x + 6561) - 1893/12500*(3*x^2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6
+ 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 11989/100000*
(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 291
6*x + 729) - 58683/625000*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 +
1080*x^2 + 810*x + 243) - 3636453/50000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*
x^3 + 216*x^2 + 216*x + 81) - 3482489/62500000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 +
36*x^2 + 54*x + 27) - 105574503/2500000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x
+ 9) - 23096493/5000000000*(3*x^2 + 5*x + 2)^(5/2)*x - 153963369/40000000000*(3
*x^2 + 5*x + 2)^(5/2) - 19795101/500000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 3
84279/800000000*(3*x^2 + 5*x + 2)^(3/2)*x + 2566277/6400000000*(3*x^2 + 5*x + 2)
^(3/2) - 39753/640000000*sqrt(3*x^2 + 5*x + 2)*x - 13251/10240000000*sqrt(5)*log
(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 251769/512
0000000*sqrt(3*x^2 + 5*x + 2)
_______________________________________________________________________________________
Fricas [A] time = 0.293369, size = 297, normalized size = 1.42 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (371791872 \, x^{9} + 5268182272 \, x^{8} + 40186580992 \, x^{7} + 148740043392 \, x^{6} + 304078211712 \, x^{5} + 372602220928 \, x^{4} + 281702072128 \, x^{3} + 128970753208 \, x^{2} + 32786922608 \, x + 3544392763\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 13251 \,{\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{20480000000 \,{\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )}} \]
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)^11,x, algorithm="fricas")
[Out]
1/20480000000*sqrt(5)*(4*sqrt(5)*(371791872*x^9 + 5268182272*x^8 + 40186580992*x
^7 + 148740043392*x^6 + 304078211712*x^5 + 372602220928*x^4 + 281702072128*x^3 +
128970753208*x^2 + 32786922608*x + 3544392763)*sqrt(3*x^2 + 5*x + 2) + 13251*(1
024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 + 244
9440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049)*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)))/(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)
_______________________________________________________________________________________
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)**11,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.355632, size = 829, normalized size = 3.97 \[ \text{result too large to display} \]
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)^11,x, algorithm="giac")
[Out]
13251/10240000000*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))) - 1/1024000000*(6784512*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^19 + 831373585
92*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 + 2689605043456*(sqrt(3)*x - s
qrt(3*x^2 + 5*x + 2))^17 + 9174489217536*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^16 - 53080570863872*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 - 89878313572262
4*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 - 13174687008250752*(sqrt(3)*x
- sqrt(3*x^2 + 5*x + 2))^13 - 40507172795248512*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^12 - 270169596727110016*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 - 458
790099197766656*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 183318353317374
3552*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 1939024456450048032*sqrt(3)*(sqrt(3
)*x - sqrt(3*x^2 + 5*x + 2))^8 - 4903074367120921776*(sqrt(3)*x - sqrt(3*x^2 + 5
*x + 2))^7 - 3280073192617110456*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 -
5164856211259534888*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 2082844158764403144
*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 1869656136275991262*(sqrt(3)*x
- sqrt(3*x^2 + 5*x + 2))^3 - 391066159205340747*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^2 - 153124376229353121*sqrt(3)*x - 9387541838830536*sqrt(3) + 153124
376229353121*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)^10