3.2458 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx\)

Optimal. Leaf size=197 \[ \frac{(266 x+269) \left (3 x^2+5 x+2\right )^{7/2}}{280 (2 x+3)^7}+\frac{3 (106 x+135) \left (3 x^2+5 x+2\right )^{5/2}}{640 (2 x+3)^5}+\frac{(30858 x+39767) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}-\frac{3 (61278 x+131465) \sqrt{3 x^2+5 x+2}}{102400 (2 x+3)}+\frac{603}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )-\frac{934161 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{204800 \sqrt{5}} \]

[Out]

(-3*(131465 + 61278*x)*Sqrt[2 + 5*x + 3*x^2])/(102400*(3 + 2*x)) + ((39767 + 308
58*x)*(2 + 5*x + 3*x^2)^(3/2))/(25600*(3 + 2*x)^3) + (3*(135 + 106*x)*(2 + 5*x +
3*x^2)^(5/2))/(640*(3 + 2*x)^5) + ((269 + 266*x)*(2 + 5*x + 3*x^2)^(7/2))/(280*
(3 + 2*x)^7) + (603*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]
)/512 - (934161*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(204800*Sq
rt[5])

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Rubi [A]  time = 0.391091, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(266 x+269) \left (3 x^2+5 x+2\right )^{7/2}}{280 (2 x+3)^7}+\frac{3 (106 x+135) \left (3 x^2+5 x+2\right )^{5/2}}{640 (2 x+3)^5}+\frac{(30858 x+39767) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}-\frac{3 (61278 x+131465) \sqrt{3 x^2+5 x+2}}{102400 (2 x+3)}+\frac{603}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )-\frac{934161 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{204800 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^8,x]

[Out]

(-3*(131465 + 61278*x)*Sqrt[2 + 5*x + 3*x^2])/(102400*(3 + 2*x)) + ((39767 + 308
58*x)*(2 + 5*x + 3*x^2)^(3/2))/(25600*(3 + 2*x)^3) + (3*(135 + 106*x)*(2 + 5*x +
3*x^2)^(5/2))/(640*(3 + 2*x)^5) + ((269 + 266*x)*(2 + 5*x + 3*x^2)^(7/2))/(280*
(3 + 2*x)^7) + (603*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]
)/512 - (934161*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(204800*Sq
rt[5])

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Rubi in Sympy [A]  time = 50.9423, size = 177, normalized size = 0.9 \[ \frac{603 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{512} + \frac{934161 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{1024000} - \frac{\left (22060080 x + 47327400\right ) \sqrt{3 x^{2} + 5 x + 2}}{12288000 \left (2 x + 3\right )} + \frac{\left (5554440 x + 7158060\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{4608000 \left (2 x + 3\right )^{3}} + \frac{\left (47700 x + 60750\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{96000 \left (2 x + 3\right )^{5}} + \frac{\left (798 x + 807\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{840 \left (2 x + 3\right )^{7}} \]

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)**8,x)

[Out]

603*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/512 + 934161*sqr
t(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/1024000 - (22060080*x
+ 47327400)*sqrt(3*x**2 + 5*x + 2)/(12288000*(2*x + 3)) + (5554440*x + 7158060)
*(3*x**2 + 5*x + 2)**(3/2)/(4608000*(2*x + 3)**3) + (47700*x + 60750)*(3*x**2 +
5*x + 2)**(5/2)/(96000*(2*x + 3)**5) + (798*x + 807)*(3*x**2 + 5*x + 2)**(7/2)/(
840*(2*x + 3)**7)

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Mathematica [A]  time = 0.212779, size = 139, normalized size = 0.71 \[ \frac{6539127 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+8442000 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{10 \sqrt{3 x^2+5 x+2} \left (9676800 x^7+338443008 x^6+2361590432 x^5+7622049520 x^4+13619671040 x^3+13975079520 x^2+7753535702 x+1810375853\right )}{(2 x+3)^7}-6539127 \sqrt{5} \log (2 x+3)}{7168000} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^8,x]

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(1810375853 + 7753535702*x + 13975079520*x^2 + 13619
671040*x^3 + 7622049520*x^4 + 2361590432*x^5 + 338443008*x^6 + 9676800*x^7))/(3
+ 2*x)^7 - 6539127*Sqrt[5]*Log[3 + 2*x] + 6539127*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[
5]*Sqrt[2 + 5*x + 3*x^2]] + 8442000*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x
^2]])/7168000

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Maple [B]  time = 0.029, size = 358, 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)^8,x)

[Out]

-934161/7000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)-934161/4000000*(3*(x+3/2)^2-4*x-19/
4)^(5/2)-311387/640000*(3*(x+3/2)^2-4*x-19/4)^(3/2)-934161/1024000*(12*(x+3/2)^2
-16*x-19)^(1/2)-13/4480/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(9/2)-3/896/(x+3/2)^6*(
3*(x+3/2)^2-4*x-19/4)^(9/2)-3/500/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-4719/56
0000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-5147/350000/(x+3/2)^3*(3*(x+3/2)^2-4
*x-19/4)^(9/2)-15267/1000000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)-78423/875000
/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+78423/1750000*(5+6*x)*(3*(x+3/2)^2-4*x-19/
4)^(7/2)+47541/1000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+14777/160000*(5+6*x)
*(3*(x+3/2)^2-4*x-19/4)^(3/2)+29661/128000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+
603/512*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+934161/10
24000*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 = 0.824911, size = 571, normalized size = 2.9 \[ \frac{45801}{1000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{35 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{14 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{24 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{125 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{4719 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{35000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{5147 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{43750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{15267 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{250000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{142623}{500000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{16659}{4000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{78423 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{350000 \,{\left (2 \, x + 3\right )}} + \frac{44331}{80000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{15847}{640000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{88983}{64000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{603}{512} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{934161}{1024000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{340941}{512000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

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)^8,x, algorithm="maxima")

[Out]

45801/1000000*(3*x^2 + 5*x + 2)^(7/2) - 13/35*(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) - 3/1
4*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2
916*x + 729) - 24/125*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080
*x^2 + 810*x + 243) - 4719/35000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*
x^2 + 216*x + 81) - 5147/43750*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x +
27) - 15267/250000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) + 142623/500000*(3
*x^2 + 5*x + 2)^(5/2)*x + 16659/4000000*(3*x^2 + 5*x + 2)^(5/2) - 78423/350000*(
3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 44331/80000*(3*x^2 + 5*x + 2)^(3/2)*x - 15847
/640000*(3*x^2 + 5*x + 2)^(3/2) + 88983/64000*sqrt(3*x^2 + 5*x + 2)*x + 603/512*
sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 934161/1024000*sqrt(5)*
log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 340941/
512000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.301236, size = 346, normalized size = 1.76 \[ \frac{\sqrt{5}{\left (1688400 \, \sqrt{5} \sqrt{3}{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 4 \, \sqrt{5}{\left (9676800 \, x^{7} + 338443008 \, x^{6} + 2361590432 \, x^{5} + 7622049520 \, x^{4} + 13619671040 \, x^{3} + 13975079520 \, x^{2} + 7753535702 \, x + 1810375853\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6539127 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} - 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{14336000 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]

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)^8,x, algorithm="fricas")

[Out]

1/14336000*sqrt(5)*(1688400*sqrt(5)*sqrt(3)*(128*x^7 + 1344*x^6 + 6048*x^5 + 151
20*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x
+ 2)*(6*x + 5) + 72*x^2 + 120*x + 49) - 4*sqrt(5)*(9676800*x^7 + 338443008*x^6 +
2361590432*x^5 + 7622049520*x^4 + 13619671040*x^3 + 13975079520*x^2 + 775353570
2*x + 1810375853)*sqrt(3*x^2 + 5*x + 2) + 6539127*(128*x^7 + 1344*x^6 + 6048*x^5
+ 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)*log((sqrt(5)*(124*x^2 + 2
12*x + 89) - 20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x^2 + 12*x + 9)))/(128*x^7 +
1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)

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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)**8,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)^8,x, algorithm="giac")

[Out]

undef