3.2460 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx\)

Optimal. Leaf size=184 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac{329 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{96000 (2 x+3)^6}+\frac{329 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1536000 (2 x+3)^4}-\frac{329 (8 x+7) \sqrt{3 x^2+5 x+2}}{20480000 (2 x+3)^2}+\frac{329 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40960000 \sqrt{5}} \]

[Out]

(-329*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000*(3 + 2*x)^2) + (329*(7 + 8*x)*(
2 + 5*x + 3*x^2)^(3/2))/(1536000*(3 + 2*x)^4) - (329*(7 + 8*x)*(2 + 5*x + 3*x^2)
^(5/2))/(96000*(3 + 2*x)^6) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(800*(3 + 2
*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) + (329*ArcTanh[(7 + 8*x)/
(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40960000*Sqrt[5])

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Rubi [A]  time = 0.253096, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac{329 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{96000 (2 x+3)^6}+\frac{329 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1536000 (2 x+3)^4}-\frac{329 (8 x+7) \sqrt{3 x^2+5 x+2}}{20480000 (2 x+3)^2}+\frac{329 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40960000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^10,x]

[Out]

(-329*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000*(3 + 2*x)^2) + (329*(7 + 8*x)*(
2 + 5*x + 3*x^2)^(3/2))/(1536000*(3 + 2*x)^4) - (329*(7 + 8*x)*(2 + 5*x + 3*x^2)
^(5/2))/(96000*(3 + 2*x)^6) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(800*(3 + 2
*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) + (329*ArcTanh[(7 + 8*x)/
(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40960000*Sqrt[5])

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Rubi in Sympy [A]  time = 44.751, size = 175, normalized size = 0.95 \[ - \frac{329 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{204800000} - \frac{329 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{20480000 \left (2 x + 3\right )^{2}} + \frac{329 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{1536000 \left (2 x + 3\right )^{4}} - \frac{329 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{96000 \left (2 x + 3\right )^{6}} + \frac{47 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{800 \left (2 x + 3\right )^{8}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{45 \left (2 x + 3\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**10,x)

[Out]

-329*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/204800000 - 3
29*(8*x + 7)*sqrt(3*x**2 + 5*x + 2)/(20480000*(2*x + 3)**2) + 329*(8*x + 7)*(3*x
**2 + 5*x + 2)**(3/2)/(1536000*(2*x + 3)**4) - 329*(8*x + 7)*(3*x**2 + 5*x + 2)*
*(5/2)/(96000*(2*x + 3)**6) + 47*(8*x + 7)*(3*x**2 + 5*x + 2)**(7/2)/(800*(2*x +
3)**8) - 13*(3*x**2 + 5*x + 2)**(9/2)/(45*(2*x + 3)**9)

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Mathematica [A]  time = 0.136947, size = 129, normalized size = 0.7 \[ -\frac{2961 \sqrt{5} (2 x+3)^9 \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-10 \sqrt{3 x^2+5 x+2} \left (28394496 x^8+2848109952 x^7+15895201728 x^6+38558367264 x^5+51825176720 x^4+41530110824 x^3+19810691268 x^2+5201574542 x+578701331\right )-2961 \sqrt{5} (2 x+3)^9 \log (2 x+3)}{1843200000 (2 x+3)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^10,x]

[Out]

-(-10*Sqrt[2 + 5*x + 3*x^2]*(578701331 + 5201574542*x + 19810691268*x^2 + 415301
10824*x^3 + 51825176720*x^4 + 38558367264*x^5 + 15895201728*x^6 + 2848109952*x^7
+ 28394496*x^8) - 2961*Sqrt[5]*(3 + 2*x)^9*Log[3 + 2*x] + 2961*Sqrt[5]*(3 + 2*x
)^9*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/(1843200000*(3 + 2*x)^9)

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Maple [B]  time = 0.055, size = 369, normalized size = 2. \[ \text{result too large to display} \]

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)^10,x)

[Out]

47/200000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+329/800000000*(3*(x+3/2)^2-4*x-19/4)^(
5/2)+329/384000000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+329/204800000*(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)^(9/2)-47/51200/(x+3/2)^8*(3
*(x+3/2)^2-4*x-19/4)^(9/2)-47/32000/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(9/2)-893/3
84000/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1457/400000/(x+3/2)^5*(3*(x+3/2)^2-
4*x-19/4)^(9/2)-90287/16000000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-259393/300
00000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-2621237/200000000/(x+3/2)^2*(3*(x+3
/2)^2-4*x-19/4)^(9/2)-491479/25000000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+49147
9/50000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-191149/200000000*(5+6*x)*(3*(x+3
/2)^2-4*x-19/4)^(5/2)+9541/96000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-329/256
00000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-329/204800000*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 = 0.821149, size = 693, normalized size = 3.77 \[ \text{result too large to display} \]

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)^10,x, algorithm="maxima")

[Out]

7863711/200000000*(3*x^2 + 5*x + 2)^(7/2) - 13/45*(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) - 47/200*(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 + 656
1) - 47/250*(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) - 893/6000*(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) - 1457/12500*(3*x
^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 9028
7/1000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 259
393/3750000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2621237/50000
000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 573447/100000000*(3*x^2 + 5*x +
2)^(5/2)*x - 3822651/800000000*(3*x^2 + 5*x + 2)^(5/2) - 491479/10000000*(3*x^2
+ 5*x + 2)^(7/2)/(2*x + 3) + 9541/16000000*(3*x^2 + 5*x + 2)^(3/2)*x + 191149/3
84000000*(3*x^2 + 5*x + 2)^(3/2) - 987/12800000*sqrt(3*x^2 + 5*x + 2)*x - 329/20
4800000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3
) - 2) - 6251/102400000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.291793, size = 277, normalized size = 1.51 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (28394496 \, x^{8} + 2848109952 \, x^{7} + 15895201728 \, x^{6} + 38558367264 \, x^{5} + 51825176720 \, x^{4} + 41530110824 \, x^{3} + 19810691268 \, x^{2} + 5201574542 \, x + 578701331\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 2961 \,{\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{3686400000 \,{\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^10,x, algorithm="fricas")

[Out]

1/3686400000*sqrt(5)*(4*sqrt(5)*(28394496*x^8 + 2848109952*x^7 + 15895201728*x^6
+ 38558367264*x^5 + 51825176720*x^4 + 41530110824*x^3 + 19810691268*x^2 + 52015
74542*x + 578701331)*sqrt(3*x^2 + 5*x + 2) + 2961*(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)*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 + 326592*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)**(7/2)/(3+2*x)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.316718, size = 760, normalized size = 4.13 \[ \text{result too large to display} \]

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)^10,x, algorithm="giac")

[Out]

329/204800000*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))
) - 1/184320000*(14930678016*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 + 2040615694
08*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 + 3866707486848*(sqrt(3)*x - s
qrt(3*x^2 + 5*x + 2))^15 + 14840812733760*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^14 + 114102022608000*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 198779998219
488*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 649357338634272*(sqrt(3)*x
- sqrt(3*x^2 + 5*x + 2))^11 + 207317438979984*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^10 - 2217334591351040*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 52479133
96815000*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 20151247122371016*(sqrt
(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 17924557725783828*sqrt(3)*(sqrt(3)*x - sqrt(3
*x^2 + 5*x + 2))^6 - 35125577732048328*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 1
6953161853593070*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 177522047264752
50*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 4253745315948057*sqrt(3)*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^2 - 1882391465118753*sqrt(3)*x - 129047626217736*sqrt(3)
+ 1882391465118753*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