3.1395 \(\int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx\)

Optimal. Leaf size=180 \[ -\frac{4741 \left (3 x^2+2\right )^{7/2}}{1800750 (2 x+3)^7}-\frac{27 \left (3 x^2+2\right )^{7/2}}{2450 (2 x+3)^8}-\frac{13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}-\frac{949 (4-9 x) \left (3 x^2+2\right )^{5/2}}{3001250 (2 x+3)^6}-\frac{2847 (4-9 x) \left (3 x^2+2\right )^{3/2}}{42017500 (2 x+3)^4}-\frac{25623 (4-9 x) \sqrt{3 x^2+2}}{1470612500 (2 x+3)^2}-\frac{76869 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{735306250 \sqrt{35}} \]

[Out]

(-25623*(4 - 9*x)*Sqrt[2 + 3*x^2])/(1470612500*(3 + 2*x)^2) - (2847*(4 - 9*x)*(2
+ 3*x^2)^(3/2))/(42017500*(3 + 2*x)^4) - (949*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(300
1250*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(315*(3 + 2*x)^9) - (27*(2 + 3*x^2)^(
7/2))/(2450*(3 + 2*x)^8) - (4741*(2 + 3*x^2)^(7/2))/(1800750*(3 + 2*x)^7) - (768
69*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(735306250*Sqrt[35])

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Rubi [A]  time = 0.283307, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{4741 \left (3 x^2+2\right )^{7/2}}{1800750 (2 x+3)^7}-\frac{27 \left (3 x^2+2\right )^{7/2}}{2450 (2 x+3)^8}-\frac{13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}-\frac{949 (4-9 x) \left (3 x^2+2\right )^{5/2}}{3001250 (2 x+3)^6}-\frac{2847 (4-9 x) \left (3 x^2+2\right )^{3/2}}{42017500 (2 x+3)^4}-\frac{25623 (4-9 x) \sqrt{3 x^2+2}}{1470612500 (2 x+3)^2}-\frac{76869 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{735306250 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(-25623*(4 - 9*x)*Sqrt[2 + 3*x^2])/(1470612500*(3 + 2*x)^2) - (2847*(4 - 9*x)*(2
+ 3*x^2)^(3/2))/(42017500*(3 + 2*x)^4) - (949*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(300
1250*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(315*(3 + 2*x)^9) - (27*(2 + 3*x^2)^(
7/2))/(2450*(3 + 2*x)^8) - (4741*(2 + 3*x^2)^(7/2))/(1800750*(3 + 2*x)^7) - (768
69*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(735306250*Sqrt[35])

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Rubi in Sympy [A]  time = 33.6988, size = 170, normalized size = 0.94 \[ - \frac{25623 \left (- 18 x + 8\right ) \sqrt{3 x^{2} + 2}}{2941225000 \left (2 x + 3\right )^{2}} - \frac{2847 \left (- 18 x + 8\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{84035000 \left (2 x + 3\right )^{4}} - \frac{949 \left (- 18 x + 8\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{6002500 \left (2 x + 3\right )^{6}} - \frac{76869 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{25735718750} - \frac{4741 \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{1800750 \left (2 x + 3\right )^{7}} - \frac{27 \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{2450 \left (2 x + 3\right )^{8}} - \frac{13 \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{315 \left (2 x + 3\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25623*(-18*x + 8)*sqrt(3*x**2 + 2)/(2941225000*(2*x + 3)**2) - 2847*(-18*x + 8)
*(3*x**2 + 2)**(3/2)/(84035000*(2*x + 3)**4) - 949*(-18*x + 8)*(3*x**2 + 2)**(5/
2)/(6002500*(2*x + 3)**6) - 76869*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*
x**2 + 2)))/25735718750 - 4741*(3*x**2 + 2)**(7/2)/(1800750*(2*x + 3)**7) - 27*(
3*x**2 + 2)**(7/2)/(2450*(2*x + 3)**8) - 13*(3*x**2 + 2)**(7/2)/(315*(2*x + 3)**
9)

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Mathematica [A]  time = 0.244611, size = 110, normalized size = 0.61 \[ \frac{-1383642 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{35 \sqrt{3 x^2+2} \left (10968696 x^8+30006612 x^7-620594352 x^6-25197346566 x^5+9750269970 x^4-11567526201 x^3+42455611758 x^2+11990965797 x+15948113036\right )}{(2 x+3)^9}+1383642 \sqrt{35} \log (2 x+3)}{463242937500} \]

Antiderivative was successfully verified.

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

[Out]

((-35*Sqrt[2 + 3*x^2]*(15948113036 + 11990965797*x + 42455611758*x^2 - 115675262
01*x^3 + 9750269970*x^4 - 25197346566*x^5 - 620594352*x^6 + 30006612*x^7 + 10968
696*x^8))/(3 + 2*x)^9 + 1383642*Sqrt[35]*Log[3 + 2*x] - 1383642*Sqrt[35]*Log[2*(
4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/463242937500

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Maple [B]  time = 0.052, size = 320, 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+2)^(5/2)/(2*x+3)^10,x)

[Out]

-106771041/157631277343750/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)+691821/514714375
00*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)-76869/25735718750*35^(1/2)*arctanh(2/35*(4-9*x
)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))[phone]/157631277343750*x*(3*(x+3/2)^
2-9*x-19/4)^(5/2)-13/161280/(x+3/2)^9*(3*(x+3/2)^2-9*x-19/4)^(7/2)-8541/16807000
00/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)-27/627200/(x+3/2)^8*(3*(x+3/2)^2-9*x-1
9/4)^(7/2)-949/96040000/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(7/2)-82563/29412250000
/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(7/2)-845559/514714375000/(x+3/2)^3*(3*(x+3/2)
^2-9*x-19/4)^(7/2)-4741/230496000/(x+3/2)^7*(3*(x+3/2)^2-9*x-19/4)^(7/2)+8993673
/1801500312500*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-9198657/9007501562500/(x+3/2)^2*(3
*(x+3/2)^2-9*x-19/4)^(7/2)+102492/450375078125*(3*(x+3/2)^2-9*x-19/4)^(3/2)+7686
9/25735718750*(12*(x+3/2)^2-36*x-19)^(1/2)+1229904/78815638671875*(3*(x+3/2)^2-9
*x-19/4)^(5/2)

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Maxima [A]  time = 0.781539, size = 586, normalized size = 3.26 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27595971/9007501562500*(3*x^2 + 2)^(5/2) - 13/315*(3*x^2 + 2)^(7/2)/(512*x^9 + 6
912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928
*x^2 + 118098*x + 19683) - 27/2450*(3*x^2 + 2)^(7/2)/(256*x^8 + 3072*x^7 + 16128
*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 4741/1
800750*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3
+ 20412*x^2 + 10206*x + 2187) - 949/1500625*(3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5
+ 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 8541/52521875*(3*x^2 + 2)^(7/
2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 82563/1838265625*(3*x
^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 845559/64339296875*(3*x
^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 9198657/2251875390625*(3*x^2 + 2)^(
7/2)/(4*x^2 + 12*x + 9) + 8993673/1801500312500*(3*x^2 + 2)^(3/2)*x + 102492/450
375078125*(3*x^2 + 2)^(3/2) - 106771041/9007501562500*(3*x^2 + 2)^(5/2)/(2*x + 3
) + 691821/51471437500*sqrt(3*x^2 + 2)*x + 76869/25735718750*sqrt(35)*arcsinh(3/
2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 76869/12867859375*sqrt(3*
x^2 + 2)

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Fricas [A]  time = 0.291889, size = 269, normalized size = 1.49 \[ -\frac{\sqrt{35}{\left (\sqrt{35}{\left (10968696 \, x^{8} + 30006612 \, x^{7} - 620594352 \, x^{6} - 25197346566 \, x^{5} + 9750269970 \, x^{4} - 11567526201 \, x^{3} + 42455611758 \, x^{2} + 11990965797 \, x + 15948113036\right )} \sqrt{3 \, x^{2} + 2} - 691821 \,{\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{463242937500 \,{\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/463242937500*sqrt(35)*(sqrt(35)*(10968696*x^8 + 30006612*x^7 - 620594352*x^6
- 25197346566*x^5 + 9750269970*x^4 - 11567526201*x^3 + 42455611758*x^2 + 1199096
5797*x + 15948113036)*sqrt(3*x^2 + 2) - 691821*(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 + 196
83)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2 + 2)*(9*x - 4))/(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+2)**(5/2)/(3+2*x)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.328409, size = 672, normalized size = 3.73 \[ \frac{76869}{25735718750} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{9 \,{\left (1093248 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{17} + 27877824 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{16} + 3126615774 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{15} - 956098170 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{14} + 3010876470 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{13} - 85987901496 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{12} - 181405205604 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} - 331045664193 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} - 68739446745 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} - 544736640510 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 854568812592 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 908850124224 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} + 271848650976 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 115517223360 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 158685613440 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 565618176 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 422125056 \, \sqrt{3} x - 17333248 \, \sqrt{3} - 422125056 \, \sqrt{3 \, x^{2} + 2}\right )}}{94119200000 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

76869/25735718750*sqrt(35)*ln(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(
3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/941192
00000*(1093248*(sqrt(3)*x - sqrt(3*x^2 + 2))^17 + 27877824*sqrt(3)*(sqrt(3)*x -
sqrt(3*x^2 + 2))^16 + 3126615774*(sqrt(3)*x - sqrt(3*x^2 + 2))^15 - 956098170*sq
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^14 + 3010876470*(sqrt(3)*x - sqrt(3*x^2 + 2)
)^13 - 85987901496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 - 181405205604*(sqrt
(3)*x - sqrt(3*x^2 + 2))^11 - 331045664193*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))
^10 - 68739446745*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 - 544736640510*sqrt(3)*(sqrt(3
)*x - sqrt(3*x^2 + 2))^8 + 854568812592*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 908850
124224*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 + 271848650976*(sqrt(3)*x - sqrt(
3*x^2 + 2))^5 - 115517223360*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 158685613
440*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 565618176*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 2))^2 + 422125056*sqrt(3)*x - 17333248*sqrt(3) - 422125056*sqrt(3*x^2 + 2))/((
sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^9