3.1396 \(\int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{11}} \, dx\)

Optimal. Leaf size=202 \[ -\frac{739619 \left (3 x^2+2\right )^{7/2}}{1260525000 (2 x+3)^7}-\frac{4393 \left (3 x^2+2\right )^{7/2}}{1715000 (2 x+3)^8}-\frac{1171 \left (3 x^2+2\right )^{7/2}}{110250 (2 x+3)^9}-\frac{13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}-\frac{73233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{1050437500 (2 x+3)^6}-\frac{219699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{14706125000 (2 x+3)^4}-\frac{1977291 (4-9 x) \sqrt{3 x^2+2}}{514714375000 (2 x+3)^2}-\frac{5931873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{257357187500 \sqrt{35}} \]

[Out]

(-1977291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(514714375000*(3 + 2*x)^2) - (219699*(4 - 9
*x)*(2 + 3*x^2)^(3/2))/(14706125000*(3 + 2*x)^4) - (73233*(4 - 9*x)*(2 + 3*x^2)^
(5/2))/(1050437500*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(350*(3 + 2*x)^10) - (1
171*(2 + 3*x^2)^(7/2))/(110250*(3 + 2*x)^9) - (4393*(2 + 3*x^2)^(7/2))/(1715000*
(3 + 2*x)^8) - (739619*(2 + 3*x^2)^(7/2))/(1260525000*(3 + 2*x)^7) - (5931873*Ar
cTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(257357187500*Sqrt[35])

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Rubi [A]  time = 0.345997, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{739619 \left (3 x^2+2\right )^{7/2}}{1260525000 (2 x+3)^7}-\frac{4393 \left (3 x^2+2\right )^{7/2}}{1715000 (2 x+3)^8}-\frac{1171 \left (3 x^2+2\right )^{7/2}}{110250 (2 x+3)^9}-\frac{13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}-\frac{73233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{1050437500 (2 x+3)^6}-\frac{219699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{14706125000 (2 x+3)^4}-\frac{1977291 (4-9 x) \sqrt{3 x^2+2}}{514714375000 (2 x+3)^2}-\frac{5931873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{257357187500 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^11,x]

[Out]

(-1977291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(514714375000*(3 + 2*x)^2) - (219699*(4 - 9
*x)*(2 + 3*x^2)^(3/2))/(14706125000*(3 + 2*x)^4) - (73233*(4 - 9*x)*(2 + 3*x^2)^
(5/2))/(1050437500*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(350*(3 + 2*x)^10) - (1
171*(2 + 3*x^2)^(7/2))/(110250*(3 + 2*x)^9) - (4393*(2 + 3*x^2)^(7/2))/(1715000*
(3 + 2*x)^8) - (739619*(2 + 3*x^2)^(7/2))/(1260525000*(3 + 2*x)^7) - (5931873*Ar
cTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(257357187500*Sqrt[35])

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Rubi in Sympy [A]  time = 37.9839, size = 190, normalized size = 0.94 \[ - \frac{1977291 \left (- 18 x + 8\right ) \sqrt{3 x^{2} + 2}}{1029428750000 \left (2 x + 3\right )^{2}} - \frac{219699 \left (- 18 x + 8\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{29412250000 \left (2 x + 3\right )^{4}} - \frac{73233 \left (- 18 x + 8\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{2100875000 \left (2 x + 3\right )^{6}} - \frac{5931873 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{9007501562500} - \frac{739619 \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{1260525000 \left (2 x + 3\right )^{7}} - \frac{4393 \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{1715000 \left (2 x + 3\right )^{8}} - \frac{1171 \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{110250 \left (2 x + 3\right )^{9}} - \frac{13 \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{350 \left (2 x + 3\right )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**11,x)

[Out]

-1977291*(-18*x + 8)*sqrt(3*x**2 + 2)/(1029428750000*(2*x + 3)**2) - 219699*(-18
*x + 8)*(3*x**2 + 2)**(3/2)/(29412250000*(2*x + 3)**4) - 73233*(-18*x + 8)*(3*x*
*2 + 2)**(5/2)/(2100875000*(2*x + 3)**6) - 5931873*sqrt(35)*atanh(sqrt(35)*(-9*x
+ 4)/(35*sqrt(3*x**2 + 2)))/9007501562500 - 739619*(3*x**2 + 2)**(7/2)/(1260525
000*(2*x + 3)**7) - 4393*(3*x**2 + 2)**(7/2)/(1715000*(2*x + 3)**8) - 1171*(3*x*
*2 + 2)**(7/2)/(110250*(2*x + 3)**9) - 13*(3*x**2 + 2)**(7/2)/(350*(2*x + 3)**10
)

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Mathematica [A]  time = 0.196787, size = 129, normalized size = 0.64 \[ -\frac{106773714 \sqrt{35} (2 x+3)^{10} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+35 \sqrt{3 x^2+2} \left (7968937464 x^9+101311348104 x^8+544524933294 x^7+1541962687104 x^6-3078520541586 x^5+11369945485836 x^4+4704132871221 x^3+18888919063956 x^2+5421307926571 x+5288003538036\right )-106773714 \sqrt{35} (2 x+3)^{10} \log (2 x+3)}{162135028125000 (2 x+3)^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^11,x]

[Out]

-(35*Sqrt[2 + 3*x^2]*(5288003538036 + 5421307926571*x + 18888919063956*x^2 + 470
4132871221*x^3 + 11369945485836*x^4 - 3078520541586*x^5 + 1541962687104*x^6 + 54
4524933294*x^7 + 101311348104*x^8 + 7968937464*x^9) - 106773714*Sqrt[35]*(3 + 2*
x)^10*Log[3 + 2*x] + 106773714*Sqrt[35]*(3 + 2*x)^10*Log[2*(4 - 9*x + Sqrt[35]*S
qrt[2 + 3*x^2])])/(162135028125000*(3 + 2*x)^10)

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Maple [B]  time = 0.075, size = 341, 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+2)^(5/2)/(2*x+3)^11,x)

[Out]

-8239371597/55170947070312500/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)[phone]/1801
5003125000*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)-5931873/9007501562500*35^(1/2)*arctanh
(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))[phone]/551709470703125
00*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)-1171/56448000/(x+3/2)^9*(3*(x+3/2)^2-9*x-19/4)
^(7/2)-659097/588245000000/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)-4393/439040000
/(x+3/2)^8*(3*(x+3/2)^2-9*x-19/4)^(7/2)-73233/33614000000/(x+3/2)^6*(3*(x+3/2)^2
-9*x-19/4)^(7/2)-6371271/10294287500000/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(7/2)-6
5250603/180150031250000/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)-739619/1613472000
00/(x+3/2)^7*(3*(x+3/2)^2-9*x-19/4)^(7/2)[phone]/630525109375000*x*(3*(x+3/2)
^2-9*x-19/4)^(3/2)-709847469/3152625546875000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(
7/2)-13/358400/(x+3/2)^10*(3*(x+3/2)^2-9*x-19/4)^(7/2)+3954582/78815638671875*(3
*(x+3/2)^2-9*x-19/4)^(3/2)+5931873/9007501562500*(12*(x+3/2)^2-36*x-19)^(1/2)+47
454984/13792736767578125*(3*(x+3/2)^2-9*x-19/4)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.788782, size = 671, normalized size = 3.32 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(5/2)*(x - 5)/(2*x + 3)^11,x, algorithm="maxima")

[Out]

2129542407/3152625546875000*(3*x^2 + 2)^(5/2) - 13/350*(3*x^2 + 2)^(7/2)/(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) - 1171/110250*(3*x^2 + 2)^(7
/2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489
888*x^3 + 314928*x^2 + 118098*x + 19683) - 4393/1715000*(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 + 349
92*x + 6561) - 739619/1260525000*(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) - 73233/525218750*(3*x^2
+ 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) -
659097/18382656250*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 8
10*x + 243) - 6371271/643392968750*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2
+ 216*x + 81) - 65250603/22518753906250*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x
+ 27) - 709847469/788156386718750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 694029
141/630525109375000*(3*x^2 + 2)^(3/2)*x + 3954582/78815638671875*(3*x^2 + 2)^(3/
2) - 8239371597/3152625546875000*(3*x^2 + 2)^(5/2)/(2*x + 3) + 53386857/18015003
125000*sqrt(3*x^2 + 2)*x + 5931873/9007501562500*sqrt(35)*arcsinh(3/2*sqrt(6)*x/
abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 5931873/4503750781250*sqrt(3*x^2 + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.296783, size = 289, normalized size = 1.43 \[ -\frac{\sqrt{35}{\left (\sqrt{35}{\left (7968937464 \, x^{9} + 101311348104 \, x^{8} + 544524933294 \, x^{7} + 1541962687104 \, x^{6} - 3078520541586 \, x^{5} + 11369945485836 \, x^{4} + 4704132871221 \, x^{3} + 18888919063956 \, x^{2} + 5421307926571 \, x + 5288003538036\right )} \sqrt{3 \, x^{2} + 2} - 53386857 \,{\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{162135028125000 \,{\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(5/2)*(x - 5)/(2*x + 3)^11,x, algorithm="fricas")

[Out]

-1/162135028125000*sqrt(35)*(sqrt(35)*(7968937464*x^9 + 101311348104*x^8 + 54452
4933294*x^7 + 1541962687104*x^6 - 3078520541586*x^5 + 11369945485836*x^4 + 47041
32871221*x^3 + 18888919063956*x^2 + 5421307926571*x + 5288003538036)*sqrt(3*x^2
+ 2) - 53386857*(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)*log(-
(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2 + 2)*(9*x - 4))/(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+2)**(5/2)/(3+2*x)**11,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.335019, size = 733, normalized size = 3.63 \[ \frac{5931873}{9007501562500} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{9 \,{\left (168728832 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{19} + 4808771712 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{18} + 180483607296 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{17} + 2449600006086 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{16} + 1950011203428 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{15} + 11324343251586 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{14} - 129748494414672 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{13} - 114750161469717 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{12} - 790683925144266 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} - 64560900263031 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} - 520582739768172 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 409007369125548 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 2437545878994816 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 775661489485344 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 927787935017088 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 53888888658816 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 63600137874432 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 6293205518848 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 1046970832896 \, \sqrt{3} x + 25185777664 \, \sqrt{3} + 1046970832896 \, \sqrt{3 \, x^{2} + 2}\right )}}{65883440000000 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(5/2)*(x - 5)/(2*x + 3)^11,x, algorithm="giac")

[Out]

5931873/9007501562500*sqrt(35)*ln(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*s
qrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/65
883440000000*(168728832*(sqrt(3)*x - sqrt(3*x^2 + 2))^19 + 4808771712*sqrt(3)*(s
qrt(3)*x - sqrt(3*x^2 + 2))^18 + 180483607296*(sqrt(3)*x - sqrt(3*x^2 + 2))^17 +
2449600006086*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^16 + 1950011203428*(sqrt(3)
*x - sqrt(3*x^2 + 2))^15 + 11324343251586*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^
14 - 129748494414672*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 - 114750161469717*sqrt(3)*
(sqrt(3)*x - sqrt(3*x^2 + 2))^12 - 790683925144266*(sqrt(3)*x - sqrt(3*x^2 + 2))
^11 - 64560900263031*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 - 520582739768172*
(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 409007369125548*sqrt(3)*(sqrt(3)*x - sqrt(3*x^
2 + 2))^8 - 2437545878994816*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 775661489485344*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 927787935017088*(sqrt(3)*x - sqrt(3*x^2
+ 2))^5 + 53888888658816*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 636001378744
32*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 6293205518848*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 2))^2 - 1046970832896*sqrt(3)*x + 25185777664*sqrt(3) + 1046970832896*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)^10