3.2428 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=149 \[ -\frac{167 \left (3 x^2+5 x+2\right )^{5/2}}{375 (2 x+3)^5}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}+\frac{1141 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{12000 (2 x+3)^4}-\frac{1141 (8 x+7) \sqrt{3 x^2+5 x+2}}{160000 (2 x+3)^2}+\frac{1141 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{320000 \sqrt{5}} \]

[Out]

(-1141*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(160000*(3 + 2*x)^2) + (1141*(7 + 8*x)*(
2 + 5*x + 3*x^2)^(3/2))/(12000*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(30*(
3 + 2*x)^6) - (167*(2 + 5*x + 3*x^2)^(5/2))/(375*(3 + 2*x)^5) + (1141*ArcTanh[(7
+ 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(320000*Sqrt[5])

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Rubi [A]  time = 0.235834, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{167 \left (3 x^2+5 x+2\right )^{5/2}}{375 (2 x+3)^5}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}+\frac{1141 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{12000 (2 x+3)^4}-\frac{1141 (8 x+7) \sqrt{3 x^2+5 x+2}}{160000 (2 x+3)^2}+\frac{1141 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{320000 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(-1141*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(160000*(3 + 2*x)^2) + (1141*(7 + 8*x)*(
2 + 5*x + 3*x^2)^(3/2))/(12000*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(30*(
3 + 2*x)^6) - (167*(2 + 5*x + 3*x^2)^(5/2))/(375*(3 + 2*x)^5) + (1141*ArcTanh[(7
+ 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(320000*Sqrt[5])

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Rubi in Sympy [A]  time = 37.9583, size = 141, normalized size = 0.95 \[ - \frac{1141 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{1600000} - \frac{1141 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{160000 \left (2 x + 3\right )^{2}} + \frac{1141 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{12000 \left (2 x + 3\right )^{4}} - \frac{167 \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{375 \left (2 x + 3\right )^{5}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{30 \left (2 x + 3\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1141*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/1600000 - 11
41*(8*x + 7)*sqrt(3*x**2 + 5*x + 2)/(160000*(2*x + 3)**2) + 1141*(8*x + 7)*(3*x*
*2 + 5*x + 2)**(3/2)/(12000*(2*x + 3)**4) - 167*(3*x**2 + 5*x + 2)**(5/2)/(375*(
2*x + 3)**5) - 13*(3*x**2 + 5*x + 2)**(5/2)/(30*(2*x + 3)**6)

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Mathematica [A]  time = 0.183138, size = 100, normalized size = 0.67 \[ \frac{-3423 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{10 \sqrt{3 x^2+5 x+2} \left (95616 x^5+799120 x^4+3065440 x^3+4479600 x^2+2526920 x+412679\right )}{(2 x+3)^6}+3423 \sqrt{5} \log (2 x+3)}{4800000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(412679 + 2526920*x + 4479600*x^2 + 3065440*x^3 + 799
120*x^4 + 95616*x^5))/(3 + 2*x)^6 + 3423*Sqrt[5]*Log[3 + 2*x] - 3423*Sqrt[5]*Log
[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/4800000

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Maple [A]  time = 0.021, size = 232, normalized size = 1.6 \[ -{\frac{13}{1920} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{167}{12000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{1141}{48000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{1141}{30000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{35371}{600000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{33089}{375000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{1141}{3000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{5705+6846\,x}{200000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{1141}{1600000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{1141\,\sqrt{5}}{1600000}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{165445+198534\,x}{750000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/1920/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(5/2)-167/12000/(x+3/2)^5*(3*(x+3/2)^2
-4*x-19/4)^(5/2)-1141/48000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1141/30000/(x
+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(5/2)-35371/600000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/
4)^(5/2)-33089/375000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+1141/3000000*(3*(x+3/
2)^2-4*x-19/4)^(3/2)-1141/200000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+1141/16000
00*(12*(x+3/2)^2-16*x-19)^(1/2)-1141/1600000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1
/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+33089/750000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3
/2)

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Maxima [A]  time = 0.778748, size = 387, normalized size = 2.6 \[ \frac{35371}{200000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{30 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{167 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{375 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{1141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{3000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{1141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{3750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{35371 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{150000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{3423}{100000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{1141}{1600000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{21679}{800000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{33089 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{150000 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35371/200000*(3*x^2 + 5*x + 2)^(3/2) - 13/30*(3*x^2 + 5*x + 2)^(5/2)/(64*x^6 + 5
76*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 167/375*(3*x^2 + 5*x +
2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 1141/3000*(3*x
^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1141/3750*(3*x^2
+ 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 35371/150000*(3*x^2 + 5*x + 2)^(
5/2)/(4*x^2 + 12*x + 9) - 3423/100000*sqrt(3*x^2 + 5*x + 2)*x - 1141/1600000*sqr
t(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 21
679/800000*sqrt(3*x^2 + 5*x + 2) - 33089/150000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3
)

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Fricas [A]  time = 0.282699, size = 216, normalized size = 1.45 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (95616 \, x^{5} + 799120 \, x^{4} + 3065440 \, x^{3} + 4479600 \, x^{2} + 2526920 \, x + 412679\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3423 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{9600000 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9600000*sqrt(5)*(4*sqrt(5)*(95616*x^5 + 799120*x^4 + 3065440*x^3 + 4479600*x^2
+ 2526920*x + 412679)*sqrt(3*x^2 + 5*x + 2) + 3423*(64*x^6 + 576*x^5 + 2160*x^4
+ 4320*x^3 + 4860*x^2 + 2916*x + 729)*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)))/(64*x^6 + 576*x^5 + 2160*x
^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x
**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(-23*x*sqrt(3*x**2
+ 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*
x**2 + 10206*x + 2187), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(128*x**7
+ 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187)
, x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5
+ 15120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x)

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GIAC/XCAS [A]  time = 0.304238, size = 554, normalized size = 3.72 \[ \frac{1141}{1600000} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{109536 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 6127344 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 70129360 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 83080800 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 3334681440 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 9802137888 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 47432214576 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 48106882440 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 94851959950 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 39436262415 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 28403540997 \, \sqrt{3} x - 3009604608 \, \sqrt{3} + 28403540997 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{480000 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1141/1600000*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)))
- 1/480000*(109536*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 6127344*sqrt(3)*(sq
rt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 70129360*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2
))^9 - 83080800*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 3334681440*(sqrt
(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 9802137888*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^6 - 47432214576*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 48106882440*sq
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 94851959950*(sqrt(3)*x - sqrt(3*x^
2 + 5*x + 2))^3 - 39436262415*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 28
403540997*sqrt(3)*x - 3009604608*sqrt(3) + 28403540997*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)^6