3.2427 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=124 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac{141 (8 x+7) \sqrt{3 x^2+5 x+2}}{16000 (2 x+3)^2}+\frac{141 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{32000 \sqrt{5}} \]

[Out]

(-141*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(16000*(3 + 2*x)^2) + (47*(7 + 8*x)*(2 +
5*x + 3*x^2)^(3/2))/(400*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(25*(3 + 2*
x)^5) + (141*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(32000*Sqrt[5
])

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Rubi [A]  time = 0.165673, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac{141 (8 x+7) \sqrt{3 x^2+5 x+2}}{16000 (2 x+3)^2}+\frac{141 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{32000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

(-141*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(16000*(3 + 2*x)^2) + (47*(7 + 8*x)*(2 +
5*x + 3*x^2)^(3/2))/(400*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(25*(3 + 2*
x)^5) + (141*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(32000*Sqrt[5
])

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Rubi in Sympy [A]  time = 30.1175, size = 117, normalized size = 0.94 \[ - \frac{141 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{160000} - \frac{141 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{16000 \left (2 x + 3\right )^{2}} + \frac{47 \left (8 x + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{400 \left (2 x + 3\right )^{4}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{25 \left (2 x + 3\right )^{5}} \]

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)**6,x)

[Out]

-141*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/160000 - 141*
(8*x + 7)*sqrt(3*x**2 + 5*x + 2)/(16000*(2*x + 3)**2) + 47*(8*x + 7)*(3*x**2 + 5
*x + 2)**(3/2)/(400*(2*x + 3)**4) - 13*(3*x**2 + 5*x + 2)**(5/2)/(25*(2*x + 3)**
5)

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Mathematica [A]  time = 0.16562, size = 95, normalized size = 0.77 \[ \frac{-141 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+\frac{10 \sqrt{3 x^2+5 x+2} \left (6336 x^4+66616 x^3+131516 x^2+90126 x+19031\right )}{(2 x+3)^5}+141 \sqrt{5} \log (2 x+3)}{160000} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(19031 + 90126*x + 131516*x^2 + 66616*x^3 + 6336*x^4)
)/(3 + 2*x)^5 + 141*Sqrt[5]*Log[3 + 2*x] - 141*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*
Sqrt[2 + 5*x + 3*x^2]])/160000

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Maple [B]  time = 0.019, size = 211, normalized size = 1.7 \[ -{\frac{13}{800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{47}{1600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{47}{1000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1457}{20000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{1363}{12500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{47}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{705+846\,x}{20000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{141}{160000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{141\,\sqrt{5}}{160000}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{6815+8178\,x}{25000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \]

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)^6,x)

[Out]

-13/800/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(5/2)-47/1600/(x+3/2)^4*(3*(x+3/2)^2-4*
x-19/4)^(5/2)-47/1000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1457/20000/(x+3/2)^
2*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1363/12500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+4
7/100000*(3*(x+3/2)^2-4*x-19/4)^(3/2)-141/20000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(
1/2)+141/160000*(12*(x+3/2)^2-16*x-19)^(1/2)-141/160000*5^(1/2)*arctanh(2/5*(-7/
2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+1363/25000*(5+6*x)*(3*(x+3/2)^2-4*x
-19/4)^(3/2)

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Maxima [A]  time = 0.772497, size = 325, normalized size = 2.62 \[ \frac{4371}{20000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{25 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{47 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{100 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{47 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{125 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1457 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{5000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{423}{10000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{141}{160000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{2679}{80000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1363 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{5000 \,{\left (2 \, x + 3\right )}} \]

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)^6,x, algorithm="maxima")

[Out]

4371/20000*(3*x^2 + 5*x + 2)^(3/2) - 13/25*(3*x^2 + 5*x + 2)^(5/2)/(32*x^5 + 240
*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 47/100*(3*x^2 + 5*x + 2)^(5/2)/(16*x^
4 + 96*x^3 + 216*x^2 + 216*x + 81) - 47/125*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*
x^2 + 54*x + 27) - 1457/5000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) - 423/10
000*sqrt(3*x^2 + 5*x + 2)*x - 141/160000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x +
2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 2679/80000*sqrt(3*x^2 + 5*x + 2) - 136
3/5000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 0.278378, size = 196, normalized size = 1.58 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (6336 \, x^{4} + 66616 \, x^{3} + 131516 \, x^{2} + 90126 \, x + 19031\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 141 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{320000 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

1/320000*sqrt(5)*(4*sqrt(5)*(6336*x^4 + 66616*x^3 + 131516*x^2 + 90126*x + 19031
)*sqrt(3*x^2 + 5*x + 2) + 141*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 2
43)*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)))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \]

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)**6,x)

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3
+ 4860*x**2 + 2916*x + 729), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(64*x**
6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x) - Integral(
-10*x**2*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 48
60*x**2 + 2916*x + 729), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(64*x**6 +
576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x)

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GIAC/XCAS [A]  time = 0.302319, size = 485, normalized size = 3.91 \[ \frac{141}{160000} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{146256 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 654456 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 415048 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 15455452 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 140042336 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 207568854 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 544555762 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 286352757 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 252454821 \, \sqrt{3} x - 31985676 \, \sqrt{3} + 252454821 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{16000 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \]

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)^6,x, algorithm="giac")

[Out]

141/160000*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))) -
1/16000*(146256*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 654456*sqrt(3)*(sqrt(3)
*x - sqrt(3*x^2 + 5*x + 2))^8 + 415048*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 1
5455452*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 140042336*(sqrt(3)*x - s
qrt(3*x^2 + 5*x + 2))^5 - 207568854*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^
4 - 544555762*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 286352757*sqrt(3)*(sqrt(3)
*x - sqrt(3*x^2 + 5*x + 2))^2 - 252454821*sqrt(3)*x - 31985676*sqrt(3) + 2524548
21*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)^5