3.2425 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=137 \[ \frac{(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}-\frac{(402 x+845) \sqrt{3 x^2+5 x+2}}{160 (2 x+3)}+\frac{51}{32} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )-\frac{1973 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{320 \sqrt{5}} \]

[Out]

-((845 + 402*x)*Sqrt[2 + 5*x + 3*x^2])/(160*(3 + 2*x)) + ((383 + 342*x)*(2 + 5*x
+ 3*x^2)^(3/2))/(120*(3 + 2*x)^3) + (51*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sq
rt[2 + 5*x + 3*x^2])])/32 - (1973*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*
x^2])])/(320*Sqrt[5])

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Rubi [A]  time = 0.247711, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}-\frac{(402 x+845) \sqrt{3 x^2+5 x+2}}{160 (2 x+3)}+\frac{51}{32} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )-\frac{1973 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{320 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4,x]

[Out]

-((845 + 402*x)*Sqrt[2 + 5*x + 3*x^2])/(160*(3 + 2*x)) + ((383 + 342*x)*(2 + 5*x
+ 3*x^2)^(3/2))/(120*(3 + 2*x)^3) + (51*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sq
rt[2 + 5*x + 3*x^2])])/32 - (1973*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*
x^2])])/(320*Sqrt[5])

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Rubi in Sympy [A]  time = 32.6102, size = 122, normalized size = 0.89 \[ \frac{51 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{32} + \frac{1973 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{1600} - \frac{\left (804 x + 1690\right ) \sqrt{3 x^{2} + 5 x + 2}}{320 \left (2 x + 3\right )} + \frac{\left (342 x + 383\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{120 \left (2 x + 3\right )^{3}} \]

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)**4,x)

[Out]

51*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/32 + 1973*sqrt(5)
*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/1600 - (804*x + 1690)*sqr
t(3*x**2 + 5*x + 2)/(320*(2*x + 3)) + (342*x + 383)*(3*x**2 + 5*x + 2)**(3/2)/(1
20*(2*x + 3)**3)

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Mathematica [A]  time = 0.108867, size = 121, normalized size = 0.88 \[ \frac{1973 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+2550 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{10 \sqrt{3 x^2+5 x+2} \left (720 x^3+13176 x^2+30878 x+19751\right )}{3 (2 x+3)^3}-1973 \sqrt{5} \log (2 x+3)}{1600} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4,x]

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(19751 + 30878*x + 13176*x^2 + 720*x^3))/(3*(3 + 2*x
)^3) - 1973*Sqrt[5]*Log[3 + 2*x] + 1973*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2
+ 5*x + 3*x^2]] + 2550*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/1600

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Maple [A]  time = 0.016, size = 200, normalized size = 1.5 \[ -{\frac{13}{120} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{37}{600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{158}{375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{1973}{3000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{605+726\,x}{400}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{51\,\sqrt{3}}{32}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }-{\frac{1973}{1600}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{1973\,\sqrt{5}}{1600}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{395+474\,x}{375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \]

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)^4,x)

[Out]

-13/120/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(5/2)-37/600/(x+3/2)^2*(3*(x+3/2)^2-4*x
-19/4)^(5/2)-158/375/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1973/3000*(3*(x+3/2)^2
-4*x-19/4)^(3/2)+121/400*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+51/32*ln(1/3*(5/2+
3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-1973/1600*(12*(x+3/2)^2-16*x-
19)^(1/2)+1973/1600*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19
)^(1/2))+79/375*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)

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Maxima [A]  time = 0.785413, size = 258, normalized size = 1.88 \[ \frac{37}{200} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{15 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{37 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{150 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{363}{200} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{51}{32} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{1973}{1600} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{763}{800} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{79 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{75 \,{\left (2 \, x + 3\right )}} \]

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)^4,x, algorithm="maxima")

[Out]

37/200*(3*x^2 + 5*x + 2)^(3/2) - 13/15*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 +
54*x + 27) - 37/150*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) + 363/200*sqrt(3
*x^2 + 5*x + 2)*x + 51/32*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2)
+ 1973/1600*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*
x + 3) - 2) - 763/800*sqrt(3*x^2 + 5*x + 2) - 79/75*(3*x^2 + 5*x + 2)^(3/2)/(2*x
+ 3)

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Fricas [A]  time = 0.290605, size = 238, normalized size = 1.74 \[ \frac{\sqrt{5}{\left (1530 \, \sqrt{5} \sqrt{3}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 4 \, \sqrt{5}{\left (720 \, x^{3} + 13176 \, x^{2} + 30878 \, x + 19751\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 5919 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} - 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{9600 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

1/9600*sqrt(5)*(1530*sqrt(5)*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(4*sqrt(3)*
sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) - 4*sqrt(5)*(720*x^3 + 13
176*x^2 + 30878*x + 19751)*sqrt(3*x^2 + 5*x + 2) + 5919*(8*x^3 + 36*x^2 + 54*x +
27)*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)))/(8*x^3 + 36*x^2 + 54*x + 27)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \]

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)**4,x)

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81)
, x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216
*x + 81), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216
*x**2 + 216*x + 81), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x
**3 + 216*x**2 + 216*x + 81), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

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)^4,x, algorithm="giac")

[Out]

undef