3.2413 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=94 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^3}+\frac{47 (8 x+7) \sqrt{3 x^2+5 x+2}}{200 (2 x+3)^2}-\frac{47 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{400 \sqrt{5}} \]

[Out]

(47*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(200*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(
3/2))/(15*(3 + 2*x)^3) - (47*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])
])/(400*Sqrt[5])

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Rubi [A]  time = 0.116682, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^3}+\frac{47 (8 x+7) \sqrt{3 x^2+5 x+2}}{200 (2 x+3)^2}-\frac{47 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{400 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(47*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(200*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(
3/2))/(15*(3 + 2*x)^3) - (47*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])
])/(400*Sqrt[5])

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Rubi in Sympy [A]  time = 23.4723, size = 88, normalized size = 0.94 \[ \frac{47 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{2000} + \frac{47 \left (8 x + 7\right ) \sqrt{3 x^{2} + 5 x + 2}}{200 \left (2 x + 3\right )^{2}} - \frac{13 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{15 \left (2 x + 3\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**4,x)

[Out]

47*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/2000 + 47*(8*x
+ 7)*sqrt(3*x**2 + 5*x + 2)/(200*(2*x + 3)**2) - 13*(3*x**2 + 5*x + 2)**(3/2)/(1
5*(2*x + 3)**3)

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Mathematica [A]  time = 0.118408, size = 85, normalized size = 0.9 \[ \frac{\frac{10 \sqrt{3 x^2+5 x+2} \left (696 x^2+2758 x+1921\right )}{(2 x+3)^3}+141 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-141 \sqrt{5} \log (2 x+3)}{6000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(1921 + 2758*x + 696*x^2))/(3 + 2*x)^3 - 141*Sqrt[5]*
Log[3 + 2*x] + 141*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/6000

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Maple [A]  time = 0.015, size = 132, normalized size = 1.4 \[ -{\frac{13}{120} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{47}{200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{47}{125} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{47}{2000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{47\,\sqrt{5}}{2000}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{235+282\,x}{250}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^4,x)

[Out]

-13/120/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(3/2)-47/200/(x+3/2)^2*(3*(x+3/2)^2-4*x
-19/4)^(3/2)-47/125/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-47/2000*(12*(x+3/2)^2-1
6*x-19)^(1/2)+47/2000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-
19)^(1/2))+47/250*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)

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Maxima [A]  time = 0.773801, size = 182, normalized size = 1.94 \[ \frac{47}{2000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{141}{200} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{15 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{47 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{50 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{47 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{50 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^4,x, algorithm="maxima")

[Out]

47/2000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3
) - 2) + 141/200*sqrt(3*x^2 + 5*x + 2) - 13/15*(3*x^2 + 5*x + 2)^(3/2)/(8*x^3 +
36*x^2 + 54*x + 27) - 47/50*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 + 12*x + 9) - 47/50*s
qrt(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 0.277445, size = 155, normalized size = 1.65 \[ \frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (696 \, x^{2} + 2758 \, x + 1921\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 141 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} - 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{12000 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^4,x, algorithm="fricas")

[Out]

1/12000*sqrt(5)*(4*sqrt(5)*(696*x^2 + 2758*x + 1921)*sqrt(3*x^2 + 5*x + 2) + 141
*(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. \[ - \int \left (- \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**4,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81),
x) - Integral(x*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 [A]  time = 0.296531, size = 347, normalized size = 3.69 \[ -\frac{47}{2000} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{1236 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 4830 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 90290 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 144945 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 287985 \, \sqrt{3} x - 69339 \, \sqrt{3} + 287985 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{600 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^4,x, algorithm="giac")

[Out]

-47/2000*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
/600*(1236*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 4830*sqrt(3)*(sqrt(3)*x - sqr
t(3*x^2 + 5*x + 2))^4 - 90290*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 144945*sqr
t(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 287985*sqrt(3)*x - 69339*sqrt(3) +
287985*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)^3