3.2504 \(\int \frac{5-x}{(3+2 x)^6 \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=164 \[ -\frac{15891 \sqrt{3 x^2+5 x+2}}{6250 (2 x+3)}-\frac{1007 \sqrt{3 x^2+5 x+2}}{600 (2 x+3)^2}-\frac{2321 \sqrt{3 x^2+5 x+2}}{1875 (2 x+3)^3}-\frac{443 \sqrt{3 x^2+5 x+2}}{500 (2 x+3)^4}-\frac{13 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)^5}+\frac{128381 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{50000 \sqrt{5}} \]

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^5) - (443*Sqrt[2 + 5*x + 3*x^2])/(500*
(3 + 2*x)^4) - (2321*Sqrt[2 + 5*x + 3*x^2])/(1875*(3 + 2*x)^3) - (1007*Sqrt[2 +
5*x + 3*x^2])/(600*(3 + 2*x)^2) - (15891*Sqrt[2 + 5*x + 3*x^2])/(6250*(3 + 2*x))
+ (128381*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(50000*Sqrt[5])

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Rubi [A]  time = 0.368024, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{15891 \sqrt{3 x^2+5 x+2}}{6250 (2 x+3)}-\frac{1007 \sqrt{3 x^2+5 x+2}}{600 (2 x+3)^2}-\frac{2321 \sqrt{3 x^2+5 x+2}}{1875 (2 x+3)^3}-\frac{443 \sqrt{3 x^2+5 x+2}}{500 (2 x+3)^4}-\frac{13 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)^5}+\frac{128381 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{50000 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^5) - (443*Sqrt[2 + 5*x + 3*x^2])/(500*
(3 + 2*x)^4) - (2321*Sqrt[2 + 5*x + 3*x^2])/(1875*(3 + 2*x)^3) - (1007*Sqrt[2 +
5*x + 3*x^2])/(600*(3 + 2*x)^2) - (15891*Sqrt[2 + 5*x + 3*x^2])/(6250*(3 + 2*x))
+ (128381*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(50000*Sqrt[5])

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Rubi in Sympy [A]  time = 47.3085, size = 153, normalized size = 0.93 \[ - \frac{128381 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{250000} - \frac{15891 \sqrt{3 x^{2} + 5 x + 2}}{6250 \left (2 x + 3\right )} - \frac{1007 \sqrt{3 x^{2} + 5 x + 2}}{600 \left (2 x + 3\right )^{2}} - \frac{2321 \sqrt{3 x^{2} + 5 x + 2}}{1875 \left (2 x + 3\right )^{3}} - \frac{443 \sqrt{3 x^{2} + 5 x + 2}}{500 \left (2 x + 3\right )^{4}} - \frac{13 \sqrt{3 x^{2} + 5 x + 2}}{25 \left (2 x + 3\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3+2*x)**6/(3*x**2+5*x+2)**(1/2),x)

[Out]

-128381*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/250000 - 1
5891*sqrt(3*x**2 + 5*x + 2)/(6250*(2*x + 3)) - 1007*sqrt(3*x**2 + 5*x + 2)/(600*
(2*x + 3)**2) - 2321*sqrt(3*x**2 + 5*x + 2)/(1875*(2*x + 3)**3) - 443*sqrt(3*x**
2 + 5*x + 2)/(500*(2*x + 3)**4) - 13*sqrt(3*x**2 + 5*x + 2)/(25*(2*x + 3)**5)

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Mathematica [A]  time = 0.166741, size = 95, normalized size = 0.58 \[ \frac{-385143 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-\frac{10 \sqrt{3 x^2+5 x+2} \left (3051072 x^4+19313432 x^3+46092332 x^2+49233702 x+19918587\right )}{(2 x+3)^5}+385143 \sqrt{5} \log (2 x+3)}{750000} \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(19918587 + 49233702*x + 46092332*x^2 + 19313432*x^3
+ 3051072*x^4))/(3 + 2*x)^5 + 385143*Sqrt[5]*Log[3 + 2*x] - 385143*Sqrt[5]*Log[
-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/750000

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Maple [A]  time = 0.017, size = 137, normalized size = 0.8 \[ -{\frac{13}{800}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{443}{8000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{2321}{15000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1007}{2400}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{15891}{12500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{128381\,\sqrt{5}}{250000}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(3+2*x)^6/(3*x^2+5*x+2)^(1/2),x)

[Out]

-13/800/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(1/2)-443/8000/(x+3/2)^4*(3*(x+3/2)^2-4
*x-19/4)^(1/2)-2321/15000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(1/2)-1007/2400/(x+3/
2)^2*(3*(x+3/2)^2-4*x-19/4)^(1/2)-15891/12500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(1/
2)-128381/250000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(
1/2))

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Maxima [A]  time = 0.803486, size = 267, normalized size = 1.63 \[ -\frac{128381}{250000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{13 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{25 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{443 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{500 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{2321 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{1875 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1007 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{600 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{15891 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{6250 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/(sqrt(3*x^2 + 5*x + 2)*(2*x + 3)^6),x, algorithm="maxima")

[Out]

-128381/250000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(
2*x + 3) - 2) - 13/25*sqrt(3*x^2 + 5*x + 2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x
^2 + 810*x + 243) - 443/500*sqrt(3*x^2 + 5*x + 2)/(16*x^4 + 96*x^3 + 216*x^2 + 2
16*x + 81) - 2321/1875*sqrt(3*x^2 + 5*x + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1007
/600*sqrt(3*x^2 + 5*x + 2)/(4*x^2 + 12*x + 9) - 15891/6250*sqrt(3*x^2 + 5*x + 2)
/(2*x + 3)

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Fricas [A]  time = 0.289613, size = 196, normalized size = 1.2 \[ -\frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (3051072 \, x^{4} + 19313432 \, x^{3} + 46092332 \, x^{2} + 49233702 \, x + 19918587\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 385143 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{1500000 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/(sqrt(3*x^2 + 5*x + 2)*(2*x + 3)^6),x, algorithm="fricas")

[Out]

-1/1500000*sqrt(5)*(4*sqrt(5)*(3051072*x^4 + 19313432*x^3 + 46092332*x^2 + 49233
702*x + 19918587)*sqrt(3*x^2 + 5*x + 2) - 385143*(32*x^5 + 240*x^4 + 720*x^3 + 1
080*x^2 + 810*x + 243)*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 + 8
10*x + 243)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)/(3+2*x)**6/(3*x**2+5*x+2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.3051, size = 485, normalized size = 2.96 \[ \frac{128381}{250000} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{6162288 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 83190888 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 1461489304 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 4863585804 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 30365807072 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 40931011758 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 107175203674 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 58461317289 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 54344360217 \, \sqrt{3} x + 7303159752 \, \sqrt{3} - 54344360217 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{75000 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/(sqrt(3*x^2 + 5*x + 2)*(2*x + 3)^6),x, algorithm="giac")

[Out]

128381/250000*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))
) - 1/75000*(6162288*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 83190888*sqrt(3)*(s
qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 1461489304*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^7 + 4863585804*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 30365807072*
(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 40931011758*sqrt(3)*(sqrt(3)*x - sqrt(3*
x^2 + 5*x + 2))^4 + 107175203674*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 5846131
7289*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 54344360217*sqrt(3)*x + 730
3159752*sqrt(3) - 54344360217*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