3.2455 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=195 \[ -\frac{(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}+\frac{7 (43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{96 (2 x+3)^3}-\frac{35 (343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)^2}+\frac{35 (2701 x+5795) \sqrt{3 x^2+5 x+2}}{1024 (2 x+3)}-\frac{744275 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{4096 \sqrt{3}}+\frac{192171 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{4096} \]

[Out]

(35*(5795 + 2701*x)*Sqrt[2 + 5*x + 3*x^2])/(1024*(3 + 2*x)) - (35*(736 + 343*x)*
(2 + 5*x + 3*x^2)^(3/2))/(768*(3 + 2*x)^2) + (7*(93 + 43*x)*(2 + 5*x + 3*x^2)^(5
/2))/(96*(3 + 2*x)^3) - ((8 + x)*(2 + 5*x + 3*x^2)^(7/2))/(8*(3 + 2*x)^4) - (744
275*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(4096*Sqrt[3]) + (1921
71*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/4096

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Rubi [A]  time = 0.377869, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}+\frac{7 (43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{96 (2 x+3)^3}-\frac{35 (343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)^2}+\frac{35 (2701 x+5795) \sqrt{3 x^2+5 x+2}}{1024 (2 x+3)}-\frac{744275 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{4096 \sqrt{3}}+\frac{192171 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{4096} \]

Antiderivative was successfully verified.

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

[Out]

(35*(5795 + 2701*x)*Sqrt[2 + 5*x + 3*x^2])/(1024*(3 + 2*x)) - (35*(736 + 343*x)*
(2 + 5*x + 3*x^2)^(3/2))/(768*(3 + 2*x)^2) + (7*(93 + 43*x)*(2 + 5*x + 3*x^2)^(5
/2))/(96*(3 + 2*x)^3) - ((8 + x)*(2 + 5*x + 3*x^2)^(7/2))/(8*(3 + 2*x)^4) - (744
275*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(4096*Sqrt[3]) + (1921
71*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/4096

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Rubi in Sympy [A]  time = 51.1847, size = 182, normalized size = 0.93 \[ - \frac{744275 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{12288} - \frac{192171 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{4096} + \frac{35 \left (1037184 x + 2225280\right ) \sqrt{3 x^{2} + 5 x + 2}}{393216 \left (2 x + 3\right )} - \frac{35 \left (65856 x + 141312\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{147456 \left (2 x + 3\right )^{2}} + \frac{7 \left (2064 x + 4464\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{4608 \left (2 x + 3\right )^{3}} - \frac{\left (8 x + 64\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{64 \left (2 x + 3\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-744275*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/12288 - 1921
71*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/4096 + 35*(1037
184*x + 2225280)*sqrt(3*x**2 + 5*x + 2)/(393216*(2*x + 3)) - 35*(65856*x + 14131
2)*(3*x**2 + 5*x + 2)**(3/2)/(147456*(2*x + 3)**2) + 7*(2064*x + 4464)*(3*x**2 +
5*x + 2)**(5/2)/(4608*(2*x + 3)**3) - (8*x + 64)*(3*x**2 + 5*x + 2)**(7/2)/(64*
(2*x + 3)**4)

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Mathematica [A]  time = 0.270287, size = 139, normalized size = 0.71 \[ \frac{-576513 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-744275 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{12 \sqrt{3 x^2+5 x+2} \left (3456 x^7-12864 x^6-38288 x^5-253688 x^4-2869312 x^3-9107922 x^2-11295211 x-4933171\right )}{(2 x+3)^4}+576513 \sqrt{5} \log (2 x+3)}{12288} \]

Antiderivative was successfully verified.

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

[Out]

((-12*Sqrt[2 + 5*x + 3*x^2]*(-4933171 - 11295211*x - 9107922*x^2 - 2869312*x^3 -
253688*x^4 - 38288*x^5 - 12864*x^6 + 3456*x^7))/(3 + 2*x)^4 + 576513*Sqrt[5]*Lo
g[3 + 2*x] - 576513*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] - 74
4275*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/12288

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Maple [A]  time = 0.02, size = 295, normalized size = 1.5 \[{\frac{27453}{4000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}+{\frac{192171}{16000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{64057}{2560} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{192171}{4096}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13}{320} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}+{\frac{3}{100} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1263}{4000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{1479}{500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{7395+8874\,x}{1000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{50505+60606\,x}{4000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{30345+36414\,x}{1280} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{122045+146454\,x}{2048}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{744275\,\sqrt{3}}{12288}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }-{\frac{192171\,\sqrt{5}}{4096}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27453/4000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+192171/16000*(3*(x+3/2)^2-4*x-19/4)^(5/2
)+64057/2560*(3*(x+3/2)^2-4*x-19/4)^(3/2)+192171/4096*(12*(x+3/2)^2-16*x-19)^(1/
2)-13/320/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)+3/100/(x+3/2)^3*(3*(x+3/2)^2-4*
x-19/4)^(9/2)-1263/4000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)+1479/500/(x+3/2)*
(3*(x+3/2)^2-4*x-19/4)^(9/2)-1479/1000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-1010
1/4000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-6069/1280*(5+6*x)*(3*(x+3/2)^2-4*x-1
9/4)^(3/2)-24409/2048*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-744275/12288*ln(1/3*(
5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-192171/4096*5^(1/2)*arcta
nh(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 = 0.823495, size = 385, normalized size = 1.97 \[ \frac{3789}{4000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{20 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} + \frac{6 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{25 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1263 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{1000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{30303}{2000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{9849}{16000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{1479 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{200 \,{\left (2 \, x + 3\right )}} - \frac{18207}{640} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{3367}{2560} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{73227}{1024} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{744275}{12288} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{192171}{4096} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{35063}{1024} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3789/4000*(3*x^2 + 5*x + 2)^(7/2) - 13/20*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x
^3 + 216*x^2 + 216*x + 81) + 6/25*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x
+ 27) - 1263/1000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 30303/2000*(3*x^
2 + 5*x + 2)^(5/2)*x - 9849/16000*(3*x^2 + 5*x + 2)^(5/2) + 1479/200*(3*x^2 + 5*
x + 2)^(7/2)/(2*x + 3) - 18207/640*(3*x^2 + 5*x + 2)^(3/2)*x + 3367/2560*(3*x^2
+ 5*x + 2)^(3/2) - 73227/1024*sqrt(3*x^2 + 5*x + 2)*x - 744275/12288*sqrt(3)*log
(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 192171/4096*sqrt(5)*log(sqrt(5)*sq
rt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 35063/1024*sqrt(3*x^2
+ 5*x + 2)

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Fricas [A]  time = 0.294616, size = 285, normalized size = 1.46 \[ \frac{\sqrt{3}{\left (192171 \, \sqrt{5} \sqrt{3}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 8 \, \sqrt{3}{\left (3456 \, x^{7} - 12864 \, x^{6} - 38288 \, x^{5} - 253688 \, x^{4} - 2869312 \, x^{3} - 9107922 \, x^{2} - 11295211 \, x - 4933171\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 744275 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{24576 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24576*sqrt(3)*(192171*sqrt(5)*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)
*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 +
12*x + 9)) - 8*sqrt(3)*(3456*x^7 - 12864*x^6 - 38288*x^5 - 253688*x^4 - 2869312
*x^3 - 9107922*x^2 - 11295211*x - 4933171)*sqrt(3*x^2 + 5*x + 2) + 744275*(16*x^
4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(sqrt(3)*(72*x^2 + 120*x + 49) - 12*sqrt(3
*x^2 + 5*x + 2)*(6*x + 5)))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{40 \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac{292 x \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac{870 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac{1339 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac{1090 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac{396 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \frac{27 x^{7} \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2
+ 810*x + 243), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4
+ 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x
+ 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-
1339*x**3*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 81
0*x + 243), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4
+ 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x
+ 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(2
7*x**7*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x
+ 243), x)

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GIAC/XCAS [A]  time = 0.670499, size = 859, normalized size = 4.41 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

744275/12288*sqrt(3)*ln(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3
) + 2*sqrt(5)/(2*x + 3))/abs(2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3
) + 2*sqrt(5)/(2*x + 3)))*sign(1/(2*x + 3)) - 192171/4096*sqrt(5)*ln(abs(sqrt(5)
*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sign(1/(2*x
+ 3)) - 1/4096*(5*(50*(13*sign(1/(2*x + 3))/(2*x + 3) - 88*sign(1/(2*x + 3)))/(2
*x + 3) + 14343*sign(1/(2*x + 3)))/(2*x + 3) - 181996*sign(1/(2*x + 3)))*sqrt(-8
/(2*x + 3) + 5/(2*x + 3)^2 + 3) - 1/2048*(479709*(sqrt(-8/(2*x + 3) + 5/(2*x + 3
)^2 + 3) + sqrt(5)/(2*x + 3))^7*sign(1/(2*x + 3)) - 499296*sqrt(5)*(sqrt(-8/(2*x
+ 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^6*sign(1/(2*x + 3)) - 3133183*(s
qrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^5*sign(1/(2*x + 3)) +
3365712*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*
sign(1/(2*x + 3)) + 7550211*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2
*x + 3))^3*sign(1/(2*x + 3)) - 8139744*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^
2 + 3) + sqrt(5)/(2*x + 3))^2*sign(1/(2*x + 3)) - 6574257*(sqrt(-8/(2*x + 3) + 5
/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))*sign(1/(2*x + 3)) + 6966000*sqrt(5)*sign(
1/(2*x + 3)))/((sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2 -
3)^4