3.1390 \(\int \frac{(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=133 \[ -\frac{(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}-\frac{(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{672 (2 x+3)^3}+\frac{3 (1917 x+6125) \sqrt{3 x^2+2}}{448 (2 x+3)}-\frac{188379 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{896 \sqrt{35}}-\frac{2625}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(3*(6125 + 1917*x)*Sqrt[2 + 3*x^2])/(448*(3 + 2*x)) - ((5003 + 5517*x)*(2 + 3*x^2)^(3/2))/(672*(3 + 2*x)^3) -
((19 + 4*x)*(2 + 3*x^2)^(5/2))/(16*(3 + 2*x)^4) - (2625*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (188379*ArcTanh[(4
- 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(896*Sqrt[35])

________________________________________________________________________________________

Rubi [A]  time = 0.0803883, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {813, 811, 844, 215, 725, 206} \[ -\frac{(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}-\frac{(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{672 (2 x+3)^3}+\frac{3 (1917 x+6125) \sqrt{3 x^2+2}}{448 (2 x+3)}-\frac{188379 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{896 \sqrt{35}}-\frac{2625}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(6125 + 1917*x)*Sqrt[2 + 3*x^2])/(448*(3 + 2*x)) - ((5003 + 5517*x)*(2 + 3*x^2)^(3/2))/(672*(3 + 2*x)^3) -
((19 + 4*x)*(2 + 3*x^2)^(5/2))/(16*(3 + 2*x)^4) - (2625*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (188379*ArcTanh[(4
- 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(896*Sqrt[35])

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx &=-\frac{(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{5}{64} \int \frac{(32-228 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx\\ &=-\frac{(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac{(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}+\frac{\int \frac{(-35904+184032 x) \sqrt{2+3 x^2}}{(3+2 x)^2} \, dx}{7168}\\ &=\frac{3 (6125+1917 x) \sqrt{2+3 x^2}}{448 (3+2 x)}-\frac{(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac{(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{\int \frac{-1472256+7056000 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{57344}\\ &=\frac{3 (6125+1917 x) \sqrt{2+3 x^2}}{448 (3+2 x)}-\frac{(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac{(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{7875}{128} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{188379}{896} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{3 (6125+1917 x) \sqrt{2+3 x^2}}{448 (3+2 x)}-\frac{(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac{(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{2625}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{188379}{896} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=\frac{3 (6125+1917 x) \sqrt{2+3 x^2}}{448 (3+2 x)}-\frac{(5003+5517 x) \left (2+3 x^2\right )^{3/2}}{672 (3+2 x)^3}-\frac{(19+4 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac{2625}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{188379 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{896 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.165737, size = 97, normalized size = 0.73 \[ \frac{-\frac{70 \sqrt{3 x^2+2} \left (3024 x^5-57456 x^4-898734 x^3-2762820 x^2-3335009 x-1421955\right )}{(2 x+3)^4}-565137 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-1929375 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{94080} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-70*Sqrt[2 + 3*x^2]*(-1421955 - 3335009*x - 2762820*x^2 - 898734*x^3 - 57456*x^4 + 3024*x^5))/(3 + 2*x)^4 -
1929375*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] - 565137*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/94080

________________________________________________________________________________________

Maple [B]  time = 0.011, size = 227, normalized size = 1.7 \begin{align*}{\frac{23}{117600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1041}{343000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{29717}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{188379}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{58629\,x}{274400} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{58491\,x}{15680}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{2625\,\sqrt{3}}{128}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{62793}{137200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{188379}{31360}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{188379\,\sqrt{35}}{31360}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }-{\frac{89151\,x}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{13}{2240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^5,x)

[Out]

23/117600/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)-1041/343000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)+29717/6002
500/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)+188379/6002500*(3*(x+3/2)^2-9*x-19/4)^(5/2)-58629/274400*x*(3*(x+3/2)
^2-9*x-19/4)^(3/2)-58491/15680*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)-2625/128*arcsinh(1/2*x*6^(1/2))*3^(1/2)+62793/13
7200*(3*(x+3/2)^2-9*x-19/4)^(3/2)+188379/31360*(12*(x+3/2)^2-36*x-19)^(1/2)-188379/31360*35^(1/2)*arctanh(2/35
*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))-89151/6002500*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)-13/2240/(x+3/2)^4
*(3*(x+3/2)^2-9*x-19/4)^(7/2)

________________________________________________________________________________________

Maxima [A]  time = 1.5661, size = 278, normalized size = 2.09 \begin{align*} \frac{3123}{343000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{140 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} + \frac{23 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{14700 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1041 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{85750 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{58629}{274400} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{62793}{137200} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{29717 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{343000 \,{\left (2 \, x + 3\right )}} - \frac{58491}{15680} \, \sqrt{3 \, x^{2} + 2} x - \frac{2625}{128} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{188379}{31360} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{188379}{15680} \, \sqrt{3 \, x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3123/343000*(3*x^2 + 2)^(5/2) - 13/140*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) + 23/14700*(
3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1041/85750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 58629/274400
*(3*x^2 + 2)^(3/2)*x + 62793/137200*(3*x^2 + 2)^(3/2) + 29717/343000*(3*x^2 + 2)^(5/2)/(2*x + 3) - 58491/15680
*sqrt(3*x^2 + 2)*x - 2625/128*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 188379/31360*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs
(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 188379/15680*sqrt(3*x^2 + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.94258, size = 525, normalized size = 3.95 \begin{align*} \frac{1929375 \, \sqrt{3}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 565137 \, \sqrt{35}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 140 \,{\left (3024 \, x^{5} - 57456 \, x^{4} - 898734 \, x^{3} - 2762820 \, x^{2} - 3335009 \, x - 1421955\right )} \sqrt{3 \, x^{2} + 2}}{188160 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/188160*(1929375*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
+ 565137*sqrt(35)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 -
36*x + 43)/(4*x^2 + 12*x + 9)) - 140*(3024*x^5 - 57456*x^4 - 898734*x^3 - 2762820*x^2 - 3335009*x - 1421955)*
sqrt(3*x^2 + 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.69472, size = 594, normalized size = 4.47 \begin{align*} -\frac{188379}{31360} \, \sqrt{35} \log \left (\sqrt{35}{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )} - 9\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + \frac{2625}{128} \, \sqrt{3} \log \left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{35}}{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{3} + \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{10752} \,{\left (\frac{7 \,{\left (\frac{35 \,{\left (\frac{1365 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 2129 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 57681 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 242979 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )} \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac{9 \,{\left (256 \,{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{3} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - 93 \, \sqrt{35}{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{2} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - 582 \,{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + 225 \, \sqrt{35} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{64 \,{\left ({\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{2} - 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-188379/31360*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)*sgn(1
/(2*x + 3)) + 2625/128*sqrt(3)*log(1/2*abs(-2*sqrt(3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35
)/(2*x + 3))/(sqrt(3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sgn(1/(2*x + 3)) - 1/1
0752*(7*(35*(1365*sgn(1/(2*x + 3))/(2*x + 3) - 2129*sgn(1/(2*x + 3)))/(2*x + 3) + 57681*sgn(1/(2*x + 3)))/(2*x
+ 3) - 242979*sgn(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) - 9/64*(256*(sqrt(-18/(2*x + 3) + 35
/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^3*sgn(1/(2*x + 3)) - 93*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2
+ 3) + sqrt(35)/(2*x + 3))^2*sgn(1/(2*x + 3)) - 582*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x
+ 3))*sgn(1/(2*x + 3)) + 225*sqrt(35)*sgn(1/(2*x + 3)))/((sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/
(2*x + 3))^2 - 3)^2