Integrand size = 24, antiderivative size = 97 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=-\frac {1}{8} (193-63 x) \sqrt {2+3 x^2}-\frac {(21+x) \left (2+3 x^2\right )^{3/2}}{6 (3+2 x)}+\frac {663}{16} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {193}{16} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \]
-1/6*(21+x)*(3*x^2+2)^(3/2)/(3+2*x)+663/16*arcsinh(1/2*x*6^(1/2))*3^(1/2)+ 193/16*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-1/8*(193-63 *x)*(3*x^2+2)^(1/2)
Time = 0.43 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.14 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=\frac {1}{48} \left (-\frac {2 \sqrt {2+3 x^2} \left (1905+599 x-126 x^2+12 x^3\right )}{3+2 x}-1158 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )-1989 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )\right ) \]
((-2*Sqrt[2 + 3*x^2]*(1905 + 599*x - 126*x^2 + 12*x^3))/(3 + 2*x) - 1158*S qrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]] - 1989*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/48
Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {681, 27, 682, 27, 719, 222, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^2} \, dx\) |
\(\Big \downarrow \) 681 |
\(\displaystyle -\frac {1}{8} \int \frac {4 (2-63 x) \sqrt {3 x^2+2}}{2 x+3}dx-\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{2} \int \frac {(2-63 x) \sqrt {3 x^2+2}}{2 x+3}dx-\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}\) |
\(\Big \downarrow \) 682 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{24} \int \frac {6 (394-1989 x)}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1}{4} \sqrt {3 x^2+2} (193-63 x)\right )-\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} \int \frac {394-1989 x}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1}{4} \sqrt {3 x^2+2} (193-63 x)\right )-\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \left (\frac {1989}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx-\frac {6755}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx\right )-\frac {1}{4} (193-63 x) \sqrt {3 x^2+2}\right )-\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \left (\frac {663}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {6755}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx\right )-\frac {1}{4} (193-63 x) \sqrt {3 x^2+2}\right )-\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \left (\frac {6755}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}+\frac {663}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )\right )-\frac {1}{4} (193-63 x) \sqrt {3 x^2+2}\right )-\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \left (\frac {663}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {193}{2} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )-\frac {1}{4} (193-63 x) \sqrt {3 x^2+2}\right )-\frac {(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}\) |
-1/6*((21 + x)*(2 + 3*x^2)^(3/2))/(3 + 2*x) + (-1/4*((193 - 63*x)*Sqrt[2 + 3*x^2]) + ((663*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/2 + (193*Sqrt[35]*ArcTanh[( 4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/2)/4)/2
3.14.74.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
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] + Simp[p/ (e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim p[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] && GtQ[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])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(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] && !IGtQ[m, 0]
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-\frac {36 x^{5}-378 x^{4}+1821 x^{3}+5463 x^{2}+1198 x +3810}{24 \left (3+2 x \right ) \sqrt {3 x^{2}+2}}+\frac {663 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{16}+\frac {193 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{16}\) | \(87\) |
trager | \(-\frac {\left (12 x^{3}-126 x^{2}+599 x +1905\right ) \sqrt {3 x^{2}+2}}{24 \left (3+2 x \right )}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-1303715\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1303715\right ) x +6755 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1303715\right )}{3+2 x}\right )}{16}+\frac {663 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{16}\) | \(110\) |
default | \(-\frac {193 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{210}+\frac {63 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{8}+\frac {663 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{16}-\frac {193 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{16}+\frac {193 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{16}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{70 \left (x +\frac {3}{2}\right )}+\frac {39 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{70}\) | \(131\) |
-1/24*(36*x^5-378*x^4+1821*x^3+5463*x^2+1198*x+3810)/(3+2*x)/(3*x^2+2)^(1/ 2)+663/16*arcsinh(1/2*x*6^(1/2))*3^(1/2)+193/16*35^(1/2)*arctanh(2/35*(4-9 *x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.25 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=\frac {1989 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 579 \, \sqrt {35} {\left (2 \, x + 3\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (12 \, x^{3} - 126 \, x^{2} + 599 \, x + 1905\right )} \sqrt {3 \, x^{2} + 2}}{96 \, {\left (2 \, x + 3\right )}} \]
1/96*(1989*sqrt(3)*(2*x + 3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 579*sqrt(35)*(2*x + 3)*log((sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 4*(12*x^3 - 126*x^2 + 599*x + 1905)*sqrt (3*x^2 + 2))/(2*x + 3)
\[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {2 x \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\, dx - \int \left (- \frac {15 x^{2} \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\, dx \]
-Integral(-10*sqrt(3*x**2 + 2)/(4*x**2 + 12*x + 9), x) - Integral(2*x*sqrt (3*x**2 + 2)/(4*x**2 + 12*x + 9), x) - Integral(-15*x**2*sqrt(3*x**2 + 2)/ (4*x**2 + 12*x + 9), x) - Integral(3*x**3*sqrt(3*x**2 + 2)/(4*x**2 + 12*x + 9), x)
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=-\frac {1}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {63}{8} \, \sqrt {3 \, x^{2} + 2} x + \frac {663}{16} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {193}{16} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {193}{8} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{4 \, {\left (2 \, x + 3\right )}} \]
-1/12*(3*x^2 + 2)^(3/2) + 63/8*sqrt(3*x^2 + 2)*x + 663/16*sqrt(3)*arcsinh( 1/2*sqrt(6)*x) - 193/16*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3* sqrt(6)/abs(2*x + 3)) - 193/8*sqrt(3*x^2 + 2) - 13/4*(3*x^2 + 2)^(3/2)/(2* x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (75) = 150\).
Time = 0.45 (sec) , antiderivative size = 475, normalized size of antiderivative = 4.90 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=\frac {193}{16} \, \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 {663}{16} \, \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 {455}{32} \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {3 \, {\left (704 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 323 \, \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 )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 1944 \, {\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 ) + 1158 \, \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 ) + 1872 \, {\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 ) - 1263 \, \sqrt {35} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{8 \, {\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 )}^{3}} \]
193/16*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + s qrt(35)/(2*x + 3)) - 9)*sgn(1/(2*x + 3)) - 663/16*sqrt(3)*log(1/2*abs(-2*s qrt(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)) - 455/32*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)*sgn( 1/(2*x + 3)) + 3/8*(704*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(3 5)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 323*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/( 2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^4*sgn(1/(2*x + 3)) - 1944*(sqrt(-18/ (2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^3*sgn(1/(2*x + 3)) + 1158*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 1872*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))*sgn(1/(2*x + 3)) - 1263*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)^3
Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx=\frac {663\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{16}-\frac {815\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{48}-\frac {193\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{16}+\frac {193\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{16}-\frac {\sqrt {3}\,x^2\,\sqrt {x^2+\frac {2}{3}}}{4}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32\,\left (x+\frac {3}{2}\right )}+3\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}} \]
(663*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/16 - (815*3^(1/2)*(x^2 + 2/3)^( 1/2))/48 - (193*35^(1/2)*log(x + 3/2))/16 + (193*35^(1/2)*log(x - (3^(1/2) *35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/16 - (3^(1/2)*x^2*(x^2 + 2/3)^(1/2) )/4 - (455*3^(1/2)*(x^2 + 2/3)^(1/2))/(32*(x + 3/2)) + 3*3^(1/2)*x*(x^2 + 2/3)^(1/2)