\(\int \frac {(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^3} \, dx\) [214]

Optimal result

Integrand size = 24, antiderivative size = 99 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=\frac {9}{140} (127-38 x) \sqrt {2+3 x^2}+\frac {(404+421 x) \left (2+3 x^2\right )^{3/2}}{140 (3+2 x)^2}-\frac {111}{8} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {1143 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8 \sqrt {35}} \] Output:

9/140*(127-38*x)*(3*x^2+2)^(1/2)+1/140*(404+421*x)*(3*x^2+2)^(3/2)/(3+2*x) 
^2-111/8*arcsinh(1/2*x*6^(1/2))*3^(1/2)-1143/280*35^(1/2)*arctanh(1/35*(4- 
9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=-\frac {\sqrt {2+3 x^2} \left (-317-328 x-48 x^2+3 x^3\right )}{4 (3+2 x)^2}+\frac {1143 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{4 \sqrt {35}}+\frac {111}{8} \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right ) \] Input:

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

Output:

-1/4*(Sqrt[2 + 3*x^2]*(-317 - 328*x - 48*x^2 + 3*x^3))/(3 + 2*x)^2 + (1143 
*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(4*Sqrt[ 
35]) + (111*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/8
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {681, 27, 681, 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.

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

\(\Big \downarrow \) 681

\(\displaystyle -\frac {3}{32} \int \frac {16 (1-12 x) \sqrt {3 x^2+2}}{(2 x+3)^2}dx-\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{2} \int \frac {(1-12 x) \sqrt {3 x^2+2}}{(2 x+3)^2}dx-\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}\)

\(\Big \downarrow \) 681

\(\displaystyle -\frac {3}{2} \left (-\frac {1}{8} \int \frac {12 (8-37 x)}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (12 x+37)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{2} \left (-\frac {3}{2} \int \frac {8-37 x}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (12 x+37)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}\)

\(\Big \downarrow \) 719

\(\displaystyle -\frac {3}{2} \left (-\frac {3}{2} \left (\frac {127}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {37}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )-\frac {\sqrt {3 x^2+2} (12 x+37)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}\)

\(\Big \downarrow \) 222

\(\displaystyle -\frac {3}{2} \left (-\frac {3}{2} \left (\frac {127}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {37 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )-\frac {\sqrt {3 x^2+2} (12 x+37)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {3}{2} \left (-\frac {3}{2} \left (-\frac {127}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {37 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )-\frac {\sqrt {3 x^2+2} (12 x+37)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {3}{2} \left (-\frac {3}{2} \left (-\frac {37 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {127 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}\right )-\frac {\sqrt {3 x^2+2} (12 x+37)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}\)

Input:

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

Output:

-1/4*((8 + x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^2 - (3*(-1/2*((37 + 12*x)*Sqrt[ 
2 + 3*x^2])/(3 + 2*x) - (3*((-37*ArcSinh[Sqrt[3/2]*x])/(2*Sqrt[3]) - (127* 
ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(2*Sqrt[35])))/2))/2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 222

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 488

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]
 

rule 681

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])
 

rule 719

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]
 

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88

method	result	size
risch	\(-\frac {9 x^{5}-144 x^{4}-978 x^{3}-1047 x^{2}-656 x -634}{4 \left (2 x +3\right )^{2} \sqrt {3 x^{2}+2}}-\frac {111 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{8}-\frac {1143 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{280}\)	\(87\)
trager	\(-\frac {\left (3 x^{3}-48 x^{2}-328 x -317\right ) \sqrt {3 x^{2}+2}}{4 \left (2 x +3\right )^{2}}-\frac {1143 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{280}-\frac {111 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{8}\)	\(111\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{280 \left (x +\frac {3}{2}\right )^{2}}+\frac {187 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4900 \left (x +\frac {3}{2}\right )}+\frac {381 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1225}-\frac {171 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{70}-\frac {111 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{8}+\frac {1143 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{280}-\frac {1143 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{280}-\frac {561 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{4900}\)	\(152\)

Input:

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

Output:

-1/4*(9*x^5-144*x^4-978*x^3-1047*x^2-656*x-634)/(2*x+3)^2/(3*x^2+2)^(1/2)- 
111/8*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-1143/280*35^(1/2)*arctanh(2/35*(4-9*x 
)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.37 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=\frac {3885 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 1143 \, \sqrt {35} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 140 \, {\left (3 \, x^{3} - 48 \, x^{2} - 328 \, x - 317\right )} \sqrt {3 \, x^{2} + 2}}{560 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \] Input:

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

Output:

1/560*(3885*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x 
^2 - 1) + 1143*sqrt(35)*(4*x^2 + 12*x + 9)*log(-(sqrt(35)*sqrt(3*x^2 + 2)* 
(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 140*(3*x^3 - 48*x^2 
- 328*x - 317)*sqrt(3*x^2 + 2))/(4*x^2 + 12*x + 9)
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {2 x \sqrt {3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx - \int \left (- \frac {15 x^{2} \sqrt {3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \] Input:

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

Output:

-Integral(-10*sqrt(3*x**2 + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integr 
al(2*x*sqrt(3*x**2 + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-15* 
x**2*sqrt(3*x**2 + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(3*x**3 
*sqrt(3*x**2 + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.23 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=\frac {39}{280} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{70 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {171}{70} \, \sqrt {3 \, x^{2} + 2} x - \frac {111}{8} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1143}{280} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {1143}{140} \, \sqrt {3 \, x^{2} + 2} + \frac {187 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{280 \, {\left (2 \, x + 3\right )}} \] Input:

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

Output:

39/280*(3*x^2 + 2)^(3/2) - 13/70*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) - 17 
1/70*sqrt(3*x^2 + 2)*x - 111/8*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 1143/280*s 
qrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 1 
143/140*sqrt(3*x^2 + 2) + 187/280*(3*x^2 + 2)^(3/2)/(2*x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (77) = 154\).

Time = 0.15 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.21 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=-\frac {3}{16} \, \sqrt {3 \, x^{2} + 2} {\left (x - 19\right )} + \frac {111}{8} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {1143}{280} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {5 \, {\left (1452 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 3013 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 6528 \, \sqrt {3} x + 1048 \, \sqrt {3} + 6528 \, \sqrt {3 \, x^{2} + 2}\right )}}{64 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \] Input:

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

Output:

-3/16*sqrt(3*x^2 + 2)*(x - 19) + 111/8*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 
 + 2)) + 1143/280*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 
2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2) 
)) + 5/64*(1452*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 3013*sqrt(3)*(sqrt(3)*x 
- sqrt(3*x^2 + 2))^2 - 6528*sqrt(3)*x + 1048*sqrt(3) + 6528*sqrt(3*x^2 + 2 
))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 
2)) - 2)^2
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.18 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=\frac {1143\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{280}+\frac {57\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{16}-\frac {111\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{8}-\frac {1143\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{280}+\frac {655\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{64\,\left (x+\frac {3}{2}\right )}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{128\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {3\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{16} \] Input:

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

Output:

(1143*35^(1/2)*log(x + 3/2))/280 + (57*3^(1/2)*(x^2 + 2/3)^(1/2))/16 - (11 
1*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/8 - (1143*35^(1/2)*log(x - (3^(1/2 
)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/280 + (655*3^(1/2)*(x^2 + 2/3)^(1/ 
2))/(64*(x + 3/2)) - (455*3^(1/2)*(x^2 + 2/3)^(1/2))/(128*(3*x + x^2 + 9/4 
)) - (3*3^(1/2)*x*(x^2 + 2/3)^(1/2))/16
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.84 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=\frac {-420 \sqrt {3 x^{2}+2}\, x^{3}+6720 \sqrt {3 x^{2}+2}\, x^{2}+45920 \sqrt {3 x^{2}+2}\, x +44380 \sqrt {3 x^{2}+2}+9144 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+27432 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +20574 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-9144 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-27432 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -20574 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )+15540 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x^{2}+46620 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x +34965 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right )-15540 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x^{2}-46620 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x -34965 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right )}{2240 x^{2}+6720 x +5040} \] Input:

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

Output:

( - 420*sqrt(3*x**2 + 2)*x**3 + 6720*sqrt(3*x**2 + 2)*x**2 + 45920*sqrt(3* 
x**2 + 2)*x + 44380*sqrt(3*x**2 + 2) + 9144*sqrt(35)*log(sqrt(3*x**2 + 2)* 
sqrt(35) + 9*x - 4)*x**2 + 27432*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 
9*x - 4)*x + 20574*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4) - 914 
4*sqrt(35)*log(2*x + 3)*x**2 - 27432*sqrt(35)*log(2*x + 3)*x - 20574*sqrt( 
35)*log(2*x + 3) + 15540*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**2 + 
46620*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x + 34965*sqrt(3)*log(sqrt 
(3*x**2 + 2) - sqrt(3)*x) - 15540*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x 
)*x**2 - 46620*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x - 34965*sqrt(3) 
*log(sqrt(3*x**2 + 2) + sqrt(3)*x))/(560*(4*x**2 + 12*x + 9))