\(\int \frac {(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^4} \, dx\) [215]

Optimal result

Integrand size = 24, antiderivative size = 106 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=-\frac {3 (385+111 x) \sqrt {2+3 x^2}}{280 (3+2 x)}+\frac {(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}+\frac {33}{16} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {11727 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{560 \sqrt {35}} \] Output:

-3*(385+111*x)*(3*x^2+2)^(1/2)/(840+560*x)+1/420*(229+456*x)*(3*x^2+2)^(3/ 
2)/(3+2*x)^3+33/16*arcsinh(1/2*x*6^(1/2))*3^(1/2)+11727/19600*35^(1/2)*arc 
tanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.07 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=-\frac {\sqrt {2+3 x^2} \left (30269+48747 x+24474 x^2+1260 x^3\right )}{840 (3+2 x)^3}-\frac {11727 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{280 \sqrt {35}}-\frac {33}{16} \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right ) \] Input:

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^4,x]
 

Output:

-1/840*(Sqrt[2 + 3*x^2]*(30269 + 48747*x + 24474*x^2 + 1260*x^3))/(3 + 2*x 
)^3 - (11727*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35 
]])/(280*Sqrt[35]) - (33*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/16
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {680, 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.

\(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^4} \, dx\)

\(\Big \downarrow \) 680

\(\displaystyle \frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}-\frac {1}{560} \int -\frac {12 (52-111 x) \sqrt {3 x^2+2}}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{140} \int \frac {(52-111 x) \sqrt {3 x^2+2}}{(2 x+3)^2}dx+\frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {3}{140} \left (-\frac {1}{8} \int \frac {12 (74-385 x)}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (111 x+385)}{2 (2 x+3)}\right )+\frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{140} \left (-\frac {3}{2} \int \frac {74-385 x}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (111 x+385)}{2 (2 x+3)}\right )+\frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {3}{140} \left (-\frac {3}{2} \left (\frac {1303}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {385}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )-\frac {\sqrt {3 x^2+2} (111 x+385)}{2 (2 x+3)}\right )+\frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {3}{140} \left (-\frac {3}{2} \left (\frac {1303}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {385 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )-\frac {\sqrt {3 x^2+2} (111 x+385)}{2 (2 x+3)}\right )+\frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {3}{140} \left (-\frac {3}{2} \left (-\frac {1303}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {385 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )-\frac {\sqrt {3 x^2+2} (111 x+385)}{2 (2 x+3)}\right )+\frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{140} \left (-\frac {3}{2} \left (-\frac {385 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {1303 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}\right )-\frac {\sqrt {3 x^2+2} (111 x+385)}{2 (2 x+3)}\right )+\frac {(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}\)

Input:

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^4,x]
 

Output:

((229 + 456*x)*(2 + 3*x^2)^(3/2))/(420*(3 + 2*x)^3) + (3*(-1/2*((385 + 111 
*x)*Sqrt[2 + 3*x^2])/(3 + 2*x) - (3*((-385*ArcSinh[Sqrt[3/2]*x])/(2*Sqrt[3 
]) - (1303*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(2*Sqrt[35])))/2 
))/140
 

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 680

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/(e^2*(m + 1)*(m 
+ 2)*(c*d^2 + a*e^2)))*((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), x] - Sim 
p[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] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3 
, 0]
 

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 = 0.82

method	result	size
risch	\(-\frac {3780 x^{5}+73422 x^{4}+148761 x^{3}+139755 x^{2}+97494 x +60538}{840 \left (2 x +3\right )^{3} \sqrt {3 x^{2}+2}}+\frac {33 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{16}+\frac {11727 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{19600}\)	\(87\)
trager	\(-\frac {\left (1260 x^{3}+24474 x^{2}+48747 x +30269\right ) \sqrt {3 x^{2}+2}}{840 \left (2 x +3\right )^{3}}-\frac {33 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{16}+\frac {11727 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{19600}\)	\(112\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{840 \left (x +\frac {3}{2}\right )^{3}}-\frac {\left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{2450 \left (x +\frac {3}{2}\right )^{2}}-\frac {446 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{42875 \left (x +\frac {3}{2}\right )}-\frac {3909 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{85750}+\frac {3933 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{9800}+\frac {33 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{16}-\frac {11727 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{19600}+\frac {11727 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{19600}+\frac {1338 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{42875}\)	\(173\)

Input:

int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/840*(3780*x^5+73422*x^4+148761*x^3+139755*x^2+97494*x+60538)/(2*x+3)^3/ 
(3*x^2+2)^(1/2)+33/16*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+11727/19600*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 = 151, normalized size of antiderivative = 1.42 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=\frac {121275 \, \sqrt {3} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 35181 \, \sqrt {35} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 140 \, {\left (1260 \, x^{3} + 24474 \, x^{2} + 48747 \, x + 30269\right )} \sqrt {3 \, x^{2} + 2}}{117600 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^4,x, algorithm="fricas")
 

Output:

1/117600*(121275*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-sqrt(3)*sqrt(3* 
x^2 + 2)*x - 3*x^2 - 1) + 35181*sqrt(35)*(8*x^3 + 36*x^2 + 54*x + 27)*log( 
(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x + 
9)) - 140*(1260*x^3 + 24474*x^2 + 48747*x + 30269)*sqrt(3*x^2 + 2))/(8*x^3 
 + 36*x^2 + 54*x + 27)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=\text {Timed out} \] Input:

integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**4,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.42 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=\frac {3}{2450} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{105 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{1225 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {3933}{9800} \, \sqrt {3 \, x^{2} + 2} x + \frac {33}{16} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {11727}{19600} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {11727}{9800} \, \sqrt {3 \, x^{2} + 2} - \frac {223 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{1225 \, {\left (2 \, x + 3\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^4,x, algorithm="maxima")
 

Output:

3/2450*(3*x^2 + 2)^(3/2) - 13/105*(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x 
 + 27) - 2/1225*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 3933/9800*sqrt(3*x^ 
2 + 2)*x + 33/16*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 11727/19600*sqrt(35)*arc 
sinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 11727/9800*s 
qrt(3*x^2 + 2) - 223/1225*(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. 265 vs. \(2 (84) = 168\).

Time = 0.16 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.50 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=-\frac {33}{16} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {11727}{19600} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3}{16} \, \sqrt {3 \, x^{2} + 2} - \frac {\sqrt {3} {\left (14792 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 189285 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 141030 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 561630 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 166480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 50144\right )}}{1120 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \] Input:

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^4,x, algorithm="giac")
 

Output:

-33/16*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 11727/19600*sqrt(35)*lo 
g(-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))) - 3/16*sqrt(3*x^2 + 2) - 1 
/1120*sqrt(3)*(14792*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 189285*(sqr 
t(3)*x - sqrt(3*x^2 + 2))^4 + 141030*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) 
^3 - 561630*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 166480*sqrt(3)*(sqrt(3)*x - 
sqrt(3*x^2 + 2)) - 50144)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sq 
rt(3)*x - sqrt(3*x^2 + 2)) - 2)^3
 

Mupad [B] (verification not implemented)

Time = 5.95 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.25 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=\frac {33\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{16}-\frac {3\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{16}-\frac {11727\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{19600}+\frac {11727\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{19600}-\frac {1567\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{560\,\left (x+\frac {3}{2}\right )}+\frac {77\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{384\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

int(-((3*x^2 + 2)^(3/2)*(x - 5))/(2*x + 3)^4,x)
 

Output:

(33*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/16 - (3*3^(1/2)*(x^2 + 2/3)^(1/2 
))/16 - (11727*35^(1/2)*log(x + 3/2))/19600 + (11727*35^(1/2)*log(x - (3^( 
1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/19600 - (1567*3^(1/2)*(x^2 + 2/ 
3)^(1/2))/(560*(x + 3/2)) + (77*3^(1/2)*(x^2 + 2/3)^(1/2))/(32*(3*x + x^2 
+ 9/4)) - (455*3^(1/2)*(x^2 + 2/3)^(1/2))/(384*((27*x)/4 + (9*x^2)/2 + x^3 
 + 27/8))
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.49 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx=\frac {-176400 \sqrt {3 x^{2}+2}\, x^{3}-3426360 \sqrt {3 x^{2}+2}\, x^{2}-6824580 \sqrt {3 x^{2}+2}\, x -4237660 \sqrt {3 x^{2}+2}+562896 \sqrt {35}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}+2533032 \sqrt {35}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+3799548 \sqrt {35}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +1899774 \sqrt {35}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-562896 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-2533032 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-3799548 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -1899774 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )-970200 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x^{3}-4365900 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x^{2}-6548850 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x -3274425 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right )+970200 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x^{3}+4365900 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x^{2}+6548850 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x +3274425 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right )}{940800 x^{3}+4233600 x^{2}+6350400 x +3175200} \] Input:

int((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^4,x)
 

Output:

( - 176400*sqrt(3*x**2 + 2)*x**3 - 3426360*sqrt(3*x**2 + 2)*x**2 - 6824580 
*sqrt(3*x**2 + 2)*x - 4237660*sqrt(3*x**2 + 2) + 562896*sqrt(35)*log( - sq 
rt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 2533032*sqrt(35)*log( - sqrt(3*x 
**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 3799548*sqrt(35)*log( - sqrt(3*x**2 + 
2)*sqrt(35) + 9*x - 4)*x + 1899774*sqrt(35)*log( - sqrt(3*x**2 + 2)*sqrt(3 
5) + 9*x - 4) - 562896*sqrt(35)*log(2*x + 3)*x**3 - 2533032*sqrt(35)*log(2 
*x + 3)*x**2 - 3799548*sqrt(35)*log(2*x + 3)*x - 1899774*sqrt(35)*log(2*x 
+ 3) - 970200*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**3 - 4365900*sqr 
t(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**2 - 6548850*sqrt(3)*log(sqrt(3*x 
**2 + 2) - sqrt(3)*x)*x - 3274425*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x 
) + 970200*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x**3 + 4365900*sqrt(3 
)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x**2 + 6548850*sqrt(3)*log(sqrt(3*x**2 
 + 2) + sqrt(3)*x)*x + 3274425*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x))/ 
(117600*(8*x**3 + 36*x**2 + 54*x + 27))