3.14.88 \(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^3} \, dx\) [1388]

3.14.88.1 Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {15}{64} (859-267 x) \sqrt {2+3 x^2}+\frac {5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac {(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {43995}{128} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {12885}{128} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \]

output

5/32*(178+29*x)*(3*x^2+2)^(3/2)/(3+2*x)-1/16*(29+2*x)*(3*x^2+2)^(5/2)/(3+2 
*x)^2-43995/128*arcsinh(1/2*x*6^(1/2))*3^(1/2)-12885/128*arctanh(1/35*(4-9 
*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)+15/64*(859-267*x)*(3*x^2+2)^(1/2)
 

3.14.88.2 Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {1}{128} \left (-\frac {2 \sqrt {2+3 x^2} \left (-126181-127403 x-19268 x^2+2826 x^3-696 x^4+72 x^5\right )}{(3+2 x)^2}+25770 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )+43995 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )\right ) \]

input

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

output

((-2*Sqrt[2 + 3*x^2]*(-126181 - 127403*x - 19268*x^2 + 2826*x^3 - 696*x^4 
+ 72*x^5))/(3 + 2*x)^2 + 25770*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 
 2*Sqrt[2 + 3*x^2])/Sqrt[35]] + 43995*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 
3*x^2]])/128
 

3.14.88.3 Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {681, 27, 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.

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

\(\Big \downarrow \) 681

\(\displaystyle -\frac {5}{64} \int \frac {4 (4-87 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^2}dx-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{16} \int \frac {(4-87 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^2}dx-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

\(\Big \downarrow \) 681

\(\displaystyle -\frac {5}{16} \left (-\frac {1}{8} \int \frac {24 (29-267 x) \sqrt {3 x^2+2}}{2 x+3}dx-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{16} \left (-3 \int \frac {(29-267 x) \sqrt {3 x^2+2}}{2 x+3}dx-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

\(\Big \downarrow \) 682

\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {1}{24} \int \frac {42 (262-1257 x)}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \int \frac {262-1257 x}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

\(\Big \downarrow \) 719

\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \left (\frac {4295}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1257}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

\(\Big \downarrow \) 222

\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \left (\frac {4295}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {419}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \left (-\frac {4295}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {419}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {5}{16} \left (-3 \left (\frac {7}{4} \left (-\frac {419}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {859}{2} \sqrt {\frac {5}{7}} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (859-267 x)\right )-\frac {(29 x+178) \left (3 x^2+2\right )^{3/2}}{2 (2 x+3)}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}\)

input

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

output

-1/16*((29 + 2*x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^2 - (5*(-1/2*((178 + 29*x)* 
(2 + 3*x^2)^(3/2))/(3 + 2*x) - 3*(((859 - 267*x)*Sqrt[2 + 3*x^2])/4 + (7*( 
(-419*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/2 - (859*Sqrt[5/7]*ArcTanh[(4 - 9*x)/( 
Sqrt[35]*Sqrt[2 + 3*x^2])])/2))/4)))/16
 

3.14.88.3.1 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 682

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])
 

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]
 

3.14.88.4 Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {216 x^{7}-2088 x^{6}+8622 x^{5}-59196 x^{4}-376557 x^{3}-417079 x^{2}-254806 x -252362}{64 \left (3+2 x \right )^{2} \sqrt {3 x^{2}+2}}-\frac {43995 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{128}-\frac {12885 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{128}\)	\(97\)
trager	\(-\frac {\left (72 x^{5}-696 x^{4}+2826 x^{3}-19268 x^{2}-127403 x -126181\right ) \sqrt {3 x^{2}+2}}{64 \left (3+2 x \right )^{2}}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-25825835\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-25825835\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-25825835\right )-30065 \sqrt {3 x^{2}+2}}{3+2 x}\right )}{128}+\frac {43995 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{128}\)	\(122\)
default	\(\frac {421 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{4900 \left (x +\frac {3}{2}\right )}+\frac {2577 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4900}-\frac {807 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{224}-\frac {4005 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{64}-\frac {43995 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{128}+\frac {859 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{112}+\frac {12885 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{128}-\frac {12885 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{128}-\frac {1263 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4900}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{280 \left (x +\frac {3}{2}\right )^{2}}\)	\(185\)

input

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

output

-1/64*(216*x^7-2088*x^6+8622*x^5-59196*x^4-376557*x^3-417079*x^2-254806*x- 
252362)/(3+2*x)^2/(3*x^2+2)^(1/2)-43995/128*arcsinh(1/2*x*6^(1/2))*3^(1/2) 
-12885/128*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^( 
1/2))
 

3.14.88.5 Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.16 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {43995 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 12885 \, \sqrt {35} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (72 \, x^{5} - 696 \, x^{4} + 2826 \, x^{3} - 19268 \, x^{2} - 127403 \, x - 126181\right )} \sqrt {3 \, x^{2} + 2}}{256 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]

input

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

output

1/256*(43995*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3* 
x^2 - 1) + 12885*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)) - 4*(72*x^5 - 696*x^ 
4 + 2826*x^3 - 19268*x^2 - 127403*x - 126181)*sqrt(3*x^2 + 2))/(4*x^2 + 12 
*x + 9)
 

3.14.88.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.14.88.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.15 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {39}{280} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{70 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {807}{224} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {859}{112} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {421 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{280 \, {\left (2 \, x + 3\right )}} - \frac {4005}{64} \, \sqrt {3 \, x^{2} + 2} x - \frac {43995}{128} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {12885}{128} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {12885}{64} \, \sqrt {3 \, x^{2} + 2} \]

input

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

output

39/280*(3*x^2 + 2)^(5/2) - 13/70*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 80 
7/224*(3*x^2 + 2)^(3/2)*x + 859/112*(3*x^2 + 2)^(3/2) + 421/280*(3*x^2 + 2 
)^(5/2)/(2*x + 3) - 4005/64*sqrt(3*x^2 + 2)*x - 43995/128*sqrt(3)*arcsinh( 
1/2*sqrt(6)*x) + 12885/128*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2 
/3*sqrt(6)/abs(2*x + 3)) + 12885/64*sqrt(3*x^2 + 2)
 

3.14.88.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (100) = 200\).

Time = 0.30 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.83 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=-\frac {1}{32} \, {\left (3 \, {\left ({\left (3 \, x - 38\right )} x + 225\right )} x - 4177\right )} \sqrt {3 \, x^{2} + 2} + \frac {43995}{128} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {12885}{128} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {35 \, {\left (11472 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 25829 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 57912 \, \sqrt {3} x + 8984 \, \sqrt {3} + 57912 \, \sqrt {3 \, x^{2} + 2}\right )}}{256 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]

input

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

output

-1/32*(3*((3*x - 38)*x + 225)*x - 4177)*sqrt(3*x^2 + 2) + 43995/128*sqrt(3 
)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 12885/128*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))) + 35/256*(11472*(sqrt(3)*x - sqrt(3*x^2 + 
 2))^3 + 25829*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 57912*sqrt(3)*x + 
 8984*sqrt(3) + 57912*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
 

3.14.88.9 Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.17 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {12885\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{128}+\frac {4177\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32}-\frac {43995\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{128}-\frac {12885\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{128}+\frac {57\,\sqrt {3}\,x^2\,\sqrt {x^2+\frac {2}{3}}}{16}-\frac {9\,\sqrt {3}\,x^3\,\sqrt {x^2+\frac {2}{3}}}{32}+\frac {39305\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{256\,\left (x+\frac {3}{2}\right )}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{512\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {675\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{32} \]

input

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

output

(12885*35^(1/2)*log(x + 3/2))/128 + (4177*3^(1/2)*(x^2 + 2/3)^(1/2))/32 - 
(43995*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/128 - (12885*35^(1/2)*log(x - 
 (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/128 + (57*3^(1/2)*x^2*(x^2 
 + 2/3)^(1/2))/16 - (9*3^(1/2)*x^3*(x^2 + 2/3)^(1/2))/32 + (39305*3^(1/2)* 
(x^2 + 2/3)^(1/2))/(256*(x + 3/2)) - (15925*3^(1/2)*(x^2 + 2/3)^(1/2))/(51 
2*(3*x + x^2 + 9/4)) - (675*3^(1/2)*x*(x^2 + 2/3)^(1/2))/32