\(\int (5-x) (3+2 x)^3 (2+3 x^2)^{5/2} \, dx\) [221]

Optimal result

Integrand size = 24, antiderivative size = 132 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{5/2} \, dx=\frac {3731}{24} x \sqrt {2+3 x^2}+\frac {3731}{72} x \left (2+3 x^2\right )^{3/2}+\frac {3731}{180} x \left (2+3 x^2\right )^{5/2}+\frac {91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac {(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac {3731 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}} \] Output:

3731/24*x*(3*x^2+2)^(1/2)+3731/72*x*(3*x^2+2)^(3/2)+3731/180*x*(3*x^2+2)^( 
5/2)+91/270*(3+2*x)^2*(3*x^2+2)^(7/2)-1/30*(3+2*x)^3*(3*x^2+2)^(7/2)+1/162 
0*(15244+4977*x)*(3*x^2+2)^(7/2)+3731/36*arcsinh(1/2*x*6^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.69 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {\sqrt {2+3 x^2} \left (-299200-1245915 x-1350240 x^2-1922805 x^3-2036880 x^4-1503522 x^5-1035720 x^6-418446 x^7-12960 x^8+23328 x^9\right )}{3240}-\frac {3731 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{12 \sqrt {3}} \] Input:

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

Output:

-1/3240*(Sqrt[2 + 3*x^2]*(-299200 - 1245915*x - 1350240*x^2 - 1922805*x^3 
- 2036880*x^4 - 1503522*x^5 - 1035720*x^6 - 418446*x^7 - 12960*x^8 + 23328 
*x^9)) - (3731*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(12*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {687, 27, 687, 27, 676, 211, 211, 211, 222}

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.

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

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{30} \int 21 (2 x+3)^2 (13 x+22) \left (3 x^2+2\right )^{5/2}dx-\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{10} \int (2 x+3)^2 (13 x+22) \left (3 x^2+2\right )^{5/2}dx-\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {7}{10} \left (\frac {1}{27} \int 2 (2 x+3) (711 x+839) \left (3 x^2+2\right )^{5/2}dx+\frac {13}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{10} \left (\frac {2}{27} \int (2 x+3) (711 x+839) \left (3 x^2+2\right )^{5/2}dx+\frac {13}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {7}{10} \left (\frac {2}{27} \left (\frac {4797}{2} \int \left (3 x^2+2\right )^{5/2}dx+\frac {237}{4} x \left (3 x^2+2\right )^{7/2}+\frac {3811}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {13}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {7}{10} \left (\frac {2}{27} \left (\frac {4797}{2} \left (\frac {5}{3} \int \left (3 x^2+2\right )^{3/2}dx+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+\frac {237}{4} x \left (3 x^2+2\right )^{7/2}+\frac {3811}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {13}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {7}{10} \left (\frac {2}{27} \left (\frac {4797}{2} \left (\frac {5}{3} \left (\frac {3}{2} \int \sqrt {3 x^2+2}dx+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+\frac {237}{4} x \left (3 x^2+2\right )^{7/2}+\frac {3811}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {13}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {7}{10} \left (\frac {2}{27} \left (\frac {4797}{2} \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+\frac {237}{4} x \left (3 x^2+2\right )^{7/2}+\frac {3811}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {13}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {7}{10} \left (\frac {2}{27} \left (\frac {4797}{2} \left (\frac {5}{3} \left (\frac {3}{2} \left (\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+\frac {237}{4} x \left (3 x^2+2\right )^{7/2}+\frac {3811}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {13}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\)

Input:

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

Output:

-1/30*((3 + 2*x)^3*(2 + 3*x^2)^(7/2)) + (7*((13*(3 + 2*x)^2*(2 + 3*x^2)^(7 
/2))/27 + (2*((3811*(2 + 3*x^2)^(7/2))/21 + (237*x*(2 + 3*x^2)^(7/2))/4 + 
(4797*((x*(2 + 3*x^2)^(5/2))/6 + (5*((x*(2 + 3*x^2)^(3/2))/4 + (3*((x*Sqrt 
[2 + 3*x^2])/2 + ArcSinh[Sqrt[3/2]*x]/Sqrt[3]))/2))/3))/2))/27))/10
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 676

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim 
p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p 
+ 3))/(c*(2*p + 3))   Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g 
, p}, x] &&  !LeQ[p, -1]
 

rule 687

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

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.53

method	result	size
risch	\(-\frac {\left (23328 x^{9}-12960 x^{8}-418446 x^{7}-1035720 x^{6}-1503522 x^{5}-2036880 x^{4}-1922805 x^{3}-1350240 x^{2}-1245915 x -299200\right ) \sqrt {3 x^{2}+2}}{3240}+\frac {3731 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{36}\)	\(70\)
trager	\(\left (-\frac {36}{5} x^{9}+4 x^{8}+\frac {2583}{20} x^{7}+\frac {959}{3} x^{6}+\frac {9281}{20} x^{5}+\frac {1886}{3} x^{4}+\frac {14243}{24} x^{3}+\frac {11252}{27} x^{2}+\frac {9229}{24} x +\frac {7480}{81}\right ) \sqrt {3 x^{2}+2}-\frac {3731 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{36}\)	\(87\)
default	\(\frac {3731 x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{180}+\frac {3731 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{72}+\frac {3731 x \sqrt {3 x^{2}+2}}{24}+\frac {3731 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{36}+\frac {935 \left (3 x^{2}+2\right )^{\frac {7}{2}}}{81}+\frac {319 x \left (3 x^{2}+2\right )^{\frac {7}{2}}}{60}+\frac {4 x^{2} \left (3 x^{2}+2\right )^{\frac {7}{2}}}{27}-\frac {4 x^{3} \left (3 x^{2}+2\right )^{\frac {7}{2}}}{15}\)	\(101\)
meijerg	\(-\frac {675 \sqrt {3}\, \left (-\frac {8 \sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {3}{8} x^{4}+\frac {13}{16} x^{2}+\frac {11}{16}\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{3}\right )}{2 \sqrt {\pi }}-\frac {20 \sqrt {2}\, \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-\frac {567}{4} x^{8}-\frac {513}{2} x^{6}-135 x^{4}-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{945}\right )}{3 \sqrt {\pi }}-\frac {210 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (162 x^{6}+306 x^{4}+177 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{720}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{24}\right )}{\sqrt {\pi }}-\frac {1215 \sqrt {2}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {8 \sqrt {\pi }\, \left (\frac {27}{4} x^{6}+\frac {27}{2} x^{4}+9 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{105}\right )}{2 \sqrt {\pi }}+\frac {80 \sqrt {3}\, \left (\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (-648 x^{8}-1134 x^{6}-558 x^{4}-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{2400}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{80}\right )}{9 \sqrt {\pi }}\)	\(277\)

Input:

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

Output:

-1/3240*(23328*x^9-12960*x^8-418446*x^7-1035720*x^6-1503522*x^5-2036880*x^ 
4-1922805*x^3-1350240*x^2-1245915*x-299200)*(3*x^2+2)^(1/2)+3731/36*arcsin 
h(1/2*6^(1/2)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.64 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{3240} \, {\left (23328 \, x^{9} - 12960 \, x^{8} - 418446 \, x^{7} - 1035720 \, x^{6} - 1503522 \, x^{5} - 2036880 \, x^{4} - 1922805 \, x^{3} - 1350240 \, x^{2} - 1245915 \, x - 299200\right )} \sqrt {3 \, x^{2} + 2} + \frac {3731}{72} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \] Input:

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

Output:

-1/3240*(23328*x^9 - 12960*x^8 - 418446*x^7 - 1035720*x^6 - 1503522*x^5 - 
2036880*x^4 - 1922805*x^3 - 1350240*x^2 - 1245915*x - 299200)*sqrt(3*x^2 + 
 2) + 3731/72*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.36 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{5/2} \, dx=- \frac {36 x^{9} \sqrt {3 x^{2} + 2}}{5} + 4 x^{8} \sqrt {3 x^{2} + 2} + \frac {2583 x^{7} \sqrt {3 x^{2} + 2}}{20} + \frac {959 x^{6} \sqrt {3 x^{2} + 2}}{3} + \frac {9281 x^{5} \sqrt {3 x^{2} + 2}}{20} + \frac {1886 x^{4} \sqrt {3 x^{2} + 2}}{3} + \frac {14243 x^{3} \sqrt {3 x^{2} + 2}}{24} + \frac {11252 x^{2} \sqrt {3 x^{2} + 2}}{27} + \frac {9229 x \sqrt {3 x^{2} + 2}}{24} + \frac {7480 \sqrt {3 x^{2} + 2}}{81} + \frac {3731 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{36} \] Input:

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

Output:

-36*x**9*sqrt(3*x**2 + 2)/5 + 4*x**8*sqrt(3*x**2 + 2) + 2583*x**7*sqrt(3*x 
**2 + 2)/20 + 959*x**6*sqrt(3*x**2 + 2)/3 + 9281*x**5*sqrt(3*x**2 + 2)/20 
+ 1886*x**4*sqrt(3*x**2 + 2)/3 + 14243*x**3*sqrt(3*x**2 + 2)/24 + 11252*x* 
*2*sqrt(3*x**2 + 2)/27 + 9229*x*sqrt(3*x**2 + 2)/24 + 7480*sqrt(3*x**2 + 2 
)/81 + 3731*sqrt(3)*asinh(sqrt(6)*x/2)/36
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {4}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{3} + \frac {4}{27} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{2} + \frac {319}{60} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {935}{81} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {3731}{180} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {3731}{72} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {3731}{24} \, \sqrt {3 \, x^{2} + 2} x + \frac {3731}{36} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \] Input:

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

Output:

-4/15*(3*x^2 + 2)^(7/2)*x^3 + 4/27*(3*x^2 + 2)^(7/2)*x^2 + 319/60*(3*x^2 + 
 2)^(7/2)*x + 935/81*(3*x^2 + 2)^(7/2) + 3731/180*(3*x^2 + 2)^(5/2)*x + 37 
31/72*(3*x^2 + 2)^(3/2)*x + 3731/24*sqrt(3*x^2 + 2)*x + 3731/36*sqrt(3)*ar 
csinh(1/2*sqrt(6)*x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.58 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{3240} \, {\left (3 \, {\left ({\left (9 \, {\left (2 \, {\left ({\left ({\left (3 \, {\left (16 \, {\left (9 \, x - 5\right )} x - 2583\right )} x - 19180\right )} x - 27843\right )} x - 37720\right )} x - 71215\right )} x - 450080\right )} x - 415305\right )} x - 299200\right )} \sqrt {3 \, x^{2} + 2} - \frac {3731}{36} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \] Input:

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

Output:

-1/3240*(3*((9*(2*(((3*(16*(9*x - 5)*x - 2583)*x - 19180)*x - 27843)*x - 3 
7720)*x - 71215)*x - 450080)*x - 415305)*x - 299200)*sqrt(3*x^2 + 2) - 373 
1/36*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))
 

Mupad [B] (verification not implemented)

Time = 5.91 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.53 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{5/2} \, dx=\frac {3731\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{36}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {108\,x^9}{5}+12\,x^8+\frac {7749\,x^7}{20}+959\,x^6+\frac {27843\,x^5}{20}+1886\,x^4+\frac {14243\,x^3}{8}+\frac {11252\,x^2}{9}+\frac {9229\,x}{8}+\frac {7480}{27}\right )}{3} \] Input:

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

Output:

(3731*3^(1/2)*asinh((6^(1/2)*x)/2))/36 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((9229 
*x)/8 + (11252*x^2)/9 + (14243*x^3)/8 + 1886*x^4 + (27843*x^5)/20 + 959*x^ 
6 + (7749*x^7)/20 + 12*x^8 - (108*x^9)/5 + 7480/27))/3
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.13 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {36 \sqrt {3 x^{2}+2}\, x^{9}}{5}+4 \sqrt {3 x^{2}+2}\, x^{8}+\frac {2583 \sqrt {3 x^{2}+2}\, x^{7}}{20}+\frac {959 \sqrt {3 x^{2}+2}\, x^{6}}{3}+\frac {9281 \sqrt {3 x^{2}+2}\, x^{5}}{20}+\frac {1886 \sqrt {3 x^{2}+2}\, x^{4}}{3}+\frac {14243 \sqrt {3 x^{2}+2}\, x^{3}}{24}+\frac {11252 \sqrt {3 x^{2}+2}\, x^{2}}{27}+\frac {9229 \sqrt {3 x^{2}+2}\, x}{24}+\frac {7480 \sqrt {3 x^{2}+2}}{81}+\frac {3731 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{36} \] Input:

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

Output:

( - 23328*sqrt(3*x**2 + 2)*x**9 + 12960*sqrt(3*x**2 + 2)*x**8 + 418446*sqr 
t(3*x**2 + 2)*x**7 + 1035720*sqrt(3*x**2 + 2)*x**6 + 1503522*sqrt(3*x**2 + 
 2)*x**5 + 2036880*sqrt(3*x**2 + 2)*x**4 + 1922805*sqrt(3*x**2 + 2)*x**3 + 
 1350240*sqrt(3*x**2 + 2)*x**2 + 1245915*sqrt(3*x**2 + 2)*x + 299200*sqrt( 
3*x**2 + 2) + 335790*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)))/ 
3240