\(\int (5-x) (3+2 x)^4 (2+3 x^2)^{3/2} \, dx\) [207]

Optimal result

Integrand size = 24, antiderivative size = 138 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{3/2} \, dx=\frac {2777}{12} x \sqrt {2+3 x^2}+\frac {2777}{36} x \left (2+3 x^2\right )^{3/2}+\frac {4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac {13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac {(661583+226755 x) \left (2+3 x^2\right )^{5/2}}{17010}+\frac {2777 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \] Output:

2777/12*x*(3*x^2+2)^(1/2)+2777/36*x*(3*x^2+2)^(3/2)+4421/2268*(3+2*x)^2*(3 
*x^2+2)^(5/2)+13/36*(3+2*x)^3*(3*x^2+2)^(5/2)-1/27*(3+2*x)^4*(3*x^2+2)^(5/ 
2)+1/17010*(661583+226755*x)*(3*x^2+2)^(5/2)+2777/18*arcsinh(1/2*x*6^(1/2) 
)*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.62 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{3/2} \, dx=-\frac {\sqrt {2+3 x^2} \left (-8598544-19683405 x-27537072 x^2-27468315 x^3-24490404 x^4-14492520 x^5-3676320 x^6+204120 x^7+181440 x^8\right )}{34020}-\frac {2777 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{6 \sqrt {3}} \] Input:

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

Output:

-1/34020*(Sqrt[2 + 3*x^2]*(-8598544 - 19683405*x - 27537072*x^2 - 27468315 
*x^3 - 24490404*x^4 - 14492520*x^5 - 3676320*x^6 + 204120*x^7 + 181440*x^8 
)) - (2777*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(6*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {687, 687, 27, 687, 27, 676, 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.

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

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{27} \int (2 x+3)^3 (234 x+421) \left (3 x^2+2\right )^{3/2}dx-\frac {1}{27} (2 x+3)^4 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{27} \left (\frac {1}{24} \int 6 (2 x+3)^2 (4421 x+4584) \left (3 x^2+2\right )^{3/2}dx+\frac {39}{4} \left (3 x^2+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \left (\frac {1}{4} \int (2 x+3)^2 (4421 x+4584) \left (3 x^2+2\right )^{3/2}dx+\frac {39}{4} \left (3 x^2+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{27} \left (\frac {1}{4} \left (\frac {1}{21} \int 2 (2 x+3) (136053 x+126712) \left (3 x^2+2\right )^{3/2}dx+\frac {4421}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )+\frac {39}{4} \left (3 x^2+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \left (\frac {1}{4} \left (\frac {2}{21} \int (2 x+3) (136053 x+126712) \left (3 x^2+2\right )^{3/2}dx+\frac {4421}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )+\frac {39}{4} \left (3 x^2+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {1}{27} \left (\frac {1}{4} \left (\frac {2}{21} \left (349902 \int \left (3 x^2+2\right )^{3/2}dx+15117 x \left (3 x^2+2\right )^{5/2}+\frac {661583}{15} \left (3 x^2+2\right )^{5/2}\right )+\frac {4421}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )+\frac {39}{4} \left (3 x^2+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {1}{27} \left (\frac {1}{4} \left (\frac {2}{21} \left (349902 \left (\frac {3}{2} \int \sqrt {3 x^2+2}dx+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+15117 x \left (3 x^2+2\right )^{5/2}+\frac {661583}{15} \left (3 x^2+2\right )^{5/2}\right )+\frac {4421}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )+\frac {39}{4} \left (3 x^2+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {1}{27} \left (\frac {1}{4} \left (\frac {2}{21} \left (349902 \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+15117 x \left (3 x^2+2\right )^{5/2}+\frac {661583}{15} \left (3 x^2+2\right )^{5/2}\right )+\frac {4421}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )+\frac {39}{4} \left (3 x^2+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{27} \left (\frac {1}{4} \left (\frac {2}{21} \left (349902 \left (\frac {3}{2} \left (\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+15117 x \left (3 x^2+2\right )^{5/2}+\frac {661583}{15} \left (3 x^2+2\right )^{5/2}\right )+\frac {4421}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )+\frac {39}{4} \left (3 x^2+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+2\right )^{5/2}\)

Input:

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

Output:

-1/27*((3 + 2*x)^4*(2 + 3*x^2)^(5/2)) + ((39*(3 + 2*x)^3*(2 + 3*x^2)^(5/2) 
)/4 + ((4421*(3 + 2*x)^2*(2 + 3*x^2)^(5/2))/21 + (2*((661583*(2 + 3*x^2)^( 
5/2))/15 + 15117*x*(2 + 3*x^2)^(5/2) + 349902*((x*(2 + 3*x^2)^(3/2))/4 + ( 
3*((x*Sqrt[2 + 3*x^2])/2 + ArcSinh[Sqrt[3/2]*x]/Sqrt[3]))/2)))/21)/4)/27
 

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 = 65, normalized size of antiderivative = 0.47

method	result	size
risch	\(-\frac {\left (181440 x^{8}+204120 x^{7}-3676320 x^{6}-14492520 x^{5}-24490404 x^{4}-27468315 x^{3}-27537072 x^{2}-19683405 x -8598544\right ) \sqrt {3 x^{2}+2}}{34020}+\frac {2777 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{18}\)	\(65\)
trager	\(\left (-\frac {16}{3} x^{8}-6 x^{7}+\frac {6808}{63} x^{6}+426 x^{5}+\frac {226763}{315} x^{4}+\frac {9689}{12} x^{3}+\frac {2294756}{2835} x^{2}+\frac {6943}{12} x +\frac {2149636}{8505}\right ) \sqrt {3 x^{2}+2}-\frac {2777 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{18}\)	\(82\)
default	\(\frac {2777 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{36}+\frac {2777 x \sqrt {3 x^{2}+2}}{12}+\frac {2777 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{18}+\frac {537409 \left (3 x^{2}+2\right )^{\frac {5}{2}}}{8505}+\frac {434 x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{9}+\frac {7256 x^{2} \left (3 x^{2}+2\right )^{\frac {5}{2}}}{567}-\frac {2 x^{3} \left (3 x^{2}+2\right )^{\frac {5}{2}}}{3}-\frac {16 x^{4} \left (3 x^{2}+2\right )^{\frac {5}{2}}}{27}\)	\(103\)
meijerg	\(\frac {405 \sqrt {3}\, \left (\frac {4 \sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {3 x^{2}}{8}+\frac {5}{8}\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{3}+\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{2 \sqrt {\pi }}-\frac {32 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (-270 x^{6}-270 x^{4}-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{480}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{16}\right )}{9 \sqrt {\pi }}+\frac {88 \sqrt {2}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {2 \sqrt {\pi }\, \left (-\frac {135}{2} x^{6}-72 x^{4}-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{105}\right )}{\sqrt {\pi }}+\frac {288 \sqrt {3}\, \left (\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (18 x^{4}+21 x^{2}+3\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{36}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{6}\right )}{\sqrt {\pi }}+\frac {999 \sqrt {2}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (\frac {9}{2} x^{4}+6 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}\right )}{2 \sqrt {\pi }}-\frac {32 \sqrt {2}\, \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (\frac {2835}{8} x^{8}+\frac {675}{2} x^{6}+\frac {27}{2} x^{4}-12 x^{2}+16\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{945}\right )}{9 \sqrt {\pi }}\)	\(301\)

Input:

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

Output:

-1/34020*(181440*x^8+204120*x^7-3676320*x^6-14492520*x^5-24490404*x^4-2746 
8315*x^3-27537072*x^2-19683405*x-8598544)*(3*x^2+2)^(1/2)+2777/18*arcsinh( 
1/2*6^(1/2)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.58 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{3/2} \, dx=-\frac {1}{34020} \, {\left (181440 \, x^{8} + 204120 \, x^{7} - 3676320 \, x^{6} - 14492520 \, x^{5} - 24490404 \, x^{4} - 27468315 \, x^{3} - 27537072 \, x^{2} - 19683405 \, x - 8598544\right )} \sqrt {3 \, x^{2} + 2} + \frac {2777}{36} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \] Input:

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

Output:

-1/34020*(181440*x^8 + 204120*x^7 - 3676320*x^6 - 14492520*x^5 - 24490404* 
x^4 - 27468315*x^3 - 27537072*x^2 - 19683405*x - 8598544)*sqrt(3*x^2 + 2) 
+ 2777/36*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 2.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.17 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{3/2} \, dx=- \frac {16 x^{8} \sqrt {3 x^{2} + 2}}{3} - 6 x^{7} \sqrt {3 x^{2} + 2} + \frac {6808 x^{6} \sqrt {3 x^{2} + 2}}{63} + 426 x^{5} \sqrt {3 x^{2} + 2} + \frac {226763 x^{4} \sqrt {3 x^{2} + 2}}{315} + \frac {9689 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {2294756 x^{2} \sqrt {3 x^{2} + 2}}{2835} + \frac {6943 x \sqrt {3 x^{2} + 2}}{12} + \frac {2149636 \sqrt {3 x^{2} + 2}}{8505} + \frac {2777 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \] Input:

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

Output:

-16*x**8*sqrt(3*x**2 + 2)/3 - 6*x**7*sqrt(3*x**2 + 2) + 6808*x**6*sqrt(3*x 
**2 + 2)/63 + 426*x**5*sqrt(3*x**2 + 2) + 226763*x**4*sqrt(3*x**2 + 2)/315 
 + 9689*x**3*sqrt(3*x**2 + 2)/12 + 2294756*x**2*sqrt(3*x**2 + 2)/2835 + 69 
43*x*sqrt(3*x**2 + 2)/12 + 2149636*sqrt(3*x**2 + 2)/8505 + 2777*sqrt(3)*as 
inh(sqrt(6)*x/2)/18
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.74 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{3/2} \, dx=-\frac {16}{27} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{4} - \frac {2}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{3} + \frac {7256}{567} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{2} + \frac {434}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {537409}{8505} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {2777}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {2777}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {2777}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \] Input:

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

Output:

-16/27*(3*x^2 + 2)^(5/2)*x^4 - 2/3*(3*x^2 + 2)^(5/2)*x^3 + 7256/567*(3*x^2 
 + 2)^(5/2)*x^2 + 434/9*(3*x^2 + 2)^(5/2)*x + 537409/8505*(3*x^2 + 2)^(5/2 
) + 2777/36*(3*x^2 + 2)^(3/2)*x + 2777/12*sqrt(3*x^2 + 2)*x + 2777/18*sqrt 
(3)*arcsinh(1/2*sqrt(6)*x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.52 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{3/2} \, dx=-\frac {1}{34020} \, {\left (3 \, {\left ({\left (9 \, {\left (4 \, {\left (10 \, {\left ({\left (21 \, {\left (8 \, x + 9\right )} x - 3404\right )} x - 13419\right )} x - 226763\right )} x - 1017345\right )} x - 9179024\right )} x - 6561135\right )} x - 8598544\right )} \sqrt {3 \, x^{2} + 2} - \frac {2777}{18} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \] Input:

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

Output:

-1/34020*(3*((9*(4*(10*((21*(8*x + 9)*x - 3404)*x - 13419)*x - 226763)*x - 
 1017345)*x - 9179024)*x - 6561135)*x - 8598544)*sqrt(3*x^2 + 2) - 2777/18 
*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.47 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{3/2} \, dx=\frac {2777\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-16\,x^8-18\,x^7+\frac {6808\,x^6}{21}+1278\,x^5+\frac {226763\,x^4}{105}+\frac {9689\,x^3}{4}+\frac {2294756\,x^2}{945}+\frac {6943\,x}{4}+\frac {2149636}{2835}\right )}{3} \] Input:

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

Output:

(2777*3^(1/2)*asinh((6^(1/2)*x)/2))/18 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((6943 
*x)/4 + (2294756*x^2)/945 + (9689*x^3)/4 + (226763*x^4)/105 + 1278*x^5 + ( 
6808*x^6)/21 - 18*x^7 - 16*x^8 + 2149636/2835))/3
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{3/2} \, dx=-\frac {16 \sqrt {3 x^{2}+2}\, x^{8}}{3}-6 \sqrt {3 x^{2}+2}\, x^{7}+\frac {6808 \sqrt {3 x^{2}+2}\, x^{6}}{63}+426 \sqrt {3 x^{2}+2}\, x^{5}+\frac {226763 \sqrt {3 x^{2}+2}\, x^{4}}{315}+\frac {9689 \sqrt {3 x^{2}+2}\, x^{3}}{12}+\frac {2294756 \sqrt {3 x^{2}+2}\, x^{2}}{2835}+\frac {6943 \sqrt {3 x^{2}+2}\, x}{12}+\frac {2149636 \sqrt {3 x^{2}+2}}{8505}+\frac {2777 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{18} \] Input:

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

Output:

( - 181440*sqrt(3*x**2 + 2)*x**8 - 204120*sqrt(3*x**2 + 2)*x**7 + 3676320* 
sqrt(3*x**2 + 2)*x**6 + 14492520*sqrt(3*x**2 + 2)*x**5 + 24490404*sqrt(3*x 
**2 + 2)*x**4 + 27468315*sqrt(3*x**2 + 2)*x**3 + 27537072*sqrt(3*x**2 + 2) 
*x**2 + 19683405*sqrt(3*x**2 + 2)*x + 8598544*sqrt(3*x**2 + 2) + 5248530*s 
qrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)))/34020