3.25.49 \(\int (5-x) (3+2 x)^2 (2+5 x+3 x^2)^{7/2} \, dx\) [2449]

3.25.49.1 Optimal result

Integrand size = 27, antiderivative size = 179 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {39389 (5+6 x) \sqrt {2+5 x+3 x^2}}{143327232}+\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac {5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac {39389 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{286654464 \sqrt {3}} \]

output

39389/17915904*(5+6*x)*(3*x^2+5*x+2)^(3/2)-39389/1866240*(5+6*x)*(3*x^2+5* 
x+2)^(5/2)+5627/25920*(5+6*x)*(3*x^2+5*x+2)^(7/2)-1/33*(3+2*x)^2*(3*x^2+5* 
x+2)^(9/2)+1/26730*(47425+20358*x)*(3*x^2+5*x+2)^(9/2)+39389/859963392*arc 
tanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-39389/143327232*(5+6 
*x)*(3*x^2+5*x+2)^(1/2)
 

3.25.49.2 Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.56 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-254668717065-2519542755670 x-10882383306360 x^2-26847121235760 x^3-41472321125760 x^4-41190616509696 x^5-25723491978240 x^6-9116575930368 x^7-1156531322880 x^8+261858852864 x^9+77396705280 x^{10}\right )+2166395 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{23648993280} \]

input

Integrate[(5 - x)*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2),x]
 

output

(-3*Sqrt[2 + 5*x + 3*x^2]*(-254668717065 - 2519542755670*x - 1088238330636 
0*x^2 - 26847121235760*x^3 - 41472321125760*x^4 - 41190616509696*x^5 - 257 
23491978240*x^6 - 9116575930368*x^7 - 1156531322880*x^8 + 261858852864*x^9 
 + 77396705280*x^10) + 2166395*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/( 
1 + x)])/23648993280
 

3.25.49.3 Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1236, 27, 1225, 1087, 1087, 1087, 1087, 1092, 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.

\(\displaystyle \int (5-x) (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2} \, dx\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{33} \int \frac {1}{2} (2 x+3) (754 x+1141) \left (3 x^2+5 x+2\right )^{7/2}dx-\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{66} \int (2 x+3) (754 x+1141) \left (3 x^2+5 x+2\right )^{7/2}dx-\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {1}{66} \left (\frac {61897}{90} \int \left (3 x^2+5 x+2\right )^{7/2}dx+\frac {1}{405} (20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{66} \left (\frac {61897}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \int \left (3 x^2+5 x+2\right )^{5/2}dx\right )+\frac {1}{405} (20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{66} \left (\frac {61897}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )\right )+\frac {1}{405} (20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{66} \left (\frac {61897}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )\right )+\frac {1}{405} (20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{66} \left (\frac {61897}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{405} (20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{66} \left (\frac {61897}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{405} (20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{66} \left (\frac {61897}{90} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{405} (20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}\right )-\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}\)

input

Int[(5 - x)*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2),x]
 

output

-1/33*((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9/2)) + (((47425 + 20358*x)*(2 + 5*x 
 + 3*x^2)^(9/2))/405 + (61897*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/48 - (7 
*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/36 - (5*(((5 + 6*x)*(2 + 5*x + 3*x^2 
)^(3/2))/24 + (-1/12*((5 + 6*x)*Sqrt[2 + 5*x + 3*x^2]) + ArcTanh[(5 + 6*x) 
/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]/(24*Sqrt[3]))/16))/72))/96))/90)/66
 

3.25.49.3.1 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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1225

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

rule 1236

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

3.25.49.4 Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.53

method	result	size
risch	\(-\frac {\left (77396705280 x^{10}[phone] x^{9}-1156531322880 x^{8}-9116575930368 x^{7}-25723491978240 x^{6}-41190616509696 x^{5}-41472321125760 x^{4}-26847121235760 x^{3}-10882383306360 x^{2}-2519542755670 x -254668717065\right ) \sqrt {3 x^{2}+5 x +2}}{7882997760}+\frac {39389 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{859963392}\)	\(95\)
trager	\(\left (-\frac {108}{11} x^{10}-\frac {1827}{55} x^{9}+\frac {9683}{66} x^{8}+\frac {18318737}{15840} x^{7}+\frac {20675389}{6336} x^{6}+\frac {5959290583}{1140480} x^{5}+\frac {7200055751}{1368576} x^{4}+\frac {37287668383}{10948608} x^{3}+\frac {90686527553}{65691648} x^{2}+\frac {251954275567}{788299776} x +\frac {16977914471}{525533184}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {39389 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{859963392}\)	\(106\)
default	\(\frac {5627 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{25920}-\frac {39389 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{1866240}+\frac {39389 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{17915904}-\frac {39389 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{143327232}+\frac {39389 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{859963392}+\frac {8027 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{5346}-\frac {4 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{33}+\frac {197 x \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{495}\)	\(153\)

input

int((5-x)*(3+2*x)^2*(3*x^2+5*x+2)^(7/2),x,method=_RETURNVERBOSE)
 

output

-1/7882997760*(77396705280*x^10+261858852864*x^9-1156531322880*x^8-9116575 
930368*x^7-25723491978240*x^6-41190616509696*x^5-41472321125760*x^4-268471 
21235760*x^3-10882383306360*x^2-2519542755670*x-254668717065)*(3*x^2+5*x+2 
)^(1/2)+39389/859963392*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1 
/2)
 

3.25.49.5 Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.58 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{7882997760} \, {\left (77396705280 \, x^{10} + 261858852864 \, x^{9} - 1156531322880 \, x^{8} - 9116575930368 \, x^{7} - 25723491978240 \, x^{6} - 41190616509696 \, x^{5} - 41472321125760 \, x^{4} - 26847121235760 \, x^{3} - 10882383306360 \, x^{2} - 2519542755670 \, x - 254668717065\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {39389}{1719926784} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]

input

integrate((5-x)*(3+2*x)^2*(3*x^2+5*x+2)^(7/2),x, algorithm="fricas")
 

output

-1/7882997760*(77396705280*x^10 + 261858852864*x^9 - 1156531322880*x^8 - 9 
116575930368*x^7 - 25723491978240*x^6 - 41190616509696*x^5 - 4147232112576 
0*x^4 - 26847121235760*x^3 - 10882383306360*x^2 - 2519542755670*x - 254668 
717065)*sqrt(3*x^2 + 5*x + 2) + 39389/1719926784*sqrt(3)*log(4*sqrt(3)*sqr 
t(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
 

3.25.49.6 Sympy [A] (verification not implemented)

Time = 0.96 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.65 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {108 x^{10}}{11} - \frac {1827 x^{9}}{55} + \frac {9683 x^{8}}{66} + \frac {18318737 x^{7}}{15840} + \frac {20675389 x^{6}}{6336} + \frac {5959290583 x^{5}}{1140480} + \frac {7200055751 x^{4}}{1368576} + \frac {37287668383 x^{3}}{10948608} + \frac {90686527553 x^{2}}{65691648} + \frac {251954275567 x}{788299776} + \frac {16977914471}{525533184}\right ) + \frac {39389 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{859963392} \]

input

integrate((5-x)*(3+2*x)**2*(3*x**2+5*x+2)**(7/2),x)
 

output

sqrt(3*x**2 + 5*x + 2)*(-108*x**10/11 - 1827*x**9/55 + 9683*x**8/66 + 1831 
8737*x**7/15840 + 20675389*x**6/6336 + 5959290583*x**5/1140480 + 720005575 
1*x**4/1368576 + 37287668383*x**3/10948608 + 90686527553*x**2/65691648 + 2 
51954275567*x/788299776 + 16977914471/525533184) + 39389*sqrt(3)*log(6*x + 
 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/859963392
 

3.25.49.7 Maxima [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.07 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {4}{33} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{2} + \frac {197}{495} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x + \frac {8027}{5346} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {5627}{4320} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {5627}{5184} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {39389}{311040} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {39389}{373248} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {39389}{2985984} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {196945}{17915904} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {39389}{23887872} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {39389}{859963392} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {196945}{143327232} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]

input

integrate((5-x)*(3+2*x)^2*(3*x^2+5*x+2)^(7/2),x, algorithm="maxima")
 

output

-4/33*(3*x^2 + 5*x + 2)^(9/2)*x^2 + 197/495*(3*x^2 + 5*x + 2)^(9/2)*x + 80 
27/5346*(3*x^2 + 5*x + 2)^(9/2) + 5627/4320*(3*x^2 + 5*x + 2)^(7/2)*x + 56 
27/5184*(3*x^2 + 5*x + 2)^(7/2) - 39389/311040*(3*x^2 + 5*x + 2)^(5/2)*x - 
 39389/373248*(3*x^2 + 5*x + 2)^(5/2) + 39389/2985984*(3*x^2 + 5*x + 2)^(3 
/2)*x + 196945/17915904*(3*x^2 + 5*x + 2)^(3/2) - 39389/23887872*sqrt(3*x^ 
2 + 5*x + 2)*x + 39389/859963392*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 
2) + 6*x + 5) - 196945/143327232*sqrt(3*x^2 + 5*x + 2)
 

3.25.49.8 Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.55 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{7882997760} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, {\left (54 \, {\left (60 \, x + 203\right )} x - 48415\right )} x - 18318737\right )} x - 103376945\right )} x - 5959290583\right )} x - 36000278755\right )} x - 186438341915\right )} x - 453432637765\right )} x - 1259771377835\right )} x - 254668717065\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {39389}{859963392} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

input

integrate((5-x)*(3+2*x)^2*(3*x^2+5*x+2)^(7/2),x, algorithm="giac")
 

output

-1/7882997760*(2*(12*(6*(8*(6*(36*(2*(48*(54*(60*x + 203)*x - 48415)*x - 1 
8318737)*x - 103376945)*x - 5959290583)*x - 36000278755)*x - 186438341915) 
*x - 453432637765)*x - 1259771377835)*x - 254668717065)*sqrt(3*x^2 + 5*x + 
 2) - 39389/859963392*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 
 5*x + 2)) - 5))
 

3.25.49.9 Mupad [F(-1)]

Timed out. \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\int {\left (2\,x+3\right )}^2\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2} \,d x \]

input

int(-(2*x + 3)^2*(x - 5)*(5*x + 3*x^2 + 2)^(7/2),x)
 

output

-int((2*x + 3)^2*(x - 5)*(5*x + 3*x^2 + 2)^(7/2), x)