3.25.33 \(\int (5-x) (3+2 x)^3 (2+5 x+3 x^2)^{5/2} \, dx\) [2433]

3.25.33.1 Optimal result

Integrand size = 27, antiderivative size = 181 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {182917 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{71663616 \sqrt {3}} \]

output

-182917/4478976*(5+6*x)*(3*x^2+5*x+2)^(3/2)+182917/466560*(5+6*x)*(3*x^2+5 
*x+2)^(5/2)+169/405*(3+2*x)^2*(3*x^2+5*x+2)^(7/2)-1/30*(3+2*x)^3*(3*x^2+5* 
x+2)^(7/2)+1/136080*(477101+213878*x)*(3*x^2+5*x+2)^(7/2)-182917/214990848 
*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)+182917/35831808* 
(5+6*x)*(3*x^2+5*x+2)^(1/2)
 

3.25.33.2 Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.53 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-73178684475-585749416130 x-1995914277480 x^2-3762746217360 x^3-4253933381760 x^4-2893044950784 x^5-1086687912960 x^6-147947046912 x^7+29262643200 x^8+9029615616 x^9\right )-6402095 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{3762339840} \]

input

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

output

(-3*Sqrt[2 + 5*x + 3*x^2]*(-73178684475 - 585749416130*x - 1995914277480*x 
^2 - 3762746217360*x^3 - 4253933381760*x^4 - 2893044950784*x^5 - 108668791 
2960*x^6 - 147947046912*x^7 + 29262643200*x^8 + 9029615616*x^9) - 6402095* 
Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/3762339840
 

3.25.33.3 Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 206, 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, 1236, 1225, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{30} \int \frac {1}{2} (2 x+3)^2 (676 x+1029) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{60} \int (2 x+3)^2 (676 x+1029) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{60} \left (\frac {1}{27} \int (2 x+3) (30554 x+42451) \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {676}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {1}{60} \left (\frac {1}{27} \left (\frac {182917}{8} \int \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {1}{84} (213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {676}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{60} \left (\frac {1}{27} \left (\frac {182917}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )+\frac {1}{84} (213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {676}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{60} \left (\frac {1}{27} \left (\frac {182917}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )+\frac {1}{84} (213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {676}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{60} \left (\frac {1}{27} \left (\frac {182917}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{84} (213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {676}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{60} \left (\frac {1}{27} \left (\frac {182917}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{84} (213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {676}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{60} \left (\frac {1}{27} \left (\frac {182917}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{84} (213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {676}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\)

input

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

output

-1/30*((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(7/2)) + ((676*(3 + 2*x)^2*(2 + 5*x + 
 3*x^2)^(7/2))/27 + (((477101 + 213878*x)*(2 + 5*x + 3*x^2)^(7/2))/84 + (1 
82917*(((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))/8)/27)/60
 

3.25.33.3.1 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])
 

3.25.33.4 Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.50

method	result	size
risch	\(-\frac {\left (9029615616 x^{9}[phone] x^{8}-147947046912 x^{7}-1086687912960 x^{6}-2893044950784 x^{5}-4253933381760 x^{4}-3762746217360 x^{3}-1995914277480 x^{2}-585749416130 x -73178684475\right ) \sqrt {3 x^{2}+5 x +2}}{1254113280}-\frac {182917 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{214990848}\)	\(90\)
trager	\(\left (-\frac {36}{5} x^{9}-\frac {70}{3} x^{8}+\frac {42469}{360} x^{7}+\frac {873431}{1008} x^{6}+\frac {418553957}{181440} x^{5}+\frac {738530101}{217728} x^{4}+\frac {5226036413}{1741824} x^{3}+\frac {16632618979}{10450944} x^{2}+\frac {58574941613}{125411328} x +\frac {4878578965}{83607552}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {182917 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{214990848}\)	\(101\)
default	\(\frac {182917 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{466560}-\frac {182917 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{4478976}+\frac {182917 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{35831808}-\frac {182917 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{214990848}+\frac {173137 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{27216}-\frac {4 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{15}+\frac {38 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{81}+\frac {46453 x \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{9720}\)	\(151\)

input

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

output

-1/1254113280*(9029615616*x^9+29262643200*x^8-147947046912*x^7-10866879129 
60*x^6-2893044950784*x^5-4253933381760*x^4-3762746217360*x^3-1995914277480 
*x^2-585749416130*x-73178684475)*(3*x^2+5*x+2)^(1/2)-182917/214990848*ln(1 
/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
 

3.25.33.5 Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.54 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{1254113280} \, {\left (9029615616 \, x^{9} + 29262643200 \, x^{8} - 147947046912 \, x^{7} - 1086687912960 \, x^{6} - 2893044950784 \, x^{5} - 4253933381760 \, x^{4} - 3762746217360 \, x^{3} - 1995914277480 \, x^{2} - 585749416130 \, x - 73178684475\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {182917}{429981696} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]

input

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

output

-1/1254113280*(9029615616*x^9 + 29262643200*x^8 - 147947046912*x^7 - 10866 
87912960*x^6 - 2893044950784*x^5 - 4253933381760*x^4 - 3762746217360*x^3 - 
 1995914277480*x^2 - 585749416130*x - 73178684475)*sqrt(3*x^2 + 5*x + 2) + 
 182917/429981696*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 
 72*x^2 + 120*x + 49)
 

3.25.33.6 Sympy [A] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.61 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {36 x^{9}}{5} - \frac {70 x^{8}}{3} + \frac {42469 x^{7}}{360} + \frac {873431 x^{6}}{1008} + \frac {418553957 x^{5}}{181440} + \frac {738530101 x^{4}}{217728} + \frac {5226036413 x^{3}}{1741824} + \frac {16632618979 x^{2}}{10450944} + \frac {58574941613 x}{125411328} + \frac {4878578965}{83607552}\right ) - \frac {182917 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{214990848} \]

input

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

output

sqrt(3*x**2 + 5*x + 2)*(-36*x**9/5 - 70*x**8/3 + 42469*x**7/360 + 873431*x 
**6/1008 + 418553957*x**5/181440 + 738530101*x**4/217728 + 5226036413*x**3 
/1741824 + 16632618979*x**2/10450944 + 58574941613*x/125411328 + 487857896 
5/83607552) - 182917*sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 
5)/214990848
 

3.25.33.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.99 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {4}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{3} + \frac {38}{81} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {46453}{9720} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {173137}{27216} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {182917}{77760} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {182917}{93312} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {182917}{746496} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {914585}{4478976} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {182917}{5971968} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {182917}{214990848} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {914585}{35831808} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]

input

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

output

-4/15*(3*x^2 + 5*x + 2)^(7/2)*x^3 + 38/81*(3*x^2 + 5*x + 2)^(7/2)*x^2 + 46 
453/9720*(3*x^2 + 5*x + 2)^(7/2)*x + 173137/27216*(3*x^2 + 5*x + 2)^(7/2) 
+ 182917/77760*(3*x^2 + 5*x + 2)^(5/2)*x + 182917/93312*(3*x^2 + 5*x + 2)^ 
(5/2) - 182917/746496*(3*x^2 + 5*x + 2)^(3/2)*x - 914585/4478976*(3*x^2 + 
5*x + 2)^(3/2) + 182917/5971968*sqrt(3*x^2 + 5*x + 2)*x - 182917/214990848 
*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 914585/35831808* 
sqrt(3*x^2 + 5*x + 2)
 

3.25.33.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.52 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{1254113280} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, {\left (54 \, x + 175\right )} x - 42469\right )} x - 4367155\right )} x - 418553957\right )} x - 3692650505\right )} x - 26130182065\right )} x - 83163094895\right )} x - 292874708065\right )} x - 73178684475\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {182917}{214990848} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

input

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

output

-1/1254113280*(2*(12*(6*(8*(6*(36*(14*(48*(54*x + 175)*x - 42469)*x - 4367 
155)*x - 418553957)*x - 3692650505)*x - 26130182065)*x - 83163094895)*x - 
292874708065)*x - 73178684475)*sqrt(3*x^2 + 5*x + 2) + 182917/214990848*sq 
rt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
 

3.25.33.9 Mupad [F(-1)]

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

input

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

output

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