\(\int (5-2 x)^{3/2} \sqrt {4+3 x} (2+3 x+x^2)^2 \, dx\) [216]

Optimal result

Integrand size = 29, antiderivative size = 182 \[ \int (5-2 x)^{3/2} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\frac {532603019 \sqrt {5-2 x} \sqrt {4+3 x}}{7962624}+\frac {23156653 (5-2 x)^{3/2} \sqrt {4+3 x}}{3981312}-\frac {1006811 (5-2 x)^{5/2} \sqrt {4+3 x}}{331776}-\frac {552611 (5-2 x)^{5/2} (4+3 x)^{3/2}}{138240}+\frac {39083 (5-2 x)^{7/2} (4+3 x)^{3/2}}{40320}-\frac {1091 (5-2 x)^{9/2} (4+3 x)^{3/2}}{12096}+\frac {1}{336} (5-2 x)^{11/2} (4+3 x)^{3/2}+\frac {12249869437 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )}{7962624 \sqrt {6}} \] Output:

532603019/7962624*(5-2*x)^(1/2)*(4+3*x)^(1/2)[phone]/3981312*(5-2*x)^(3/ 
2)*(4+3*x)^(1/2)-1006811/331776*(5-2*x)^(5/2)*(4+3*x)^(1/2)-552611/138240* 
(5-2*x)^(5/2)*(4+3*x)^(3/2)+39083/40320*(5-2*x)^(7/2)*(4+3*x)^(3/2)-1091/1 
2096*(5-2*x)^(9/2)*(4+3*x)^(3/2)+1/336*(5-2*x)^(11/2)*(4+3*x)^(3/2)+122498 
69437/47775744*arcsin(1/23*46^(1/2)*(4+3*x)^(1/2))*6^(1/2)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.49 \[ \int (5-2 x)^{3/2} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\frac {-\frac {6 \sqrt {5-2 x} \left (92824905540+73953516515 x-16848668076 x^2-28113980256 x^3-11074972032 x^4+297879552 x^5+1270702080 x^6+238878720 x^7\right )}{\sqrt {4+3 x}}-428745430295 \sqrt {6} \arctan \left (\frac {\sqrt {\frac {15}{2}-3 x}}{\sqrt {4+3 x}}\right )}{1672151040} \] Input:

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

Output:

((-6*Sqrt[5 - 2*x]*(92824905540 + 73953516515*x - 16848668076*x^2 - 281139 
80256*x^3 - 11074972032*x^4 + 297879552*x^5 + 1270702080*x^6 + 238878720*x 
^7))/Sqrt[4 + 3*x] - 428745430295*Sqrt[6]*ArcTan[Sqrt[15/2 - 3*x]/Sqrt[4 + 
 3*x]])/1672151040
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1194, 27, 2125, 27, 1194, 27, 90, 60, 60, 60, 64, 223}

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.

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

\(\Big \downarrow \) 1194

\(\displaystyle \frac {1}{336} \int \frac {1}{2} (5-2 x)^{3/2} \sqrt {3 x+4} \left (8728 x^3-1284 x^2+12114 x+8063\right )dx+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{672} \int (5-2 x)^{3/2} \sqrt {3 x+4} \left (8728 x^3-1284 x^2+12114 x+8063\right )dx+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {1}{672} \left (-\frac {1}{144} \int 36 (5-2 x)^{3/2} \sqrt {3 x+4} \left (-156332 x^2+104284 x+49573\right )dx-\frac {1091}{18} (3 x+4)^{3/2} (5-2 x)^{9/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{672} \left (-\frac {1}{4} \int (5-2 x)^{3/2} \sqrt {3 x+4} \left (-156332 x^2+104284 x+49573\right )dx-\frac {1091}{18} (3 x+4)^{3/2} (5-2 x)^{9/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 1194

\(\displaystyle \frac {1}{672} \left (\frac {1}{4} \left (\frac {39083}{15} (5-2 x)^{7/2} (3 x+4)^{3/2}-\frac {1}{60} \int -14 (5-2 x)^{3/2} \sqrt {3 x+4} (1105222 x+94625)dx\right )-\frac {1091}{18} (5-2 x)^{9/2} (3 x+4)^{3/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{672} \left (\frac {1}{4} \left (\frac {7}{30} \int (5-2 x)^{3/2} \sqrt {3 x+4} (1105222 x+94625)dx+\frac {39083}{15} (3 x+4)^{3/2} (5-2 x)^{7/2}\right )-\frac {1091}{18} (5-2 x)^{9/2} (3 x+4)^{3/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{672} \left (\frac {1}{4} \left (\frac {7}{30} \left (\frac {5034055}{24} \int (5-2 x)^{3/2} \sqrt {3 x+4}dx-\frac {552611}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )+\frac {39083}{15} (3 x+4)^{3/2} (5-2 x)^{7/2}\right )-\frac {1091}{18} (5-2 x)^{9/2} (3 x+4)^{3/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{672} \left (\frac {1}{4} \left (\frac {7}{30} \left (\frac {5034055}{24} \left (\frac {23}{12} \int \frac {(5-2 x)^{3/2}}{\sqrt {3 x+4}}dx-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )-\frac {552611}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )+\frac {39083}{15} (3 x+4)^{3/2} (5-2 x)^{7/2}\right )-\frac {1091}{18} (5-2 x)^{9/2} (3 x+4)^{3/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{672} \left (\frac {1}{4} \left (\frac {7}{30} \left (\frac {5034055}{24} \left (\frac {23}{12} \left (\frac {23}{4} \int \frac {\sqrt {5-2 x}}{\sqrt {3 x+4}}dx+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )-\frac {552611}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )+\frac {39083}{15} (3 x+4)^{3/2} (5-2 x)^{7/2}\right )-\frac {1091}{18} (5-2 x)^{9/2} (3 x+4)^{3/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{672} \left (\frac {1}{4} \left (\frac {7}{30} \left (\frac {5034055}{24} \left (\frac {23}{12} \left (\frac {23}{4} \left (\frac {23}{6} \int \frac {1}{\sqrt {5-2 x} \sqrt {3 x+4}}dx+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )-\frac {552611}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )+\frac {39083}{15} (3 x+4)^{3/2} (5-2 x)^{7/2}\right )-\frac {1091}{18} (5-2 x)^{9/2} (3 x+4)^{3/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{672} \left (\frac {1}{4} \left (\frac {7}{30} \left (\frac {5034055}{24} \left (\frac {23}{12} \left (\frac {23}{4} \left (\frac {23}{9} \int \frac {1}{\sqrt {\frac {23}{3}-\frac {2}{3} (3 x+4)}}d\sqrt {3 x+4}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )-\frac {552611}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )+\frac {39083}{15} (3 x+4)^{3/2} (5-2 x)^{7/2}\right )-\frac {1091}{18} (5-2 x)^{9/2} (3 x+4)^{3/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{672} \left (\frac {1}{4} \left (\frac {7}{30} \left (\frac {5034055}{24} \left (\frac {23}{12} \left (\frac {23}{4} \left (\frac {23 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{3 \sqrt {6}}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{6} \sqrt {3 x+4} (5-2 x)^{3/2}\right )-\frac {1}{6} (5-2 x)^{5/2} \sqrt {3 x+4}\right )-\frac {552611}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )+\frac {39083}{15} (3 x+4)^{3/2} (5-2 x)^{7/2}\right )-\frac {1091}{18} (5-2 x)^{9/2} (3 x+4)^{3/2}\right )+\frac {1}{336} (3 x+4)^{3/2} (5-2 x)^{11/2}\)

Input:

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

Output:

((5 - 2*x)^(11/2)*(4 + 3*x)^(3/2))/336 + ((-1091*(5 - 2*x)^(9/2)*(4 + 3*x) 
^(3/2))/18 + ((39083*(5 - 2*x)^(7/2)*(4 + 3*x)^(3/2))/15 + (7*((-552611*(5 
 - 2*x)^(5/2)*(4 + 3*x)^(3/2))/12 + (5034055*(-1/6*((5 - 2*x)^(5/2)*Sqrt[4 
 + 3*x]) + (23*(((5 - 2*x)^(3/2)*Sqrt[4 + 3*x])/6 + (23*((Sqrt[5 - 2*x]*Sq 
rt[4 + 3*x])/3 + (23*ArcSin[Sqrt[2/23]*Sqrt[4 + 3*x]])/(3*Sqrt[6])))/4))/1 
2))/24))/30)/4)/672
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 64

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp 
[2/b   Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] 
 /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] 
 || PosQ[b])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 1194

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

rule 2125

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x 
)^(m + q)*((c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1))), x] + Simp[1/(d*b^q*( 
m + n + q + 1))   Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q 
+ 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x) 
^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c, d, m, n}, 
 x] && PolyQ[Px, x]
 

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {\left (79626240 x^{6}[phone] x^{5}-323905536 x^{4}-3259783296 x^{3}-5024949024 x^{2}[phone] x [phone]\right ) \sqrt {3 x +4}\, \left (-5+2 x \right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{278691840 \sqrt {-\left (3 x +4\right ) \left (-5+2 x \right )}\, \sqrt {5-2 x}}+\frac {12249869437 \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{95551488 \sqrt {5-2 x}\, \sqrt {3 x +4}}\)	\(118\)
default	\(\frac {\sqrt {5-2 x}\, \sqrt {3 x +4}\, \left (-955514880 x^{6} \sqrt {-6 x^{2}+7 x +20}-3808788480 x^{5} \sqrt {-6 x^{2}+7 x +20}[phone] x^{4} \sqrt {-6 x^{2}+7 x +20}[phone] x^{3} \sqrt {-6 x^{2}+7 x +20}[phone] x^{2} \sqrt {-6 x^{2}+7 x +20}[phone] \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right )-13004512080 x \sqrt {-6 x^{2}+7 x +20}-278474716620 \sqrt {-6 x^{2}+7 x +20}\right )}{3344302080 \sqrt {-6 x^{2}+7 x +20}}\)	\(155\)

Input:

int((5-2*x)^(3/2)*(3*x+4)^(1/2)*(x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
 

Output:

1/278691840*(79626240*x^6+317399040*x^5-323905536*x^4-3259783296*x^3-50249 
49024*x^2+1083709340*x[phone])*(3*x+4)^(1/2)*(-5+2*x)/(-(3*x+4)*(-5+2 
*x))^(1/2)*((5-2*x)*(3*x+4))^(1/2)/(5-2*x)^(1/2)[phone]/95551488*6^(1 
/2)*arcsin(12/23*x-7/23)*((5-2*x)*(3*x+4))^(1/2)/(5-2*x)^(1/2)/(3*x+4)^(1/ 
2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.49 \[ \int (5-2 x)^{3/2} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=-\frac {1}{278691840} \, {\left (79626240 \, x^{6} + 317399040 \, x^{5} - 323905536 \, x^{4} - 3259783296 \, x^{3} - 5024949024 \, x^{2} + 1083709340 \, x + 23206226385\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} - \frac {12249869437}{95551488} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} {\left (12 \, x - 7\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{12 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) \] Input:

integrate((5-2*x)^(3/2)*(4+3*x)^(1/2)*(x^2+3*x+2)^2,x, algorithm="fricas")
 

Output:

-1/278691840*(79626240*x^6 + 317399040*x^5 - 323905536*x^4 - 3259783296*x^ 
3 - 5024949024*x^2 + 1083709340*x + 23206226385)*sqrt(3*x + 4)*sqrt(-2*x + 
 5) - 12249869437/95551488*sqrt(6)*arctan(1/12*sqrt(6)*(12*x - 7)*sqrt(3*x 
 + 4)*sqrt(-2*x + 5)/(6*x^2 - 7*x - 20))
 

Sympy [F]

\[ \int (5-2 x)^{3/2} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\int \left (5 - 2 x\right )^{\frac {3}{2}} \left (x + 1\right )^{2} \left (x + 2\right )^{2} \sqrt {3 x + 4}\, dx \] Input:

integrate((5-2*x)**(3/2)*(4+3*x)**(1/2)*(x**2+3*x+2)**2,x)
 

Output:

Integral((5 - 2*x)**(3/2)*(x + 1)**2*(x + 2)**2*sqrt(3*x + 4), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.66 \[ \int (5-2 x)^{3/2} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\frac {1}{21} \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} x^{4} + \frac {53}{216} \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} x^{3} + \frac {2533}{10080} \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} x^{2} - \frac {57949}{69120} \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} x - \frac {10958029}{3483648} \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} + \frac {23156653}{663552} \, \sqrt {-6 \, x^{2} + 7 \, x + 20} x - \frac {12249869437}{95551488} \, \sqrt {6} \arcsin \left (-\frac {12}{23} \, x + \frac {7}{23}\right ) - \frac {162096571}{7962624} \, \sqrt {-6 \, x^{2} + 7 \, x + 20} \] Input:

integrate((5-2*x)^(3/2)*(4+3*x)^(1/2)*(x^2+3*x+2)^2,x, algorithm="maxima")
 

Output:

1/21*(-6*x^2 + 7*x + 20)^(3/2)*x^4 + 53/216*(-6*x^2 + 7*x + 20)^(3/2)*x^3 
+ 2533/10080*(-6*x^2 + 7*x + 20)^(3/2)*x^2 - 57949/69120*(-6*x^2 + 7*x + 2 
0)^(3/2)*x - 10958029/3483648*(-6*x^2 + 7*x + 20)^(3/2) + 23156653/663552* 
sqrt(-6*x^2 + 7*x + 20)*x - 12249869437/95551488*sqrt(6)*arcsin(-12/23*x + 
 7/23) - 162096571/7962624*sqrt(-6*x^2 + 7*x + 20)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (131) = 262\).

Time = 0.35 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.45 \[ \int (5-2 x)^{3/2} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((5-2*x)^(3/2)*(4+3*x)^(1/2)*(x^2+3*x+2)^2,x, algorithm="giac")
 

Output:

-1/15049359360*sqrt(3)*(2*(4*(8*(4*(16*(20*(72*x - 599)*(3*x + 4) + 170917 
)*(3*x + 4) - 18917781)*(3*x + 4) + 315126553)*(3*x + 4) - 7029935045)*(3* 
x + 4) + 29323924455)*sqrt(3*x + 4)*sqrt(-6*x + 15) + 622092990015*sqrt(2) 
*arcsin(1/23*sqrt(46)*sqrt(3*x + 4))) - 29/1074954240*sqrt(3)*(2*(4*(8*(12 
*(16*(20*x - 141)*(3*x + 4) + 27693)*(3*x + 4) - 1890227)*(3*x + 4) + 4804 
4695)*(3*x + 4) - 554161485)*sqrt(3*x + 4)*sqrt(-6*x + 15) + 1169435115*sq 
rt(2)*arcsin(1/23*sqrt(46)*sqrt(3*x + 4))) - 1/933120*sqrt(3)*(2*(4*(8*(12 
*(16*x - 93)*(3*x + 4) + 11459)*(3*x + 4) - 423015)*(3*x + 4) + 3579965)*s 
qrt(3*x + 4)*sqrt(-6*x + 15) + 38247045*sqrt(2)*arcsin(1/23*sqrt(46)*sqrt( 
3*x + 4))) + 139/746496*sqrt(3)*(2*(4*(8*(36*x - 167)*(3*x + 4) + 10987)*( 
3*x + 4) - 188073)*sqrt(3*x + 4)*sqrt(-6*x + 15) - 196305*sqrt(2)*arcsin(1 
/23*sqrt(46)*sqrt(3*x + 4))) + 5/81*sqrt(3)*(2*(4*(8*x - 29)*(3*x + 4) + 7 
19)*sqrt(3*x + 4)*sqrt(-6*x + 15) + 7015*sqrt(2)*arcsin(1/23*sqrt(46)*sqrt 
(3*x + 4))) + 67/36*sqrt(3)*(2*(4*x - 13)*sqrt(3*x + 4)*sqrt(-6*x + 15) - 
69*sqrt(2)*arcsin(1/23*sqrt(46)*sqrt(3*x + 4))) + 40/9*sqrt(3)*(23*sqrt(2) 
*arcsin(1/23*sqrt(46)*sqrt(3*x + 4)) + 2*sqrt(3*x + 4)*sqrt(-6*x + 15))
 

Mupad [F(-1)]

Timed out. \[ \int (5-2 x)^{3/2} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\int {\left (5-2\,x\right )}^{3/2}\,\sqrt {3\,x+4}\,{\left (x^2+3\,x+2\right )}^2 \,d x \] Input:

int((5 - 2*x)^(3/2)*(3*x + 4)^(1/2)*(3*x + x^2 + 2)^2,x)
                                                                                    
                                                                                    
 

Output:

int((5 - 2*x)^(3/2)*(3*x + 4)^(1/2)*(3*x + x^2 + 2)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.73 \[ \int (5-2 x)^{3/2} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=-\frac {12249869437 \sqrt {6}\, \mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{47775744}-\frac {2 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{6}}{7}-\frac {41 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{5}}{36}+\frac {17573 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{4}}{15120}+\frac {404239 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{3}}{34560}+\frac {52343219 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{2}}{2903040}-\frac {7740781 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x}{1990656}-\frac {1547081759 \sqrt {3 x +4}\, \sqrt {-2 x +5}}{18579456} \] Input:

int((5-2*x)^(3/2)*(4+3*x)^(1/2)*(x^2+3*x+2)^2,x)
 

Output:

( - 428745430295*sqrt(6)*asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23)) - 47775 
7440*sqrt(3*x + 4)*sqrt( - 2*x + 5)*x**6 - 1904394240*sqrt(3*x + 4)*sqrt( 
- 2*x + 5)*x**5 + 1943433216*sqrt(3*x + 4)*sqrt( - 2*x + 5)*x**4 + 1955869 
9776*sqrt(3*x + 4)*sqrt( - 2*x + 5)*x**3 + 30149694144*sqrt(3*x + 4)*sqrt( 
 - 2*x + 5)*x**2 - 6502256040*sqrt(3*x + 4)*sqrt( - 2*x + 5)*x - 139237358 
310*sqrt(3*x + 4)*sqrt( - 2*x + 5))/1672151040