\(\int \sqrt {5-2 x} \sqrt {4+3 x} (2+3 x+x^2)^2 \, dx\) [217]

Optimal result

Integrand size = 29, antiderivative size = 160 \[ \int \sqrt {5-2 x} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\frac {5917417 \sqrt {5-2 x} \sqrt {4+3 x}}{147456}-\frac {257279 (5-2 x)^{3/2} \sqrt {4+3 x}}{24576}-\frac {55079 (5-2 x)^{3/2} (4+3 x)^{3/2}}{9216}+\frac {643}{512} (5-2 x)^{5/2} (4+3 x)^{3/2}-\frac {7}{64} (5-2 x)^{7/2} (4+3 x)^{3/2}+\frac {1}{288} (5-2 x)^{9/2} (4+3 x)^{3/2}+\frac {136100591 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )}{147456 \sqrt {6}} \] Output:

5917417/147456*(5-2*x)^(1/2)*(4+3*x)^(1/2)-257279/24576*(5-2*x)^(3/2)*(4+3 
*x)^(1/2)-55079/9216*(5-2*x)^(3/2)*(4+3*x)^(3/2)+643/512*(5-2*x)^(5/2)*(4+ 
3*x)^(3/2)-7/64*(5-2*x)^(7/2)*(4+3*x)^(3/2)+1/288*(5-2*x)^(9/2)*(4+3*x)^(3 
/2)[phone]/884736*arcsin(1/23*46^(1/2)*(4+3*x)^(1/2))*6^(1/2)
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.52 \[ \int \sqrt {5-2 x} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\frac {\frac {6 \sqrt {5-2 x} \left (-30767332-28941979 x-3523188 x^2+2374240 x^3+1983872 x^4+620544 x^5+73728 x^6\right )}{\sqrt {4+3 x}}-136100591 \sqrt {6} \arctan \left (\frac {\sqrt {\frac {15}{2}-3 x}}{\sqrt {4+3 x}}\right )}{884736} \] Input:

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

Output:

((6*Sqrt[5 - 2*x]*(-30767332 - 28941979*x - 3523188*x^2 + 2374240*x^3 + 19 
83872*x^4 + 620544*x^5 + 73728*x^6))/Sqrt[4 + 3*x] - 136100591*Sqrt[6]*Arc 
Tan[Sqrt[15/2 - 3*x]/Sqrt[4 + 3*x]])/884736
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1194, 27, 2125, 27, 1194, 27, 90, 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.

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

\(\Big \downarrow \) 1194

\(\displaystyle \frac {1}{288} \int \frac {9}{2} \sqrt {5-2 x} \sqrt {3 x+4} \left (840 x^3-188 x^2+1318 x+631\right )dx+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{64} \int \sqrt {5-2 x} \sqrt {3 x+4} \left (840 x^3-188 x^2+1318 x+631\right )dx+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {1}{64} \left (-\frac {1}{120} \int 180 \sqrt {5-2 x} \sqrt {3 x+4} \left (-2572 x^2+2108 x+221\right )dx-7 (3 x+4)^{3/2} (5-2 x)^{7/2}\right )+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{64} \left (-\frac {3}{2} \int \sqrt {5-2 x} \sqrt {3 x+4} \left (-2572 x^2+2108 x+221\right )dx-7 (3 x+4)^{3/2} (5-2 x)^{7/2}\right )+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 1194

\(\displaystyle \frac {1}{64} \left (-\frac {3}{2} \left (\frac {1}{48} \int 2 (21379-110158 x) \sqrt {5-2 x} \sqrt {3 x+4}dx-\frac {643}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )-7 (3 x+4)^{3/2} (5-2 x)^{7/2}\right )+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{64} \left (-\frac {3}{2} \left (\frac {1}{24} \int (21379-110158 x) \sqrt {5-2 x} \sqrt {3 x+4}dx-\frac {643}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )-7 (3 x+4)^{3/2} (5-2 x)^{7/2}\right )+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{64} \left (-\frac {3}{2} \left (\frac {1}{24} \left (\frac {55079}{9} (5-2 x)^{3/2} (3 x+4)^{3/2}-\frac {257279}{6} \int \sqrt {5-2 x} \sqrt {3 x+4}dx\right )-\frac {643}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )-7 (3 x+4)^{3/2} (5-2 x)^{7/2}\right )+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{64} \left (-\frac {3}{2} \left (\frac {1}{24} \left (\frac {55079}{9} (5-2 x)^{3/2} (3 x+4)^{3/2}-\frac {257279}{6} \left (\frac {23}{8} \int \frac {\sqrt {5-2 x}}{\sqrt {3 x+4}}dx-\frac {1}{4} (5-2 x)^{3/2} \sqrt {3 x+4}\right )\right )-\frac {643}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )-7 (3 x+4)^{3/2} (5-2 x)^{7/2}\right )+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{64} \left (-\frac {3}{2} \left (\frac {1}{24} \left (\frac {55079}{9} (5-2 x)^{3/2} (3 x+4)^{3/2}-\frac {257279}{6} \left (\frac {23}{8} \left (\frac {23}{6} \int \frac {1}{\sqrt {5-2 x} \sqrt {3 x+4}}dx+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )-\frac {1}{4} (5-2 x)^{3/2} \sqrt {3 x+4}\right )\right )-\frac {643}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )-7 (3 x+4)^{3/2} (5-2 x)^{7/2}\right )+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{64} \left (-\frac {3}{2} \left (\frac {1}{24} \left (\frac {55079}{9} (5-2 x)^{3/2} (3 x+4)^{3/2}-\frac {257279}{6} \left (\frac {23}{8} \left (\frac {23}{9} \int \frac {1}{\sqrt {\frac {23}{3}-\frac {2}{3} (3 x+4)}}d\sqrt {3 x+4}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )-\frac {1}{4} (5-2 x)^{3/2} \sqrt {3 x+4}\right )\right )-\frac {643}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )-7 (3 x+4)^{3/2} (5-2 x)^{7/2}\right )+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{64} \left (-\frac {3}{2} \left (\frac {1}{24} \left (\frac {55079}{9} (5-2 x)^{3/2} (3 x+4)^{3/2}-\frac {257279}{6} \left (\frac {23}{8} \left (\frac {23 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{3 \sqrt {6}}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )-\frac {1}{4} (5-2 x)^{3/2} \sqrt {3 x+4}\right )\right )-\frac {643}{12} (5-2 x)^{5/2} (3 x+4)^{3/2}\right )-7 (3 x+4)^{3/2} (5-2 x)^{7/2}\right )+\frac {1}{288} (3 x+4)^{3/2} (5-2 x)^{9/2}\)

Input:

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

Output:

((5 - 2*x)^(9/2)*(4 + 3*x)^(3/2))/288 + (-7*(5 - 2*x)^(7/2)*(4 + 3*x)^(3/2 
) - (3*((-643*(5 - 2*x)^(5/2)*(4 + 3*x)^(3/2))/12 + ((55079*(5 - 2*x)^(3/2 
)*(4 + 3*x)^(3/2))/9 - (257279*(-1/4*((5 - 2*x)^(3/2)*Sqrt[4 + 3*x]) + (23 
*((Sqrt[5 - 2*x]*Sqrt[4 + 3*x])/3 + (23*ArcSin[Sqrt[2/23]*Sqrt[4 + 3*x]])/ 
(3*Sqrt[6])))/8))/6)/24))/2)/64
 

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 = 113, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {\left (24576 x^{5}+174080 x^{4}+429184 x^{3}+219168 x^{2}-1466620 x -7691833\right ) \sqrt {3 x +4}\, \left (-5+2 x \right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{147456 \sqrt {-\left (3 x +4\right ) \left (-5+2 x \right )}\, \sqrt {5-2 x}}+\frac {136100591 \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{1769472 \sqrt {5-2 x}\, \sqrt {3 x +4}}\)	\(113\)
default	\(\frac {\sqrt {5-2 x}\, \sqrt {3 x +4}\, \left (294912 x^{5} \sqrt {-6 x^{2}+7 x +20}+2088960 x^{4} \sqrt {-6 x^{2}+7 x +20}+5150208 x^{3} \sqrt {-6 x^{2}+7 x +20}+2630016 x^{2} \sqrt {-6 x^{2}+7 x +20}[phone] \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right )-17599440 x \sqrt {-6 x^{2}+7 x +20}-92301996 \sqrt {-6 x^{2}+7 x +20}\right )}{1769472 \sqrt {-6 x^{2}+7 x +20}}\)	\(138\)

Input:

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

Output:

-1/147456*(24576*x^5+174080*x^4+429184*x^3+219168*x^2-1466620*x-7691833)*( 
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]/1769472*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 = 84, normalized size of antiderivative = 0.52 \[ \int \sqrt {5-2 x} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\frac {1}{147456} \, {\left (24576 \, x^{5} + 174080 \, x^{4} + 429184 \, x^{3} + 219168 \, x^{2} - 1466620 \, x - 7691833\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} - \frac {136100591}{1769472} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} {\left (12 \, x - 7\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{12 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) \] Input:

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

Output:

1/147456*(24576*x^5 + 174080*x^4 + 429184*x^3 + 219168*x^2 - 1466620*x - 7 
691833)*sqrt(3*x + 4)*sqrt(-2*x + 5) - 136100591/1769472*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]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.65 \[ \int \sqrt {5-2 x} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=-\frac {1}{36} \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} x^{3} - \frac {11}{48} \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} x^{2} - \frac {649}{768} \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} x - \frac {18409}{9216} \, {\left (-6 \, x^{2} + 7 \, x + 20\right )}^{\frac {3}{2}} + \frac {257279}{12288} \, \sqrt {-6 \, x^{2} + 7 \, x + 20} x - \frac {136100591}{1769472} \, \sqrt {6} \arcsin \left (-\frac {12}{23} \, x + \frac {7}{23}\right ) - \frac {1800953}{147456} \, \sqrt {-6 \, x^{2} + 7 \, x + 20} \] Input:

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

Output:

-1/36*(-6*x^2 + 7*x + 20)^(3/2)*x^3 - 11/48*(-6*x^2 + 7*x + 20)^(3/2)*x^2 
- 649/768*(-6*x^2 + 7*x + 20)^(3/2)*x - 18409/9216*(-6*x^2 + 7*x + 20)^(3/ 
2) + 257279/12288*sqrt(-6*x^2 + 7*x + 20)*x - 136100591/1769472*sqrt(6)*ar 
csin(-12/23*x + 7/23) - 1800953/147456*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. 356 vs. \(2 (115) = 230\).

Time = 0.32 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.22 \[ \int \sqrt {5-2 x} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\frac {1}{358318080} \, \sqrt {3} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (16 \, {\left (20 \, x - 141\right )} {\left (3 \, x + 4\right )} + 27693\right )} {\left (3 \, x + 4\right )} - 1890227\right )} {\left (3 \, x + 4\right )} + 48044695\right )} {\left (3 \, x + 4\right )} - 554161485\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15} + 1169435115 \, \sqrt {2} \arcsin \left (\frac {1}{23} \, \sqrt {46} \sqrt {3 \, x + 4}\right )\right )} + \frac {11}{7464960} \, \sqrt {3} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (16 \, x - 93\right )} {\left (3 \, x + 4\right )} + 11459\right )} {\left (3 \, x + 4\right )} - 423015\right )} {\left (3 \, x + 4\right )} + 3579965\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15} + 38247045 \, \sqrt {2} \arcsin \left (\frac {1}{23} \, \sqrt {46} \sqrt {3 \, x + 4}\right )\right )} + \frac {7}{82944} \, \sqrt {3} {\left (2 \, {\left (4 \, {\left (8 \, {\left (36 \, x - 167\right )} {\left (3 \, x + 4\right )} + 10987\right )} {\left (3 \, x + 4\right )} - 188073\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15} - 196305 \, \sqrt {2} \arcsin \left (\frac {1}{23} \, \sqrt {46} \sqrt {3 \, x + 4}\right )\right )} + \frac {11}{648} \, \sqrt {3} {\left (2 \, {\left (4 \, {\left (8 \, x - 29\right )} {\left (3 \, x + 4\right )} + 719\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15} + 7015 \, \sqrt {2} \arcsin \left (\frac {1}{23} \, \sqrt {46} \sqrt {3 \, x + 4}\right )\right )} + \frac {5}{12} \, \sqrt {3} {\left (2 \, {\left (4 \, x - 13\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15} - 69 \, \sqrt {2} \arcsin \left (\frac {1}{23} \, \sqrt {46} \sqrt {3 \, x + 4}\right )\right )} + \frac {8}{9} \, \sqrt {3} {\left (23 \, \sqrt {2} \arcsin \left (\frac {1}{23} \, \sqrt {46} \sqrt {3 \, x + 4}\right ) + 2 \, \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15}\right )} \] Input:

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

Output:

1/358318080*sqrt(3)*(2*(4*(8*(12*(16*(20*x - 141)*(3*x + 4) + 27693)*(3*x 
+ 4) - 1890227)*(3*x + 4) + 48044695)*(3*x + 4) - 554161485)*sqrt(3*x + 4) 
*sqrt(-6*x + 15) + 1169435115*sqrt(2)*arcsin(1/23*sqrt(46)*sqrt(3*x + 4))) 
 + 11/7464960*sqrt(3)*(2*(4*(8*(12*(16*x - 93)*(3*x + 4) + 11459)*(3*x + 4 
) - 423015)*(3*x + 4) + 3579965)*sqrt(3*x + 4)*sqrt(-6*x + 15) + 38247045* 
sqrt(2)*arcsin(1/23*sqrt(46)*sqrt(3*x + 4))) + 7/82944*sqrt(3)*(2*(4*(8*(3 
6*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))) + 11/648*sqrt( 
3)*(2*(4*(8*x - 29)*(3*x + 4) + 719)*sqrt(3*x + 4)*sqrt(-6*x + 15) + 7015* 
sqrt(2)*arcsin(1/23*sqrt(46)*sqrt(3*x + 4))) + 5/12*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))) + 8/9*sqrt(3)*(23*sqrt(2)*arcsin(1/23*sqrt(46)*sqrt(3*x + 4)) + 2*sq 
rt(3*x + 4)*sqrt(-6*x + 15))
 

Mupad [B] (verification not implemented)

Time = 31.49 (sec) , antiderivative size = 1056, normalized size of antiderivative = 6.60 \[ \int \sqrt {5-2 x} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

(136100591*6^(1/2)*atan((6^(1/2)*(5^(1/2) - (5 - 2*x)^(1/2)))/(2*((3*x + 4 
)^(1/2) - 2))))/442368 - ((112507631*(5^(1/2) - (5 - 2*x)^(1/2)))/(1913187 
6*((3*x + 4)^(1/2) - 2)) - (1464234323*(5^(1/2) - (5 - 2*x)^(1/2))^3)/(382 
63752*((3*x + 4)^(1/2) - 2)^3) - (11190572219*(5^(1/2) - (5 - 2*x)^(1/2))^ 
5)/(8503056*((3*x + 4)^(1/2) - 2)^5) + (65144353189*(5^(1/2) - (5 - 2*x)^( 
1/2))^7)/(5668704*((3*x + 4)^(1/2) - 2)^7) + (1124203249763*(5^(1/2) - (5 
- 2*x)^(1/2))^9)/(153055008*((3*x + 4)^(1/2) - 2)^9) - (2464444087391*(5^( 
1/2) - (5 - 2*x)^(1/2))^11)/(34012224*((3*x + 4)^(1/2) - 2)^11) + (2464444 
087391*(5^(1/2) - (5 - 2*x)^(1/2))^13)/(22674816*((3*x + 4)^(1/2) - 2)^13) 
 - (1124203249763*(5^(1/2) - (5 - 2*x)^(1/2))^15)/(45349632*((3*x + 4)^(1/ 
2) - 2)^15) - (65144353189*(5^(1/2) - (5 - 2*x)^(1/2))^17)/(746496*((3*x + 
 4)^(1/2) - 2)^17) + (11190572219*(5^(1/2) - (5 - 2*x)^(1/2))^19)/(497664* 
((3*x + 4)^(1/2) - 2)^19) + (1464234323*(5^(1/2) - (5 - 2*x)^(1/2))^21)/(9 
95328*((3*x + 4)^(1/2) - 2)^21) - (112507631*(5^(1/2) - (5 - 2*x)^(1/2))^2 
3)/(221184*((3*x + 4)^(1/2) - 2)^23) + (4390912*5^(1/2)*(5^(1/2) - (5 - 2* 
x)^(1/2))^2)/(531441*((3*x + 4)^(1/2) - 2)^2) + (64552960*5^(1/2)*(5^(1/2) 
 - (5 - 2*x)^(1/2))^4)/(177147*((3*x + 4)^(1/2) - 2)^4) + (1547927552*5^(1 
/2)*(5^(1/2) - (5 - 2*x)^(1/2))^6)/(1594323*((3*x + 4)^(1/2) - 2)^6) - (46 
78762496*5^(1/2)*(5^(1/2) - (5 - 2*x)^(1/2))^8)/(531441*((3*x + 4)^(1/2) - 
 2)^8) + (23324127232*5^(1/2)*(5^(1/2) - (5 - 2*x)^(1/2))^10)/(531441*(...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.72 \[ \int \sqrt {5-2 x} \sqrt {4+3 x} \left (2+3 x+x^2\right )^2 \, dx=-\frac {136100591 \sqrt {6}\, \mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{884736}+\frac {\sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{5}}{6}+\frac {85 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{4}}{72}+\frac {3353 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{3}}{1152}+\frac {761 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x^{2}}{512}-\frac {366655 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x}{36864}-\frac {7691833 \sqrt {3 x +4}\, \sqrt {-2 x +5}}{147456} \] Input:

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

Output:

( - 136100591*sqrt(6)*asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23)) + 147456*s 
qrt(3*x + 4)*sqrt( - 2*x + 5)*x**5 + 1044480*sqrt(3*x + 4)*sqrt( - 2*x + 5 
)*x**4 + 2575104*sqrt(3*x + 4)*sqrt( - 2*x + 5)*x**3 + 1315008*sqrt(3*x + 
4)*sqrt( - 2*x + 5)*x**2 - 8799720*sqrt(3*x + 4)*sqrt( - 2*x + 5)*x - 4615 
0998*sqrt(3*x + 4)*sqrt( - 2*x + 5))/884736