\(\int (5-x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2} \, dx\) [1035]

Optimal result

Integrand size = 29, antiderivative size = 170 \[ \int (5-x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2} \, dx=-\frac {1039}{945} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {1}{105} (118-15 x) (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}-\frac {697 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{270 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {1039 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{378 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:

-1039/945*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)+1/105*(118-15*x)*(3+2*x)^(3/2) 
*(3*x^2+5*x+2)^(1/2)-697/810*(-3*x^2-5*x-2)^(1/2)*EllipticE((1+x)^(1/2)*3^ 
(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+5*x+2)^(1/2)+1039/1134*(-3*x^2-5*x-2)^ 
(1/2)*EllipticF((1+x)^(1/2)*3^(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+5*x+2)^( 
1/2)
 

Mathematica [A] (verified)

Time = 30.12 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.16 \[ \int (5-x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2} \, dx=-\frac {2 \sqrt {3+2 x} \left (-28888-128926 x-200865 x^2-121239 x^3-15552 x^4+4860 x^5\right )+4879 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-1762 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{5670 (3+2 x) \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

-1/5670*(2*Sqrt[3 + 2*x]*(-28888 - 128926*x - 200865*x^2 - 121239*x^3 - 15 
552*x^4 + 4860*x^5) + 4879*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqr 
t[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 1 
762*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]* 
EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 + 5*x + 
 3*x^2])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1236, 27, 1231, 1269, 1172, 27, 321, 327}

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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{21} \int \frac {(241 x+364) \sqrt {3 x^2+5 x+2}}{2 \sqrt {2 x+3}}dx-\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \int \frac {(241 x+364) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx-\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{21} \left (\frac {1}{45} \sqrt {2 x+3} (2169 x+2327) \sqrt {3 x^2+5 x+2}-\frac {1}{90} \int \frac {4879 x+4721}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{21} \left (\frac {1}{90} \left (\frac {5195}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {4879}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (2169 x+2327)\right )-\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{21} \left (\frac {1}{90} \left (\frac {5195 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {4879 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (2169 x+2327)\right )-\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \left (\frac {1}{90} \left (\frac {5195 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {4879 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (2169 x+2327)\right )-\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{21} \left (\frac {1}{90} \left (\frac {5195 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {4879 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (2169 x+2327)\right )-\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{21} \left (\frac {1}{90} \left (\frac {5195 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {4879 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (2169 x+2327)\right )-\frac {2}{21} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

Input:

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

Output:

(-2*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(3/2))/21 + ((Sqrt[3 + 2*x]*(2327 + 21 
69*x)*Sqrt[2 + 5*x + 3*x^2])/45 + ((-4879*Sqrt[-2 - 5*x - 3*x^2]*EllipticE 
[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (51 
95*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(S 
qrt[3]*Sqrt[2 + 5*x + 3*x^2]))/90)/21
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

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

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1231

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

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])
 

rule 1269

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

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (145800 x^{5}+474 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-4879 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-466560 x^{4}-3637170 x^{3}-6465060 x^{2}-4599630 x -1159380\right )}{85050 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\)	\(146\)
risch	\(-\frac {\left (270 x^{2}-1719 x -2147\right ) \sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}}{945}-\frac {\left (\frac {4721 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{28350 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {697 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{4050 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(203\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {2 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{7}+\frac {191 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{105}+\frac {2147 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{945}+\frac {4721 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{28350 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {697 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{4050 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(231\)

Input:

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

Output:

-1/85050*(2*x+3)^(1/2)*(3*x^2+5*x+2)^(1/2)*(145800*x^5+474*(-30*x-20)^(1/2 
)*(3+3*x)^(1/2)*15^(1/2)*(2*x+3)^(1/2)*EllipticF(1/5*(-30*x-20)^(1/2),1/2* 
10^(1/2))-4879*(-30*x-20)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(2*x+3)^(1/2)*Ellip 
ticE(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))-466560*x^4-3637170*x^3-6465060*x^2 
-4599630*x-1159380)/(6*x^3+19*x^2+19*x+6)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.34 \[ \int (5-x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{945} \, {\left (270 \, x^{2} - 1719 \, x - 2147\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {7723}{102060} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {697}{810} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \] Input:

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

Output:

-1/945*(270*x^2 - 1719*x - 2147)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3) + 772 
3/102060*sqrt(6)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 697/810* 
sqrt(6)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729 
, x + 19/18))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-integrate(sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3)*(x - 5), x)
 

Giac [F]

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

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

Output:

integrate(-sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3)*(x - 5), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (5-x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2} \, dx=-\frac {2 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{7}+\frac {191 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x}{105}+\frac {2841 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{1330}+\frac {697 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{570}-\frac {169 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{140} \] Input:

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

Output:

( - 2280*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2 + 14516*sqrt(2*x + 3)*s 
qrt(3*x**2 + 5*x + 2)*x + 17046*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 975 
8*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(6*x**3 + 19*x**2 + 19*x 
 + 6),x) - 9633*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(6*x**3 + 19*x* 
*2 + 19*x + 6),x))/7980