\(\int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx\) [507]

Optimal result

Integrand size = 27, antiderivative size = 140 \[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\frac {697}{105} \sqrt {5-2 x} \sqrt {2+3 x+x^2}-\frac {1}{35} (5-2 x)^{3/2} (67+15 x) \sqrt {2+3 x+x^2}-\frac {5483 \sqrt {-2-3 x-x^2} E\left (\arcsin \left (\sqrt {2+x}\right )|\frac {2}{9}\right )}{70 \sqrt {2+3 x+x^2}}+\frac {697 \sqrt {-2-3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{10 \sqrt {2+3 x+x^2}} \] Output:

697/105*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)-1/35*(5-2*x)^(3/2)*(67+15*x)*(x^2+ 
3*x+2)^(1/2)-5483/70*(-x^2-3*x-2)^(1/2)*EllipticE((2+x)^(1/2),1/3*2^(1/2)) 
/(x^2+3*x+2)^(1/2)+697/10*(-x^2-3*x-2)^(1/2)*EllipticF((2+x)^(1/2),1/3*2^( 
1/2))/(x^2+3*x+2)^(1/2)
 

Mathematica [A] (verified)

Time = 29.21 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.21 \[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\frac {2 \sqrt {5-2 x} \left (-7886-14831 x-8638 x^2-1429 x^3+444 x^4+180 x^5\right )-16449 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} E\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right )|\frac {7}{9}\right )+1812 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right ),\frac {7}{9}\right )}{210 (-5+2 x) \sqrt {2+3 x+x^2}} \] Input:

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

Output:

(2*Sqrt[5 - 2*x]*(-7886 - 14831*x - 8638*x^2 - 1429*x^3 + 444*x^4 + 180*x^ 
5) - 16449*(5 - 2*x)^2*Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[(2 + x)/(-5 + 2*x)]*E 
llipticE[ArcSin[3/Sqrt[5 - 2*x]], 7/9] + 1812*(5 - 2*x)^2*Sqrt[(1 + x)/(-5 
 + 2*x)]*Sqrt[(2 + x)/(-5 + 2*x)]*EllipticF[ArcSin[3/Sqrt[5 - 2*x]], 7/9]) 
/(210*(-5 + 2*x)*Sqrt[2 + 3*x + x^2])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1236, 27, 1231, 25, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{7} \int \frac {(31 x+17) \sqrt {x^2+3 x+2}}{2 \sqrt {5-2 x}}dx+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \int \frac {(31 x+17) \sqrt {x^2+3 x+2}}{\sqrt {5-2 x}}dx+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{7} \left (-\frac {1}{30} \int -\frac {5483 x+8248}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {1}{15} \sqrt {5-2 x} \sqrt {x^2+3 x+2} (93 x+488)\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \int \frac {5483 x+8248}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {43911}{2} \int \frac {1}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {5483}{2} \int \frac {\sqrt {5-2 x}}{\sqrt {x^2+3 x+2}}dx\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {14637 \sqrt {-x^2-3 x-2} \int \frac {3}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {16449 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{3 \sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {43911 \sqrt {-x^2-3 x-2} \int \frac {1}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {5483 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {14637 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {5483 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{7} \left (\frac {1}{30} \left (\frac {14637 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {16449 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (93 x+488) \sqrt {x^2+3 x+2}\right )+\frac {6}{7} \sqrt {5-2 x} \left (x^2+3 x+2\right )^{3/2}\)

Input:

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

Output:

(6*Sqrt[5 - 2*x]*(2 + 3*x + x^2)^(3/2))/7 + (-1/15*(Sqrt[5 - 2*x]*(488 + 9 
3*x)*Sqrt[2 + 3*x + x^2]) + ((-16449*Sqrt[-2 - 3*x - x^2]*EllipticE[ArcSin 
[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2] + (14637*Sqrt[-2 - 3*x - x^2]*Ell 
ipticF[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2])/30)/7
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 = 1.89 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}\, \left (720 x^{5}+790 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+5483 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+1776 x^{4}-5716 x^{3}-12620 x^{2}+6472 x +12320\right )}{840 x^{3}+420 x^{2}-4620 x -4200}\)	\(136\)
elliptic	\(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (\frac {59 x \sqrt {-2 x^{3}-x^{2}+11 x +10}}{35}-\frac {44 \sqrt {-2 x^{3}-x^{2}+11 x +10}}{15}-\frac {4124 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{735 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {5483 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{1470 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {6 x^{2} \sqrt {-2 x^{3}-x^{2}+11 x +10}}{7}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(228\)
risch	\(-\frac {\left (90 x^{2}+177 x -308\right ) \left (-5+2 x \right ) \sqrt {x^{2}+3 x +2}\, \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{105 \sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \sqrt {5-2 x}}-\frac {\left (\frac {4124 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2205 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {5483 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \left (\frac {9 \operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2}-2 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )\right )}{4410 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right ) \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(235\)

Input:

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

Output:

1/420*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)*(720*x^5+790*(5-2*x)^(1/2)*(14*x+14) 
^(1/2)*(2*x+4)^(1/2)*EllipticF(1/3*(5-2*x)^(1/2),3/7*7^(1/2))+5483*(5-2*x) 
^(1/2)*(14*x+14)^(1/2)*(2*x+4)^(1/2)*EllipticE(1/3*(5-2*x)^(1/2),3/7*7^(1/ 
2))+1776*x^4-5716*x^3-12620*x^2+6472*x+12320)/(2*x^3+x^2-11*x-10)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.39 \[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\frac {1}{105} \, {\left (90 \, x^{2} + 177 \, x - 308\right )} \sqrt {x^{2} + 3 \, x + 2} \sqrt {-2 \, x + 5} - \frac {8801}{252} \, \sqrt {-2} {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right ) + \frac {5483}{210} \, \sqrt {-2} {\rm weierstrassZeta}\left (\frac {67}{3}, \frac {440}{27}, {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right )\right ) \] Input:

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

Output:

1/105*(90*x^2 + 177*x - 308)*sqrt(x^2 + 3*x + 2)*sqrt(-2*x + 5) - 8801/252 
*sqrt(-2)*weierstrassPInverse(67/3, 440/27, x + 1/6) + 5483/210*sqrt(-2)*w 
eierstrassZeta(67/3, 440/27, weierstrassPInverse(67/3, 440/27, x + 1/6))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(sqrt(x^2 + 3*x + 2)*(3*x + 4)*sqrt(-2*x + 5), x)
 

Giac [F]

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

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

Output:

integrate(sqrt(x^2 + 3*x + 2)*(3*x + 4)*sqrt(-2*x + 5), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {5-2 x} (4+3 x) \sqrt {2+3 x+x^2} \, dx=\frac {6 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{2}}{7}+\frac {59 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x}{35}-\frac {2033 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{70}+\frac {5483 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{70}-\frac {25603 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{140} \] Input:

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

Output:

(120*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x**2 + 236*sqrt( - 2*x + 5)*sqr 
t(x**2 + 3*x + 2)*x - 4066*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2) + 10966*i 
nt((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x**2)/(2*x**3 + x**2 - 11*x - 10 
),x) - 25603*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2))/(2*x**3 + x**2 - 
11*x - 10),x))/140