\(\int \frac {(4+3 x)^3 \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx\) [512]

Optimal result

Integrand size = 29, antiderivative size = 165 \[ \int \frac {(4+3 x)^3 \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx=-\frac {1}{210} \sqrt {5-2 x} (82249+12069 x) \sqrt {2+3 x+x^2}-\frac {333}{14} \sqrt {5-2 x} \left (2+3 x+x^2\right )^{3/2}+\frac {3}{2} (5-2 x)^{3/2} \left (2+3 x+x^2\right )^{3/2}-\frac {1014259 \sqrt {-2-3 x-x^2} E\left (\arcsin \left (\sqrt {2+x}\right )|\frac {2}{9}\right )}{140 \sqrt {2+3 x+x^2}}+\frac {128291 \sqrt {-2-3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{20 \sqrt {2+3 x+x^2}} \] Output:

-1/210*(5-2*x)^(1/2)*(82249+12069*x)*(x^2+3*x+2)^(1/2)-333/14*(5-2*x)^(1/2 
)*(x^2+3*x+2)^(3/2)+3/2*(5-2*x)^(3/2)*(x^2+3*x+2)^(3/2)-1014259/140*(-x^2- 
3*x-2)^(1/2)*EllipticE((2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)+128291/2 
0*(-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 = 32.70 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.06 \[ \int \frac {(4+3 x)^3 \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx=\frac {-2 \sqrt {5-2 x} \left (1137628+1826908 x+790769 x^2+137057 x^3+45558 x^4+11250 x^5+1260 x^6\right )-3042777 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} E\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right )|\frac {7}{9}\right )+348666 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right ),\frac {7}{9}\right )}{420 (-5+2 x) \sqrt {2+3 x+x^2}} \] Input:

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

Output:

(-2*Sqrt[5 - 2*x]*(1137628 + 1826908*x + 790769*x^2 + 137057*x^3 + 45558*x 
^4 + 11250*x^5 + 1260*x^6) - 3042777*(5 - 2*x)^2*Sqrt[(1 + x)/(-5 + 2*x)]* 
Sqrt[(2 + x)/(-5 + 2*x)]*EllipticE[ArcSin[3/Sqrt[5 - 2*x]], 7/9] + 348666* 
(5 - 2*x)^2*Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[(2 + x)/(-5 + 2*x)]*EllipticF[Ar 
cSin[3/Sqrt[5 - 2*x]], 7/9])/(420*(-5 + 2*x)*Sqrt[2 + 3*x + x^2])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.24, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1273, 2184, 27, 2184, 27, 2184, 27, 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.

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

\(\Big \downarrow \) 1273

\(\displaystyle \frac {1}{18} \int \frac {(3 x+4)^2 \left (186 x^2+547 x+368\right )}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {1}{18} \left (\frac {1}{56} \int \frac {2 \left (410148 x^3+669198 x^2+32002 x+60239\right )}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \int \frac {410148 x^3+669198 x^2+32002 x+60239}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \left (\frac {205074}{5} (5-2 x)^{3/2} \sqrt {x^2+3 x+2}-\frac {1}{20} \int \frac {8 \left (-2800902 x^2-3976411 x+618430\right )}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx\right )-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \left (\frac {205074}{5} (5-2 x)^{3/2} \sqrt {x^2+3 x+2}-\frac {2}{5} \int \frac {-2800902 x^2-3976411 x+618430}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx\right )-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \left (\frac {205074}{5} (5-2 x)^{3/2} \sqrt {x^2+3 x+2}-\frac {2}{5} \left (\frac {1}{6} \int -\frac {18 (1014259 x+1505519)}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx+933634 \sqrt {5-2 x} \sqrt {x^2+3 x+2}\right )\right )-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \left (\frac {205074}{5} (5-2 x)^{3/2} \sqrt {x^2+3 x+2}-\frac {2}{5} \left (933634 \sqrt {5-2 x} \sqrt {x^2+3 x+2}-3 \int \frac {1014259 x+1505519}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx\right )\right )-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \left (\frac {205074}{5} (5-2 x)^{3/2} \sqrt {x^2+3 x+2}-\frac {2}{5} \left (933634 \sqrt {5-2 x} \sqrt {x^2+3 x+2}-3 \left (\frac {8082333}{2} \int \frac {1}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {1014259}{2} \int \frac {\sqrt {5-2 x}}{\sqrt {x^2+3 x+2}}dx\right )\right )\right )-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \left (\frac {205074}{5} (5-2 x)^{3/2} \sqrt {x^2+3 x+2}-\frac {2}{5} \left (933634 \sqrt {5-2 x} \sqrt {x^2+3 x+2}-3 \left (\frac {2694111 \sqrt {-x^2-3 x-2} \int \frac {3}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {3042777 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{3 \sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )\right )\right )-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \left (\frac {205074}{5} (5-2 x)^{3/2} \sqrt {x^2+3 x+2}-\frac {2}{5} \left (933634 \sqrt {5-2 x} \sqrt {x^2+3 x+2}-3 \left (\frac {8082333 \sqrt {-x^2-3 x-2} \int \frac {1}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {1014259 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )\right )\right )-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \left (\frac {205074}{5} (5-2 x)^{3/2} \sqrt {x^2+3 x+2}-\frac {2}{5} \left (933634 \sqrt {5-2 x} \sqrt {x^2+3 x+2}-3 \left (\frac {2694111 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {1014259 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )\right )\right )-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{18} \left (\frac {1}{28} \left (\frac {205074}{5} (5-2 x)^{3/2} \sqrt {x^2+3 x+2}-\frac {2}{5} \left (933634 \sqrt {5-2 x} \sqrt {x^2+3 x+2}-3 \left (\frac {2694111 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {3042777 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )\right )\right )-\frac {837}{14} (5-2 x)^{5/2} \sqrt {x^2+3 x+2}\right )-\frac {1}{9} \sqrt {5-2 x} (3 x+4)^3 \sqrt {x^2+3 x+2}\)

Input:

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

Output:

-1/9*(Sqrt[5 - 2*x]*(4 + 3*x)^3*Sqrt[2 + 3*x + x^2]) + ((-837*(5 - 2*x)^(5 
/2)*Sqrt[2 + 3*x + x^2])/14 + ((205074*(5 - 2*x)^(3/2)*Sqrt[2 + 3*x + x^2] 
)/5 - (2*(933634*Sqrt[5 - 2*x]*Sqrt[2 + 3*x + x^2] - 3*((-3042777*Sqrt[-2 
- 3*x - x^2]*EllipticE[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2] + (2 
694111*Sqrt[-2 - 3*x - x^2]*EllipticF[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 
3*x + x^2])))/5)/28)/18
 

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 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]
 

rule 1273

Int[(((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2])/Sq 
rt[(f_.) + (g_.)*(x_)], x_Symbol] :> Simp[2*(d + e*x)^m*Sqrt[f + g*x]*(Sqrt 
[a + b*x + c*x^2]/(g*(2*m + 3))), x] - Simp[1/(g*(2*m + 3))   Int[((d + e*x 
)^(m - 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*d*f + 2*a*(e*f*m - 
d*g*(m + 1)) + (2*c*d*f - 2*a*e*g + b*(e*f - d*g)*(2*m + 1))*x - (b*e*g + 2 
*c*(d*g*m - e*f*(m + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x 
] && IntegerQ[2*m] && GtQ[m, 0]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\sqrt {x^{2}+3 x +2}\, \sqrt {5-2 x}\, \left (-5040 x^{6}-45000 x^{5}+140360 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+1014259 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )-182232 x^{4}-548228 x^{3}+893960 x^{2}+4863476 x +3563560\right )}{1680 x^{3}+840 x^{2}-9240 x -8400}\)	\(141\)
risch	\(\frac {\left (630 x^{3}+5310 x^{2}+23589 x +89089\right ) \left (-5+2 x \right ) \sqrt {x^{2}+3 x +2}\, \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{210 \sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \sqrt {5-2 x}}+\frac {\left (-\frac {1505519 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{8820 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {1014259 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \left (\frac {9 \operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2}-2 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )\right )}{8820 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right ) \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(239\)
elliptic	\(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (-\frac {7863 x \sqrt {-2 x^{3}-x^{2}+11 x +10}}{70}-\frac {12727 \sqrt {-2 x^{3}-x^{2}+11 x +10}}{30}-\frac {1505519 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2940 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {1014259 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{2940 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {177 x^{2} \sqrt {-2 x^{3}-x^{2}+11 x +10}}{7}-3 x^{3} \sqrt {-2 x^{3}-x^{2}+11 x +10}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(250\)

Input:

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

Output:

1/840*(x^2+3*x+2)^(1/2)*(5-2*x)^(1/2)*(-5040*x^6-45000*x^5+140360*(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 
))+1014259*(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))-182232*x^4-548228*x^3+893960*x^2+4863476*x+3563560)/ 
(2*x^3+x^2-11*x-10)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.36 \[ \int \frac {(4+3 x)^3 \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx=-\frac {1}{210} \, {\left (630 \, x^{3} + 5310 \, x^{2} + 23589 \, x + 89089\right )} \sqrt {x^{2} + 3 \, x + 2} \sqrt {-2 \, x + 5} - \frac {1603771}{504} \, \sqrt {-2} {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right ) + \frac {1014259}{420} \, \sqrt {-2} {\rm weierstrassZeta}\left (\frac {67}{3}, \frac {440}{27}, {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right )\right ) \] Input:

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

Output:

-1/210*(630*x^3 + 5310*x^2 + 23589*x + 89089)*sqrt(x^2 + 3*x + 2)*sqrt(-2* 
x + 5) - 1603771/504*sqrt(-2)*weierstrassPInverse(67/3, 440/27, x + 1/6) + 
 1014259/420*sqrt(-2)*weierstrassZeta(67/3, 440/27, weierstrassPInverse(67 
/3, 440/27, x + 1/6))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(4+3 x)^3 \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx=-3 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{3}-\frac {177 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{2}}{7}-\frac {7863 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x}{70}-\frac {397479 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{140}+\frac {1014259 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{140}-\frac {4722629 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{280} \] Input:

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

Output:

( - 840*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x**3 - 7080*sqrt( - 2*x + 5) 
*sqrt(x**2 + 3*x + 2)*x**2 - 31452*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x 
 - 794958*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2) + 2028518*int((sqrt( - 2*x 
 + 5)*sqrt(x**2 + 3*x + 2)*x**2)/(2*x**3 + x**2 - 11*x - 10),x) - 4722629* 
int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2))/(2*x**3 + x**2 - 11*x - 10),x) 
)/280