\(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{\sqrt {3+2 x}} \, dx\) [1054]

Optimal result

Integrand size = 29, antiderivative size = 207 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt {3+2 x}} \, dx=\frac {(34372-676791 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{324324}-\frac {5 \sqrt {3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac {1}{429} (224-33 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {651617 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{92664 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {5983645 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{648648 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:

1/324324*(34372-676791*x)*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)-5/18018*(3+2*x 
)^(1/2)*(563+4669*x)*(3*x^2+5*x+2)^(3/2)+1/429*(224-33*x)*(3+2*x)^(1/2)*(3 
*x^2+5*x+2)^(5/2)-651617/277992*(-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)+5983645/1945944*(-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 = 31.42 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.03 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt {3+2 x}} \, dx=-\frac {2 \sqrt {3+2 x} \left (-1864706-39284147 x-170798082 x^2-349849791 x^3-406647648 x^4-268524558 x^5-83553120 x^6-1163484 x^7+4041576 x^8\right )+4561319 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-971132 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{1945944 (3+2 x) \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

-1/1945944*(2*Sqrt[3 + 2*x]*(-1864706 - 39284147*x - 170798082*x^2 - 34984 
9791*x^3 - 406647648*x^4 - 268524558*x^5 - 83553120*x^6 - 1163484*x^7 + 40 
41576*x^8) + 4561319*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 
 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 971132* 
Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*Elli 
pticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 + 5*x + 3*x 
^2])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {1231, 1231, 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.

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \int \frac {(2001 x+1744) \left (3 x^2+5 x+2\right )^{3/2}}{\sqrt {2 x+3}}dx\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{21} \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{126} \int -\frac {3 (75199 x+62881) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \int \frac {(75199 x+62881) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx+\frac {1}{21} \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}\right )\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (-\frac {1}{90} \int -\frac {4561319 x+3850156}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (34372-676791 x)\right )+\frac {1}{21} \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \int \frac {4561319 x+3850156}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} (34372-676791 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}\right )\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4561319}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {5983645}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{45} (34372-676791 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}\right )\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4561319 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {5983645 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (34372-676791 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4561319 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {5983645 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (34372-676791 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}\right )\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4561319 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {5983645 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (34372-676791 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}\right )\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4561319 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {5983645 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (34372-676791 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}\right )\)

Input:

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

Output:

((224 - 33*x)*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(5/2))/429 - (5*((Sqrt[3 + 2 
*x]*(563 + 4669*x)*(2 + 5*x + 3*x^2)^(3/2))/21 + (-1/45*((34372 - 676791*x 
)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2]) + ((4561319*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]) - (5983645*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x] 
], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]))/90)/42))/858
 

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 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 = 1.79 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.78

method	result	size
default	\(-\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}\, \left (121247280 x^{8}-34904520 x^{7}-2506593600 x^{6}-8055736740 x^{5}+2133489 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-4561319 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-12199429440 x^{4}-10495493730 x^{3}-5534461170 x^{2}-1862722260 x -329620320\right )}{29189160 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\)	\(161\)
risch	\(-\frac {\left (224532 x^{5}-775656 x^{4}-2896614 x^{3}-3513708 x^{2}-1516527 x -610408\right ) \sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}}{324324}-\frac {\left (\frac {962539 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{2432430 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {651617 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{1389960 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(218\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {9 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{5}}{13}+\frac {342 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{143}+\frac {7663 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{858}+\frac {8873 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{819}+\frac {168503 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{36036}+\frac {152602 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{81081}+\frac {962539 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{2432430 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {651617 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{1389960 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(297\)

Input:

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

Output:

-1/29189160*(3*x^2+5*x+2)^(1/2)*(2*x+3)^(1/2)*(121247280*x^8-34904520*x^7- 
2506593600*x^6-8055736740*x^5+2133489*(-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))-4561319*(-3 
0*x-20)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(2*x+3)^(1/2)*EllipticE(1/5*(-30*x-20 
)^(1/2),1/2*10^(1/2))-12199429440*x^4-10495493730*x^3-5534461170*x^2-18627 
22260*x-329620320)/(6*x^3+19*x^2+19*x+6)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.35 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt {3+2 x}} \, dx=-\frac {1}{324324} \, {\left (224532 \, x^{5} - 775656 \, x^{4} - 2896614 \, x^{3} - 3513708 \, x^{2} - 1516527 \, x - 610408\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {17362253}{35026992} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {651617}{277992} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \] Input:

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

Output:

-1/324324*(224532*x^5 - 775656*x^4 - 2896614*x^3 - 3513708*x^2 - 1516527*x 
 - 610408)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3) + 17362253/35026992*sqrt(6) 
*weierstrassPInverse(19/27, -28/729, x + 19/18) + 651617/277992*sqrt(6)*we 
ierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/1 
8))
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt {3+2 x}} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\, dx \] Input:

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

Output:

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(-96*x*sq 
rt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(-165*x**2*sqrt(3*x**2 + 
5*x + 2)/sqrt(2*x + 3), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/sqr 
t(2*x + 3), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x 
) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt {3+2 x}} \, dx=-\frac {9 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{5}}{13}+\frac {342 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{4}}{143}+\frac {7663 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{3}}{858}+\frac {8873 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{819}+\frac {168503 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x}{36036}+\frac {690155 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{456456}+\frac {651617 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{195624}-\frac {271 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{112} \] Input:

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

Output:

( - 1896048*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**5 + 6549984*sqrt(2*x + 
 3)*sqrt(3*x**2 + 5*x + 2)*x**4 + 24460296*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x 
 + 2)*x**3 + 29671312*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2 + 12806228 
*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x + 4140930*sqrt(2*x + 3)*sqrt(3*x** 
2 + 5*x + 2) + 9122638*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(6* 
x**3 + 19*x**2 + 19*x + 6),x) - 6626763*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5 
*x + 2))/(6*x**3 + 19*x**2 + 19*x + 6),x))/2738736