\(\int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx\) [1040]

Optimal result

Integrand size = 29, antiderivative size = 197 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx=\frac {183 \sqrt {2+5 x+3 x^2}}{875 (3+2 x)^{3/2}}+\frac {159 \sqrt {2+5 x+3 x^2}}{625 \sqrt {3+2 x}}+\frac {(46+139 x) \sqrt {2+5 x+3 x^2}}{175 (3+2 x)^{7/2}}-\frac {159 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1250 \sqrt {2+5 x+3 x^2}}+\frac {183 \sqrt {3} \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{1750 \sqrt {2+5 x+3 x^2}} \] Output:

183/875*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(3/2)+159/625*(3*x^2+5*x+2)^(1/2)/(3+2 
*x)^(1/2)+1/175*(46+139*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(7/2)-159/1250*(-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)+183/1750*(-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 = 31.46 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.10 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx=-\frac {-4 \left (2+5 x+3 x^2\right ) \left (39436+74557 x+43728 x^2+8904 x^3\right )+6 (3+2 x)^3 \left (742 \left (2+5 x+3 x^2\right )+371 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-188 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{17500 (3+2 x)^{7/2} \sqrt {2+5 x+3 x^2}} \] Input:

Integrate[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^(9/2),x]
 

Output:

-1/17500*(-4*(2 + 5*x + 3*x^2)*(39436 + 74557*x + 43728*x^2 + 8904*x^3) + 
6*(3 + 2*x)^3*(742*(2 + 5*x + 3*x^2) + 371*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)] 
*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt 
[3 + 2*x]], 3/5] - 188*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqr 
t[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/( 
(3 + 2*x)^(7/2)*Sqrt[2 + 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1229, 27, 1237, 27, 1237, 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.

\(\displaystyle \int \frac {(5-x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{9/2}} \, dx\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {(139 x+46) \sqrt {3 x^2+5 x+2}}{175 (2 x+3)^{7/2}}-\frac {1}{350} \int -\frac {3 (121 x+90)}{(2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{350} \int \frac {121 x+90}{(2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {3}{350} \left (\frac {122 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)^{3/2}}-\frac {2}{15} \int -\frac {3 (183 x+89)}{2 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{350} \left (\frac {1}{5} \int \frac {183 x+89}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {122 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {3}{350} \left (\frac {1}{5} \left (\frac {742 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {2}{5} \int \frac {3 (371 x+404)}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {122 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{350} \left (\frac {1}{5} \left (\frac {742 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \int \frac {371 x+404}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {122 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {3}{350} \left (\frac {1}{5} \left (\frac {742 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {371}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {305}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )\right )+\frac {122 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {3}{350} \left (\frac {1}{5} \left (\frac {742 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {371 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {305 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {122 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{350} \left (\frac {1}{5} \left (\frac {742 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {371 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {305 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )+\frac {122 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {3}{350} \left (\frac {1}{5} \left (\frac {742 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {371 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {305 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {122 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {3}{350} \left (\frac {1}{5} \left (\frac {742 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {371 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {305 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {122 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}\)

Input:

Int[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^(9/2),x]
 

Output:

((46 + 139*x)*Sqrt[2 + 5*x + 3*x^2])/(175*(3 + 2*x)^(7/2)) + (3*((122*Sqrt 
[2 + 5*x + 3*x^2])/(5*(3 + 2*x)^(3/2)) + ((742*Sqrt[2 + 5*x + 3*x^2])/(5*S 
qrt[3 + 2*x]) - (3*((371*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*S 
qrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (305*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])))/5)/5))/350
 

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 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[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 = 1.71 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.37

method	result	size
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {13 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{224 \left (x +\frac {3}{2}\right )^{4}}+\frac {139 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2800 \left (x +\frac {3}{2}\right )^{3}}+\frac {183 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{3500 \left (x +\frac {3}{2}\right )^{2}}+\frac {\frac {477}{625} x^{2}+\frac {159}{125} x +\frac {318}{625}}{\sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}+\frac {606 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{21875 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {159 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{6250 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(270\)
default	\(\frac {792 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}+2968 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}+3564 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+13356 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+5346 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+20034 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+2673 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )+10017 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )+267120 x^{5}+1757040 x^{4}+4601190 x^{3}+5785490 x^{2}+3462940 x +788720}{43750 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {7}{2}}}\)	\(389\)

Input:

int((5-x)*(3*x^2+5*x+2)^(1/2)/(2*x+3)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

((3*x^2+5*x+2)*(2*x+3))^(1/2)/(2*x+3)^(1/2)/(3*x^2+5*x+2)^(1/2)*(-13/224*( 
6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^4+139/2800*(6*x^3+19*x^2+19*x+6)^(1/2)/ 
(x+3/2)^3+183/3500*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^2+159/1250*(6*x^2+1 
0*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)+606/21875*(-30*x-20)^(1/2)*(3+3*x)^( 
1/2)*(30*x+45)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*EllipticF(1/5*(-30*x-20)^ 
(1/2),1/2*10^(1/2))+159/6250*(-30*x-20)^(1/2)*(3+3*x)^(1/2)*(30*x+45)^(1/2 
)/(6*x^3+19*x^2+19*x+6)^(1/2)*(1/3*EllipticE(1/5*(-30*x-20)^(1/2),1/2*10^( 
1/2))-EllipticF(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.64 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx=-\frac {223 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 6678 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 12 \, {\left (8904 \, x^{3} + 43728 \, x^{2} + 74557 \, x + 39436\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{52500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(9/2),x, algorithm="fricas")
 

Output:

-1/52500*(223*sqrt(6)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*weierstrass 
PInverse(19/27, -28/729, x + 19/18) - 6678*sqrt(6)*(16*x^4 + 96*x^3 + 216* 
x^2 + 216*x + 81)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/2 
7, -28/729, x + 19/18)) - 12*(8904*x^3 + 43728*x^2 + 74557*x + 39436)*sqrt 
(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)
 

Sympy [F]

\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\, dx \] Input:

integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**(9/2),x)
 

Output:

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**3*sqrt( 
2*x + 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2*x + 3) + 81*sqrt(2*x + 3) 
), x) - Integral(x*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**3 
*sqrt(2*x + 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2*x + 3) + 81*sqrt(2* 
x + 3)), x)
 

Maxima [F]

\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(9/2),x, algorithm="maxima")
 

Output:

-integrate(sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^(9/2), x)
 

Giac [F]

\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(9/2),x, algorithm="giac")
 

Output:

integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{{\left (2\,x+3\right )}^{9/2}} \,d x \] Input:

int(-((x - 5)*(5*x + 3*x^2 + 2)^(1/2))/(2*x + 3)^(9/2),x)
 

Output:

-int(((x - 5)*(5*x + 3*x^2 + 2)^(1/2))/(2*x + 3)^(9/2), x)
 

Reduce [F]

\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx=\frac {84 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x -226 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}-11232 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right ) x^{4}-67392 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right ) x^{3}-151632 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right ) x^{2}-151632 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right ) x -56862 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right )+8752 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right ) x^{4}+52512 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right ) x^{3}+118152 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right ) x^{2}+118152 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right ) x +44307 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{96 x^{7}+880 x^{6}+3424 x^{5}+7320 x^{4}+9270 x^{3}+6939 x^{2}+2835 x +486}d x \right )}{4032 x^{4}+24192 x^{3}+54432 x^{2}+54432 x +20412} \] Input:

int((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(9/2),x)
 

Output:

(84*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x - 226*sqrt(2*x + 3)*sqrt(3*x**2 
 + 5*x + 2) - 11232*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(96*x* 
*7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x + 4 
86),x)*x**4 - 67392*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(96*x* 
*7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x + 4 
86),x)*x**3 - 151632*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(96*x 
**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x + 
486),x)*x**2 - 151632*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(96* 
x**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x + 
 486),x)*x - 56862*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(96*x** 
7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x + 48 
6),x) + 8752*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(96*x**7 + 880*x** 
6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x + 486),x)*x**4 
+ 52512*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(96*x**7 + 880*x**6 + 3 
424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x + 486),x)*x**3 + 118 
152*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(96*x**7 + 880*x**6 + 3424* 
x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x + 486),x)*x**2 + 118152* 
int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(96*x**7 + 880*x**6 + 3424*x**5 
 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x + 486),x)*x + 44307*int((sqr 
t(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(96*x**7 + 880*x**6 + 3424*x**5 + 73...