\(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^{11/2}} \, dx\) [1049]

Optimal result

Integrand size = 29, antiderivative size = 202 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=-\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {23 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {7 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:

-23/11250*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2)-1/2250*(189+211*x)*(3*x^2+5*x+ 
2)^(1/2)/(3+2*x)^(5/2)+1/45*(44+51*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(9/2)+23 
/22500*(-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)+7/4500*(-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 = 31.52 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\frac {53980+373610 x+998860 x^2+1297210 x^3+822160 x^4+204180 x^5+23 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{11/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-44 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{11/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{22500 (3+2 x)^{9/2} \sqrt {2+5 x+3 x^2}} \] Input:

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(11/2),x]
 

Output:

(53980 + 373610*x + 998860*x^2 + 1297210*x^3 + 822160*x^4 + 204180*x^5 + 2 
3*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(11/2)*Sqrt[(2 + 3*x)/(3 + 2*x 
)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 44*Sqrt[5]*Sqrt[(1 + 
x)/(3 + 2*x)]*(3 + 2*x)^(11/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[ 
Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(22500*(3 + 2*x)^(9/2)*Sqrt[2 + 5*x + 3*x^ 
2])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 213, 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 = {1229, 27, 1229, 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.

\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{11/2}} \, dx\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac {1}{210} \int \frac {7 (4-3 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{7/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac {1}{30} \int \frac {(4-3 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{7/2}}dx\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{30} \left (\frac {1}{150} \int \frac {21 x+43}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{5/2}}\right )+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{30} \left (\frac {1}{150} \left (-\frac {2}{5} \int -\frac {3 (23 x+52)}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {46 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{5/2}}\right )+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{30} \left (\frac {1}{150} \left (\frac {3}{5} \int \frac {23 x+52}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {46 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{5/2}}\right )+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{30} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {35}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx+\frac {23}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx\right )-\frac {46 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{5/2}}\right )+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{30} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {35 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {23 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {46 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{5/2}}\right )+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{30} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {35 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}+\frac {23 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {46 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{5/2}}\right )+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{30} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {23 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}+\frac {35 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {46 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{5/2}}\right )+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{30} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {35 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {23 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {46 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{5/2}}\right )+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}\)

Input:

Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(11/2),x]
 

Output:

((44 + 51*x)*(2 + 5*x + 3*x^2)^(3/2))/(45*(3 + 2*x)^(9/2)) + (-1/75*((189 
+ 211*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^(5/2) + ((-46*Sqrt[2 + 5*x + 3*x 
^2])/(5*Sqrt[3 + 2*x]) + (3*((23*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[S 
qrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (35*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)/150)/30
 

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 = 1.76 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.46

method	result	size
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2304 \left (x +\frac {3}{2}\right )^{5}}+\frac {155 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1152 \left (x +\frac {3}{2}\right )^{4}}-\frac {971 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{4800 \left (x +\frac {3}{2}\right )^{3}}+\frac {3403 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{36000 \left (x +\frac {3}{2}\right )^{2}}-\frac {23 \left (6 x^{2}+10 x +4\right )}{22500 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {13 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{28125 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {23 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{112500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(294\)
default	\(-\frac {1392 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+368 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+8352 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}+2208 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}+18792 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+4968 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+18792 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+4968 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+33120 x^{6}+7047 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )+1863 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-2808780 x^{5}-11532000 x^{4}-18133350 x^{3}-13771950 x^{2}-5026620 x -697920}{337500 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {9}{2}}}\)	\(482\)

Input:

int((5-x)*(3*x^2+5*x+2)^(3/2)/(2*x+3)^(11/2),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)*(-65/2304* 
(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^5+155/1152*(6*x^3+19*x^2+19*x+6)^(1/2) 
/(x+3/2)^4-971/4800*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^3+3403/36000*(6*x^ 
3+19*x^2+19*x+6)^(1/2)/(x+3/2)^2-23/22500*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+1 
0*x+4))^(1/2)-13/28125*(-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))-23/11 
2500*(-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 = 0.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\frac {499 \, \sqrt {6} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 414 \, \sqrt {6} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 36 \, {\left (368 \, x^{4} - 31822 \, x^{3} - 75342 \, x^{2} - 54697 \, x - 11632\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{405000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2),x, algorithm="fricas")
 

Output:

1/405000*(499*sqrt(6)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243 
)*weierstrassPInverse(19/27, -28/729, x + 19/18) - 414*sqrt(6)*(32*x^5 + 2 
40*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*weierstrassZeta(19/27, -28/729, 
 weierstrassPInverse(19/27, -28/729, x + 19/18)) - 36*(368*x^4 - 31822*x^3 
 - 75342*x^2 - 54697*x - 11632)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(32*x 
^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\, dx \] Input:

integrate((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**(11/2),x)
 

Output:

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(32*x**5*sqrt(2*x + 3) + 240*x**4*sqr 
t(2*x + 3) + 720*x**3*sqrt(2*x + 3) + 1080*x**2*sqrt(2*x + 3) + 810*x*sqrt 
(2*x + 3) + 243*sqrt(2*x + 3)), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2) 
/(32*x**5*sqrt(2*x + 3) + 240*x**4*sqrt(2*x + 3) + 720*x**3*sqrt(2*x + 3) 
+ 1080*x**2*sqrt(2*x + 3) + 810*x*sqrt(2*x + 3) + 243*sqrt(2*x + 3)), x) - 
 Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(32*x**5*sqrt(2*x + 3) + 240*x** 
4*sqrt(2*x + 3) + 720*x**3*sqrt(2*x + 3) + 1080*x**2*sqrt(2*x + 3) + 810*x 
*sqrt(2*x + 3) + 243*sqrt(2*x + 3)), x) - Integral(3*x**3*sqrt(3*x**2 + 5* 
x + 2)/(32*x**5*sqrt(2*x + 3) + 240*x**4*sqrt(2*x + 3) + 720*x**3*sqrt(2*x 
 + 3) + 1080*x**2*sqrt(2*x + 3) + 810*x*sqrt(2*x + 3) + 243*sqrt(2*x + 3)) 
, x)
 

Maxima [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2),x, algorithm="maxima")
 

Output:

-integrate((3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(11/2), x)
 

Giac [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2),x, algorithm="giac")
 

Output:

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{11/2}} \,d x \] Input:

int(-((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^(11/2),x)
 

Output:

-int(((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^(11/2), x)
 

Reduce [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx =\text {Too large to display} \] Input:

int((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2),x)
 

Output:

(11160*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**3 + 26040*sqrt(2*x + 3)*sqr 
t(3*x**2 + 5*x + 2)*x**2 + 18724*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x + 
3694*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) - 29376*int((sqrt(2*x + 3)*sqrt( 
3*x**2 + 5*x + 2)*x**2)/(192*x**8 + 2048*x**7 + 9488*x**6 + 24912*x**5 + 4 
0500*x**4 + 41688*x**3 + 26487*x**2 + 9477*x + 1458),x)*x**5 - 220320*int( 
(sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(192*x**8 + 2048*x**7 + 9488*x 
**6 + 24912*x**5 + 40500*x**4 + 41688*x**3 + 26487*x**2 + 9477*x + 1458),x 
)*x**4 - 660960*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(192*x**8 
+ 2048*x**7 + 9488*x**6 + 24912*x**5 + 40500*x**4 + 41688*x**3 + 26487*x** 
2 + 9477*x + 1458),x)*x**3 - 991440*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 
 2)*x**2)/(192*x**8 + 2048*x**7 + 9488*x**6 + 24912*x**5 + 40500*x**4 + 41 
688*x**3 + 26487*x**2 + 9477*x + 1458),x)*x**2 - 743580*int((sqrt(2*x + 3) 
*sqrt(3*x**2 + 5*x + 2)*x**2)/(192*x**8 + 2048*x**7 + 9488*x**6 + 24912*x* 
*5 + 40500*x**4 + 41688*x**3 + 26487*x**2 + 9477*x + 1458),x)*x - 223074*i 
nt((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(192*x**8 + 2048*x**7 + 948 
8*x**6 + 24912*x**5 + 40500*x**4 + 41688*x**3 + 26487*x**2 + 9477*x + 1458 
),x) + 26976*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(192*x**8 + 2048*x 
**7 + 9488*x**6 + 24912*x**5 + 40500*x**4 + 41688*x**3 + 26487*x**2 + 9477 
*x + 1458),x)*x**5 + 202320*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(19 
2*x**8 + 2048*x**7 + 9488*x**6 + 24912*x**5 + 40500*x**4 + 41688*x**3 +...