\(\int (5-x) (3+2 x)^{3/2} (2+5 x+3 x^2)^{3/2} \, dx\) [1042]

Optimal result

Integrand size = 29, antiderivative size = 229 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {332459 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{2432430}-\frac {31487 (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}}{270270}-\frac {(3+2 x)^{5/2} (20395+40677 x) \sqrt {2+5 x+3 x^2}}{54054}+\frac {1}{143} (72-11 x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {152657 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{694980 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {332459 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{972972 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:

-332459/2432430*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)-31487/270270*(3+2*x)^(3/ 
2)*(3*x^2+5*x+2)^(1/2)-1/54054*(3+2*x)^(5/2)*(20395+40677*x)*(3*x^2+5*x+2) 
^(1/2)+1/143*(72-11*x)*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(3/2)-152657/2084940*(- 
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)+332459/2918916*(-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 = 31.43 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.93 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {2 \sqrt {3+2 x} \left (-224705588-1668494576 x-5141306625 x^2-8470029969 x^3-7944858702 x^4-4079217510 x^5-895236300 x^6+52050600 x^7+40415760 x^8\right )+1068599 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-71222 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{14594580 (3+2 x) \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

-1/14594580*(2*Sqrt[3 + 2*x]*(-224705588 - 1668494576*x - 5141306625*x^2 - 
 8470029969*x^3 - 7944858702*x^4 - 4079217510*x^5 - 895236300*x^6 + 520506 
00*x^7 + 40415760*x^8) + 1068599*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] - 71222*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 
 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 
+ 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {1236, 27, 1236, 27, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{39} \int \frac {1}{2} \sqrt {2 x+3} (443 x+672) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{39} \int \sqrt {2 x+3} (443 x+672) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{39} \left (\frac {2}{33} \int \frac {(21759 x+31531) \left (3 x^2+5 x+2\right )^{3/2}}{2 \sqrt {2 x+3}}dx+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \int \frac {(21759 x+31531) \left (3 x^2+5 x+2\right )^{3/2}}{\sqrt {2 x+3}}dx+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{21} \sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{126} \int \frac {3 (87079 x+107926) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx\right )+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{21} \sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{42} \int \frac {(87079 x+107926) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx\right )+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{42} \left (\frac {1}{90} \int -\frac {1068599 x+771751}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} \sqrt {2 x+3} (783711 x+486863) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{42} \left (-\frac {1}{90} \int \frac {1068599 x+771751}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (783711 x+486863)\right )+\frac {1}{21} \sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {1662295}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1068599}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{45} \sqrt {2 x+3} (783711 x+486863) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {1662295 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {1068599 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (783711 x+486863) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {1662295 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {1068599 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (783711 x+486863) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {1662295 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {1068599 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (783711 x+486863) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{39} \left (\frac {1}{33} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {1662295 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {1068599 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (783711 x+486863) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {886}{33} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\)

Input:

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

Output:

(-2*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(5/2))/39 + ((886*Sqrt[3 + 2*x]*(2 + 
 5*x + 3*x^2)^(5/2))/33 + ((Sqrt[3 + 2*x]*(43822 + 50771*x)*(2 + 5*x + 3*x 
^2)^(3/2))/21 + (-1/45*(Sqrt[3 + 2*x]*(486863 + 783711*x)*Sqrt[2 + 5*x + 3 
*x^2]) + ((-1068599*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]) + (1662295*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)/33)/39
 

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 = 1.81 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (1212472800 x^{8}[phone] x^{7}-26857089000 x^{6}-122376525300 x^{5}+890544 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-1068599 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-238345761060 x^{4}-254100899070 x^{3}-154335372660 x^{2}-50215127130 x -6805283580\right )}{218918700 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\)	\(161\)
risch	\(-\frac {\left (2245320 x^{5}-4218480 x^{4}-43487010 x^{3}-77801130 x^{2}-53083449 x -12602377\right ) \sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}}{2432430}-\frac {\left (\frac {771751 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{72972900 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {152657 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{10424700 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(218\)
elliptic	\(\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}\, \sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {12 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{5}}{13}+\frac {248 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{143}+\frac {23009 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1287}+\frac {78587 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2457}+\frac {5898161 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{270270}+\frac {12602377 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2432430}+\frac {771751 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{72972900 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {152657 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{10424700 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{6 x^{3}+19 x^{2}+19 x +6}\)	\(314\)

Input:

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

Output:

-1/218918700*(2*x+3)^(1/2)*(3*x^2+5*x+2)^(1/2)*(1212472800*x^8+1561518000* 
x^7-26857089000*x^6-122376525300*x^5+890544*(-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))-10685 
99*(-30*x-20)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(2*x+3)^(1/2)*EllipticE(1/5*(-3 
0*x-20)^(1/2),1/2*10^(1/2))-238345761060*x^4-254100899070*x^3-154335372660 
*x^2-50215127130*x-6805283580)/(6*x^3+19*x^2+19*x+6)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.31 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{2432430} \, {\left (2245320 \, x^{5} - 4218480 \, x^{4} - 43487010 \, x^{3} - 77801130 \, x^{2} - 53083449 \, x - 12602377\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {6411863}{262702440} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {152657}{2084940} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \] Input:

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

Output:

-1/2432430*(2245320*x^5 - 4218480*x^4 - 43487010*x^3 - 77801130*x^2 - 5308 
3449*x - 12602377)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3) + 6411863/262702440 
*sqrt(6)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 152657/2084940*s 
qrt(6)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, 
 x + 19/18))
 

Sympy [F]

\[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=- \int \left (- 30 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 89 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 76 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 11 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 6 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \] Input:

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

Output:

-Integral(-30*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-89*x*sq 
rt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-76*x**2*sqrt(2*x + 3)*s 
qrt(3*x**2 + 5*x + 2), x) - Integral(-11*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 
5*x + 2), x) - Integral(6*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {12 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{5}}{13}+\frac {248 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{4}}{143}+\frac {23009 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{3}}{1287}+\frac {78587 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{2457}+\frac {5898161 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x}{270270}+\frac {17697101 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{3423420}+\frac {152657 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{1467180}-\frac {41 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{840} \] Input:

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

Output:

( - 18960480*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**5 + 35622720*sqrt(2*x 
 + 3)*sqrt(3*x**2 + 5*x + 2)*x**4 + 367223640*sqrt(2*x + 3)*sqrt(3*x**2 + 
5*x + 2)*x**3 + 656987320*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2 + 4482 
60236*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x + 106182606*sqrt(2*x + 3)*sqr 
t(3*x**2 + 5*x + 2) + 2137198*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x* 
*2)/(6*x**3 + 19*x**2 + 19*x + 6),x) - 1002573*int((sqrt(2*x + 3)*sqrt(3*x 
**2 + 5*x + 2))/(6*x**3 + 19*x**2 + 19*x + 6),x))/20540520