\(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{13/2}} \, dx\) [1060]

Optimal result

Integrand size = 29, antiderivative size = 207 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx=\frac {(1301762+948443 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{3/2}}+\frac {(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac {(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac {107857 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{33000 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {198109 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{46200 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:

1/346500*(1301762+948443*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(3/2)+1/34650*(241 
61+18699*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(7/2)+1/99*(114+115*x)*(3*x^2+5*x+ 
2)^(5/2)/(3+2*x)^(11/2)-107857/99000*(-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)+198109/138600*(-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 = 31.55 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.10 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx=-\frac {-8 \left (2+5 x+3 x^2\right ) \left (111387702+387631385 x+544712540 x^2+387989550 x^3+140915480 x^4+21041468 x^5\right )+4 (3+2 x)^5 \left (1509998 \left (2+5 x+3 x^2\right )+754999 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-160672 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{2772000 (3+2 x)^{11/2} \sqrt {2+5 x+3 x^2}} \] Input:

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(13/2),x]
 

Output:

-1/2772000*(-8*(2 + 5*x + 3*x^2)*(111387702 + 387631385*x + 544712540*x^2 
+ 387989550*x^3 + 140915480*x^4 + 21041468*x^5) + 4*(3 + 2*x)^5*(1509998*( 
2 + 5*x + 3*x^2) + 754999*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] 
- 160672*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 
 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/((3 + 2*x)^(11/ 
2)*Sqrt[2 + 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.65 (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 = {1229, 1229, 25, 1229, 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.

\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^{13/2}} \, dx\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}-\frac {1}{198} \int \frac {(321 x+326) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{9/2}}dx\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{198} \left (\frac {3}{350} \int -\frac {(37437 x+31975) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{5/2}}dx+\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}\right )+\frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{198} \left (\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \int \frac {(37437 x+31975) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{5/2}}dx\right )+\frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{198} \left (\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (-\frac {1}{30} \int -\frac {3 (754999 x+637226)}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {\sqrt {3 x^2+5 x+2} (948443 x+1301762)}{15 (2 x+3)^{3/2}}\right )\right )+\frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{198} \left (\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \int \frac {754999 x+637226}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {(948443 x+1301762) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\right )+\frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{198} \left (\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {754999}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {990545}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(948443 x+1301762) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\right )+\frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{198} \left (\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {754999 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {990545 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {(948443 x+1301762) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\right )+\frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{198} \left (\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {754999 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {990545 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {(948443 x+1301762) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\right )+\frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{198} \left (\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {754999 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {990545 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {(948443 x+1301762) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\right )+\frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{198} \left (\frac {(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {754999 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {990545 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {(948443 x+1301762) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\right )+\frac {(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}\)

Input:

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(13/2),x]
 

Output:

((114 + 115*x)*(2 + 5*x + 3*x^2)^(5/2))/(99*(3 + 2*x)^(11/2)) + (((24161 + 
 18699*x)*(2 + 5*x + 3*x^2)^(3/2))/(175*(3 + 2*x)^(7/2)) - (3*(-1/15*((130 
1762 + 948443*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^(3/2) + ((754999*Sqrt[-2 
 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqr 
t[2 + 5*x + 3*x^2]) - (990545*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]))/10))/350)/198
 

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 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 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 = 1.94 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.54

method	result	size
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {325 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{22528 \left (x +\frac {3}{2}\right )^{6}}+\frac {12235 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{101376 \left (x +\frac {3}{2}\right )^{5}}-\frac {821 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2016 \left (x +\frac {3}{2}\right )^{4}}+\frac {53099 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{73920 \left (x +\frac {3}{2}\right )^{3}}-\frac {1689553 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2217600 \left (x +\frac {3}{2}\right )^{2}}+\frac {\frac {751481}{132000} x^{2}+\frac {751481}{79200} x +\frac {751481}{198000}}{\sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}+\frac {318613 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{1732500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {107857 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{495000 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(318\)
default	\(\frac {24159968 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-11306208 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}[phone] \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}-84796560 \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}[phone] \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}-254389680 \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}[phone] \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}-381584520 \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}[phone] x^{7}[phone] \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}-286188390 \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}[phone] x^{6}[phone] \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 )-85856517 \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 )[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{10395000 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {11}{2}}}\)	\(575\)

Input:

int((5-x)*(3*x^2+5*x+2)^(5/2)/(2*x+3)^(13/2),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)*(-325/2252 
8*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^6+12235/101376*(6*x^3+19*x^2+19*x+6) 
^(1/2)/(x+3/2)^5-821/2016*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^4+53099/7392 
0*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^3-1689553/2217600*(6*x^3+19*x^2+19*x 
+6)^(1/2)/(x+3/2)^2+751481/792000*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^ 
(1/2)+318613/1732500*(-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))+107857/ 
495000*(-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 = 166, normalized size of antiderivative = 0.80 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx=\frac {2874913 \, \sqrt {6} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 13589982 \, \sqrt {6} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (21041468 \, x^{5} + 140915480 \, x^{4} + 387989550 \, x^{3} + 544712540 \, x^{2} + 387631385 \, x + 111387702\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{12474000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(13/2),x, algorithm="fricas")
 

Output:

1/12474000*(2874913*sqrt(6)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860 
*x^2 + 2916*x + 729)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 1358 
9982*sqrt(6)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 
 729)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, 
x + 19/18)) + 36*(21041468*x^5 + 140915480*x^4 + 387989550*x^3 + 544712540 
*x^2 + 387631385*x + 111387702)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(64*x 
^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**(13/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {13}{2}}} \,d x } \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(13/2),x, algorithm="maxima")
 

Output:

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(13/2), x)
 

Giac [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {13}{2}}} \,d x } \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(13/2),x, algorithm="giac")
 

Output:

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(13/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{13/2}} \,d x \] Input:

int(-((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(13/2),x)
 

Output:

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(13/2), x)
 

Reduce [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx =\text {Too large to display} \] Input:

int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(13/2),x)
 

Output:

( - 123984*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**5 - 2231712*sqrt(2*x + 
3)*sqrt(3*x**2 + 5*x + 2)*x**4 - 10393992*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x 
+ 2)*x**3 - 21485968*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2 - 21394948* 
sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x - 8734914*sqrt(2*x + 3)*sqrt(3*x**2 
 + 5*x + 2) - 162195840*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(3 
84*x**9 + 4672*x**8 + 25120*x**7 + 78288*x**6 + 155736*x**5 + 204876*x**4 
+ 178038*x**3 + 98415*x**2 + 31347*x + 4374),x)*x**6 - 1459762560*int((sqr 
t(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(384*x**9 + 4672*x**8 + 25120*x**7 
 + 78288*x**6 + 155736*x**5 + 204876*x**4 + 178038*x**3 + 98415*x**2 + 313 
47*x + 4374),x)*x**5 - 5474109600*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2 
)*x**2)/(384*x**9 + 4672*x**8 + 25120*x**7 + 78288*x**6 + 155736*x**5 + 20 
4876*x**4 + 178038*x**3 + 98415*x**2 + 31347*x + 4374),x)*x**4 - 109482192 
00*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(384*x**9 + 4672*x**8 + 
 25120*x**7 + 78288*x**6 + 155736*x**5 + 204876*x**4 + 178038*x**3 + 98415 
*x**2 + 31347*x + 4374),x)*x**3 - 12316746600*int((sqrt(2*x + 3)*sqrt(3*x* 
*2 + 5*x + 2)*x**2)/(384*x**9 + 4672*x**8 + 25120*x**7 + 78288*x**6 + 1557 
36*x**5 + 204876*x**4 + 178038*x**3 + 98415*x**2 + 31347*x + 4374),x)*x**2 
 - 7390047960*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(384*x**9 + 
4672*x**8 + 25120*x**7 + 78288*x**6 + 155736*x**5 + 204876*x**4 + 178038*x 
**3 + 98415*x**2 + 31347*x + 4374),x)*x - 1847511990*int((sqrt(2*x + 3)...