3.27.4 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{15/2}} \, dx\) [2604]

3.27.4.1 Optimal result

Integrand size = 29, antiderivative size = 234 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {5083 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{165000 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {9421 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{231000 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

output

1/6930*(73-33*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(9/2)+1/11*(8+9*x)*(3*x^2+5*x 
+2)^(5/2)/(3+2*x)^(13/2)+5083/495000*EllipticE(3^(1/2)*(1+x)^(1/2),1/3*I*6 
^(1/2))*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)-9421/693000*Ellip 
ticF(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^ 
2+5*x+2)^(1/2)+1/346500*(21492+17833*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(5/2)- 
5083/247500*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2)
 

3.27.4.2 Mathematica [A] (verified)

Time = 31.42 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.99 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {8 \left (2+5 x+3 x^2\right ) \left (11865789+41339721 x+54318160 x^2+33648370 x^3+12953760 x^4+6409516 x^5+2277184 x^6\right )-4 (3+2 x)^6 \left (71162 \left (2+5 x+3 x^2\right )+35581 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-7318 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{13860000 (3+2 x)^{13/2} \sqrt {2+5 x+3 x^2}} \]

input

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]
 

output

-1/13860000*(8*(2 + 5*x + 3*x^2)*(11865789 + 41339721*x + 54318160*x^2 + 3 
3648370*x^3 + 12953760*x^4 + 6409516*x^5 + 2277184*x^6) - 4*(3 + 2*x)^6*(7 
1162*(2 + 5*x + 3*x^2) + 35581*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] - 7318*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)^( 
13/2)*Sqrt[2 + 5*x + 3*x^2])
 

3.27.4.3 Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 250, 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 = {1229, 27, 1229, 25, 1229, 25, 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.

\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^{15/2}} \, dx\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}-\frac {1}{286} \int \frac {13 (3 x+8) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{11/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}-\frac {1}{22} \int \frac {(3 x+8) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{11/2}}dx\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \int -\frac {(489 x+383) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{7/2}}dx+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{22} \left (\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}-\frac {1}{210} \int \frac {(489 x+383) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{7/2}}dx\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \left (\frac {1}{150} \int -\frac {28263 x+24604}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {\sqrt {3 x^2+5 x+2} (17833 x+21492)}{75 (2 x+3)^{5/2}}\right )+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \left (\frac {(17833 x+21492) \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^{5/2}}-\frac {1}{150} \int \frac {28263 x+24604}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \left (\frac {1}{150} \left (\frac {2}{5} \int \frac {3 (35581 x+29819)}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {71162 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (17833 x+21492)}{75 (2 x+3)^{5/2}}\right )+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \left (\frac {1}{150} \left (\frac {3}{5} \int \frac {35581 x+29819}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {71162 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (17833 x+21492)}{75 (2 x+3)^{5/2}}\right )+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {35581}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {47105}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {71162 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (17833 x+21492)}{75 (2 x+3)^{5/2}}\right )+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {35581 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {47105 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {71162 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (17833 x+21492)}{75 (2 x+3)^{5/2}}\right )+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {35581 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {47105 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {71162 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (17833 x+21492)}{75 (2 x+3)^{5/2}}\right )+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {35581 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {47105 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {71162 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (17833 x+21492)}{75 (2 x+3)^{5/2}}\right )+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{22} \left (\frac {1}{210} \left (\frac {1}{150} \left (\frac {3}{5} \left (\frac {35581 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {47105 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {71162 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (17833 x+21492)}{75 (2 x+3)^{5/2}}\right )+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{315 (2 x+3)^{9/2}}\right )+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}\)

input

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]
 

output

((8 + 9*x)*(2 + 5*x + 3*x^2)^(5/2))/(11*(3 + 2*x)^(13/2)) + (((73 - 33*x)* 
(2 + 5*x + 3*x^2)^(3/2))/(315*(3 + 2*x)^(9/2)) + (((21492 + 17833*x)*Sqrt[ 
2 + 5*x + 3*x^2])/(75*(3 + 2*x)^(5/2)) + ((-71162*Sqrt[2 + 5*x + 3*x^2])/( 
5*Sqrt[3 + 2*x]) + (3*((35581*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]) - (47105*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)/210)/22
 

3.27.4.3.1 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 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]
 

3.27.4.4 Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.46

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {25 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{4096 \left (x +\frac {3}{2}\right )^{7}}+\frac {1105 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{22528 \left (x +\frac {3}{2}\right )^{6}}-\frac {3929 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{25344 \left (x +\frac {3}{2}\right )^{5}}+\frac {487 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2016 \left (x +\frac {3}{2}\right )^{4}}-\frac {278249 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1478400 \left (x +\frac {3}{2}\right )^{3}}+\frac {704257 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{11088000 \left (x +\frac {3}{2}\right )^{2}}-\frac {5083 \left (6 x^{2}+10 x +4\right )}{495000 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {29819 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{17325000 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {5083 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{2475000 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(342\)
default	\(\frac {1106304 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x^{6} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-2277184 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{6} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+9956736 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-20-30 x}\, \sqrt {3+3 x}-20494656 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-20-30 x}\, \sqrt {3+3 x}[phone] \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-76854960 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}[phone] \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-153709920 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-204946560 x^{8}[phone] \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-172923660 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-918434040 x^{7}[phone] F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-103754196 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-2263896840 x^{6}[phone] \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-25938549 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-5355988260 x^{5}-10713115500 x^{4}-13887201090 x^{3}-10527968760 x^{2}-4260251610 x -711947340}{51975000 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {13}{2}}}\)	\(668\)

input

int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2),x,method=_RETURNVERBOSE)
 

output

((3+2*x)*(3*x^2+5*x+2))^(1/2)/(3+2*x)^(1/2)/(3*x^2+5*x+2)^(1/2)*(-25/4096* 
(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^7+1105/22528*(6*x^3+19*x^2+19*x+6)^(1/ 
2)/(x+3/2)^6-3929/25344*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^5+487/2016*(6* 
x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^4-278249/1478400*(6*x^3+19*x^2+19*x+6)^(1 
/2)/(x+3/2)^3+704257/11088000*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^2-5083/4 
95000*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)-29819/17325000*(-20-30 
*x)^(1/2)*(3+3*x)^(1/2)*(45+30*x)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*Ellipt 
icF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))-5083/2475000*(-20-30*x)^(1/2)*(3+3* 
x)^(1/2)*(45+30*x)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*(1/3*EllipticE(1/5*(- 
20-30*x)^(1/2),1/2*10^(1/2))-EllipticF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))) 
)
 

3.27.4.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.79 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {139297 \, \sqrt {6} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 640458 \, \sqrt {6} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (2277184 \, x^{6} + 6409516 \, x^{5} + 12953760 \, x^{4} + 33648370 \, x^{3} + 54318160 \, x^{2} + 41339721 \, x + 11865789\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{62370000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]

input

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2),x, algorithm="fricas")
 

output

-1/62370000*(139297*sqrt(6)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 2 
2680*x^3 + 20412*x^2 + 10206*x + 2187)*weierstrassPInverse(19/27, -28/729, 
 x + 19/18) + 640458*sqrt(6)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 
22680*x^3 + 20412*x^2 + 10206*x + 2187)*weierstrassZeta(19/27, -28/729, we 
ierstrassPInverse(19/27, -28/729, x + 19/18)) + 36*(2277184*x^6 + 6409516* 
x^5 + 12953760*x^4 + 33648370*x^3 + 54318160*x^2 + 41339721*x + 11865789)* 
sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(128*x^7 + 1344*x^6 + 6048*x^5 + 1512 
0*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)
 

3.27.4.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=\text {Timed out} \]

input

integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**(15/2),x)
 

output

Timed out
 

3.27.4.7 Maxima [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {15}{2}}} \,d x } \]

input

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2),x, algorithm="maxima")
 

output

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2), x)
 

3.27.4.8 Giac [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {15}{2}}} \,d x } \]

input

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2),x, algorithm="giac")
 

output

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2), x)
 

3.27.4.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{15/2}} \,d x \]

input

int(-((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(15/2),x)
 

output

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(15/2), x)