Integrand size = 29, antiderivative size = 288 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {16503475 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{269374248 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {142149125 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{1885619736 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
430/969*(3+2*x)^(3/2)*(3*x^2+5*x+2)^(7/2)-2/57*(3+2*x)^(5/2)*(3*x^2+5*x+2) ^(7/2)-125/52378326*(64006+79583*x)*(3*x^2+5*x+2)^(3/2)*(3+2*x)^(1/2)+25/1 247103*(72737+86493*x)*(3*x^2+5*x+2)^(5/2)*(3+2*x)^(1/2)+2350/2907*(3*x^2+ 5*x+2)^(7/2)*(3+2*x)^(1/2)-16503475/808122744*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)+14214912 5/5656859208*EllipticF(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)+25/942809868*(749099+216603*x)*(3+2*x)^(1 /2)*(3*x^2+5*x+2)^(1/2)
Time = 31.31 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.79 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {2 \sqrt {3+2 x} \left (-341519551612-3595384785664 x-16735272462363 x^2-45255052994607 x^3-78460508136978 x^4-90580760151282 x^5-69684837178068 x^6-34294970344572 x^7-9445976815968 x^8-694795413312 x^9+311460012864 x^{10}+64309557312 x^{11}\right )+115524325 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-30234850 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{5656859208 (3+2 x) \sqrt {2+5 x+3 x^2}} \]
-1/5656859208*(2*Sqrt[3 + 2*x]*(-341519551612 - 3595384785664*x - 16735272 462363*x^2 - 45255052994607*x^3 - 78460508136978*x^4 - 90580760151282*x^5 - 69684837178068*x^6 - 34294970344572*x^7 - 9445976815968*x^8 - 6947954133 12*x^9 + 311460012864*x^10 + 64309557312*x^11) + 115524325*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sq rt[5/3]/Sqrt[3 + 2*x]], 3/5] - 30234850*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])
Time = 0.54 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {1236, 27, 1236, 27, 1236, 27, 1231, 1231, 27, 1231, 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.
| \(\displaystyle \int (5-x) (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{57} \int \frac {5}{2} (2 x+3)^{3/2} (129 x+196) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{57} \int (2 x+3)^{3/2} (129 x+196) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {5}{57} \left (\frac {2}{51} \int \frac {45}{2} \sqrt {2 x+3} (235 x+331) \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \int \sqrt {2 x+3} (235 x+331) \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {2}{45} \int \frac {5 (2621 x+3814) \left (3 x^2+5 x+2\right )^{5/2}}{2 \sqrt {2 x+3}}dx+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \int \frac {(2621 x+3814) \left (3 x^2+5 x+2\right )^{5/2}}{\sqrt {2 x+3}}dx+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \left (\frac {1}{429} \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \int \frac {(34107 x+47863) \left (3 x^2+5 x+2\right )^{3/2}}{\sqrt {2 x+3}}dx\right )+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \left (\frac {1}{429} \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{21} \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{126} \int \frac {3 (24067 x+70798) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx\right )\right )+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \left (\frac {1}{429} \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{21} \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{42} \int \frac {(24067 x+70798) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx\right )\right )+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \left (\frac {1}{429} \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \int \frac {4620973 x+4088477}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} \sqrt {2 x+3} (216603 x+749099) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \left (\frac {1}{429} \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4620973}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {5685965}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{45} \sqrt {2 x+3} (216603 x+749099) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \left (\frac {1}{429} \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4620973 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {5685965 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (216603 x+749099) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \left (\frac {1}{429} \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4620973 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {5685965 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (216603 x+749099) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \left (\frac {1}{429} \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4620973 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {5685965 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (216603 x+749099) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {5}{57} \left (\frac {15}{17} \left (\frac {1}{9} \left (\frac {1}{429} \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {4620973 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {5685965 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (216603 x+749099) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {94}{9} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {86}{17} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
(-2*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(7/2))/57 + (5*((86*(3 + 2*x)^(3/2)* (2 + 5*x + 3*x^2)^(7/2))/17 + (15*((94*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(7/ 2))/9 + ((Sqrt[3 + 2*x]*(72737 + 86493*x)*(2 + 5*x + 3*x^2)^(5/2))/429 - ( 5*((Sqrt[3 + 2*x]*(64006 + 79583*x)*(2 + 5*x + 3*x^2)^(3/2))/21 + (-1/45*( Sqrt[3 + 2*x]*(749099 + 216603*x)*Sqrt[2 + 5*x + 3*x^2]) + ((4620973*Sqrt[ -2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*S qrt[2 + 5*x + 3*x^2]) - (5685965*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[S qrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]))/90)/42))/858) /9))/17))/57
3.26.94.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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])
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])
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]
Time = 0.41 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(-\frac {\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}\, \left (385857343872 x^{11}+1868760077184 x^{10}-4168772479872 x^{9}-56675860895808 x^{8}-205769822067432 x^{7}-418109023068408 x^{6}-543484560907692 x^{5}+7987440 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-23104865 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-470763048821868 x^{4}-271530317967642 x^{3}-100413714212028 x^{2}-21575774443734 x -2050503601572\right )}{16970577624 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(176\) |
| risch | \(-\frac {\left (3572753184 x^{8}+5989615632 x^{7}-68880579768 x^{6}-329194523196 x^{5}-650694586500 x^{4}-699517082754 x^{3}-427399643682 x^{2}-139652898507 x -18986144459\right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{942809868}-\frac {\left (\frac {1075915 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{297729432 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {3300695 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {E\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{808122744 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(233\) |
| elliptic | \(\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {72 x^{8} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{19}-\frac {108 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{7}}{17}+\frac {1242 x^{6} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{17}+\frac {4398409 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{5}}{12597}+\frac {5033375 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{7293}+\frac {97398647 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{131274}+\frac {2158584059 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{4761666}+\frac {816683617 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5513508}+\frac {58780633 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2918916}+\frac {1075915 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{297729432 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {3300695 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{808122744 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{6 x^{3}+19 x^{2}+19 x +6}\) | \(380\) |
-1/16970577624*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(385857343872*x^11+186876 0077184*x^10-4168772479872*x^9-56675860895808*x^8-205769822067432*x^7-4181 09023068408*x^6-543484560907692*x^5+7987440*(-20-30*x)^(1/2)*(3+3*x)^(1/2) *15^(1/2)*(3+2*x)^(1/2)*EllipticF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))-23104 865*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(3+2*x)^(1/2)*EllipticE(1/5*(- 20-30*x)^(1/2),1/2*10^(1/2))-470763048821868*x^4-271530317967642*x^3-10041 3714212028*x^2-21575774443734*x-2050503601572)/(6*x^3+19*x^2+19*x+6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.30 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{942809868} \, {\left (3572753184 \, x^{8} + 5989615632 \, x^{7} - 68880579768 \, x^{6} - 329194523196 \, x^{5} - 650694586500 \, x^{4} - 699517082754 \, x^{3} - 427399643682 \, x^{2} - 139652898507 \, x - 18986144459\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {18691975}{5359129776} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {16503475}{808122744} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
-1/942809868*(3572753184*x^8 + 5989615632*x^7 - 68880579768*x^6 - 32919452 3196*x^5 - 650694586500*x^4 - 699517082754*x^3 - 427399643682*x^2 - 139652 898507*x - 18986144459)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3) + 18691975/535 9129776*sqrt(6)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 16503475/ 808122744*sqrt(6)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/2 7, -28/729, x + 19/18))
\[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=- \int \left (- 180 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1104 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2717 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3381 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2151 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 551 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 48 x^{6} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 36 x^{7} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
-Integral(-180*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1104*x *sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-2717*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3381*x**3*sqrt(2*x + 3)*sqrt(3* x**2 + 5*x + 2), x) - Integral(-2151*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-551*x**5*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - I ntegral(48*x**6*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(36*x** 7*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x)
\[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}} {\left (x - 5\right )} \,d x } \]
\[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}} {\left (x - 5\right )} \,d x } \]
Timed out. \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\int {\left (2\,x+3\right )}^{5/2}\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]