Integrand size = 29, antiderivative size = 261 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {(12174838-22593339 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{744323580}-\frac {\sqrt {3+2 x} (949997+1332121 x) \left (2+5 x+3 x^2\right )^{3/2}}{8270262}+\frac {\sqrt {3+2 x} (1063774+1253571 x) \left (2+5 x+3 x^2\right )^{5/2}}{984555}+\frac {1166 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2295}-\frac {2}{51} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {34355693 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{212663880 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {62005241 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{297729432 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
-2/51*(3+2*x)^(3/2)*(3*x^2+5*x+2)^(7/2)-1/8270262*(949997+1332121*x)*(3*x^ 2+5*x+2)^(3/2)*(3+2*x)^(1/2)+1/984555*(1063774+1253571*x)*(3*x^2+5*x+2)^(5 /2)*(3+2*x)^(1/2)+1166/2295*(3*x^2+5*x+2)^(7/2)*(3+2*x)^(1/2)-34355693/637 991640*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)+62005241/893188296*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)+1/744323 580*(12174838-22593339*x)*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)
Time = 31.30 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.85 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {2 \sqrt {3+2 x} \left (-96409103246-956384897657 x-4138096653600 x^2-10225439632143 x^3-15827034250764 x^4-15763726088406 x^5-9891695193912 x^6-3544442250300 x^7-474654810168 x^8+90474720336 x^9+28371863520 x^{10}\right )+240489851 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-54474128 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{4465941480 (3+2 x) \sqrt {2+5 x+3 x^2}} \]
-1/4465941480*(2*Sqrt[3 + 2*x]*(-96409103246 - 956384897657*x - 4138096653 600*x^2 - 10225439632143*x^3 - 15827034250764*x^4 - 15763726088406*x^5 - 9 891695193912*x^6 - 3544442250300*x^7 - 474654810168*x^8 + 90474720336*x^9 + 28371863520*x^10) + 240489851*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 ] - 54474128*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 = 0.49 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {1236, 27, 1236, 27, 1231, 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.
| \(\displaystyle \int (5-x) (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{51} \int \frac {1}{2} \sqrt {2 x+3} (583 x+882) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{51} \int \sqrt {2 x+3} (583 x+882) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{51} \left (\frac {2}{45} \int \frac {(37987 x+55523) \left (3 x^2+5 x+2\right )^{5/2}}{2 \sqrt {2 x+3}}dx+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \int \frac {(37987 x+55523) \left (3 x^2+5 x+2\right )^{5/2}}{\sqrt {2 x+3}}dx+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \int \frac {(570909 x+760706) \left (3 x^2+5 x+2\right )^{3/2}}{\sqrt {2 x+3}}dx\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{21} \sqrt {2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{126} \int -\frac {3 (2510371 x+1363999) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx\right )\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \int \frac {(2510371 x+1363999) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx+\frac {1}{21} \sqrt {2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (-\frac {1}{90} \int -\frac {240489851 x+205721674}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (12174838-22593339 x)\right )+\frac {1}{21} \sqrt {2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \int \frac {240489851 x+205721674}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} (12174838-22593339 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {240489851}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {310026205}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{45} (12174838-22593339 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {240489851 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {310026205 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (12174838-22593339 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {240489851 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {310026205 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (12174838-22593339 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {240489851 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {310026205 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (12174838-22593339 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{51} \left (\frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {240489851 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {310026205 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (12174838-22593339 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1166}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}\) |
(-2*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(7/2))/51 + ((1166*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(7/2))/45 + ((Sqrt[3 + 2*x]*(1063774 + 1253571*x)*(2 + 5*x + 3*x^2)^(5/2))/429 - (5*((Sqrt[3 + 2*x]*(949997 + 1332121*x)*(2 + 5*x + 3 *x^2)^(3/2))/21 + (-1/45*((12174838 - 22593339*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5 *x + 3*x^2]) + ((240489851*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]) - (310026205*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqr t[2 + 5*x + 3*x^2]))/90)/42))/858)/45)/51
3.26.95.3.1 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 = 0.40 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.66
| method | result | size |
| default | \(-\frac {\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}\, \left (851155905600 x^{10}+2714241610080 x^{9}-14239644305040 x^{8}-106333267509000 x^{7}-296750855817360 x^{6}-472911782652180 x^{5}+104304531 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-240489851 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-474811027522920 x^{4}-306763188964290 x^{3}-124164543694590 x^{2}-28727620407360 x -2906702488440\right )}{66989122200 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(171\) |
| risch | \(-\frac {\left (1576214640 x^{7}+35026992 x^{6}-31471976844 x^{5}-98939331792 x^{4}-136604504862 x^{3}-98401767552 x^{2}-36153819495 x -5382782386\right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{744323580}-\frac {\left (\frac {102860837 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{11164853700 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {34355693 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {E\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{3189958200 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(228\) |
| elliptic | \(\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {36 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{7}}{17}-\frac {4 x^{6} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{85}+\frac {140167 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{5}}{3315}+\frac {4847116 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{36465}+\frac {361387579 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1969110}+\frac {248489312 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1879605}+\frac {803418211 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{16540524}+\frac {158317129 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{21891870}+\frac {102860837 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{11164853700 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {34355693 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{3189958200 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{6 x^{3}+19 x^{2}+19 x +6}\) | \(358\) |
-1/66989122200*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(851155905600*x^10+271424 1610080*x^9-14239644305040*x^8-106333267509000*x^7-296750855817360*x^6-472 911782652180*x^5+104304531*(-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))-240489851*(-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))-474811027522920*x^4-306763188964290*x^3-124164543694590*x^2-2 8727620407360*x-2906702488440)/(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 = 82, normalized size of antiderivative = 0.31 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{744323580} \, {\left (1576214640 \, x^{7} + 35026992 \, x^{6} - 31471976844 \, x^{5} - 98939331792 \, x^{4} - 136604504862 \, x^{3} - 98401767552 \, x^{2} - 36153819495 \, x - 5382782386\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {866317037}{80386946640} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {34355693}{637991640} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
-1/744323580*(1576214640*x^7 + 35026992*x^6 - 31471976844*x^5 - 9893933179 2*x^4 - 136604504862*x^3 - 98401767552*x^2 - 36153819495*x - 5382782386)*s qrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3) + 866317037/80386946640*sqrt(6)*weierst rassPInverse(19/27, -28/729, x + 19/18) + 34355693/637991640*sqrt(6)*weier strassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/18))
\[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=- \int \left (- 60 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 328 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 687 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 669 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 271 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 18 x^{6} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
-Integral(-60*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-328*x*s qrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-687*x**2*sqrt(2*x + 3) *sqrt(3*x**2 + 5*x + 2), x) - Integral(-669*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-271*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3*x**5*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral (18*x**6*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x)
\[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {3}{2}} {\left (x - 5\right )} \,d x } \]
\[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x + 3\right )}^{\frac {3}{2}} {\left (x - 5\right )} \,d x } \]
Timed out. \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\int {\left (2\,x+3\right )}^{3/2}\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]