Integrand size = 29, antiderivative size = 234 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {(287729-2667537 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{14594580}-\frac {\sqrt {3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac {\sqrt {3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac {2}{45} \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2742319 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{4169880 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {5021353 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{5837832 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
-1/162162*(15076+34643*x)*(3*x^2+5*x+2)^(3/2)*(3+2*x)^(1/2)+1/19305*(15467 +17193*x)*(3*x^2+5*x+2)^(5/2)*(3+2*x)^(1/2)-2/45*(3*x^2+5*x+2)^(7/2)*(3+2* x)^(1/2)-2742319/12509640*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)+5021353/17513496*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)+1/14594580*(287729-2667537*x)*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)
Time = 31.35 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.93 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {2 \sqrt {3+2 x} \left (-666434848-6298405666 x-25296672765 x^2-56607962679 x^3-76896556902 x^4-63978029658 x^5-30512259036 x^6-6333945660 x^7+468822816 x^8+315242928 x^9\right )+19196233 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-4132174 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{87567480 (3+2 x) \sqrt {2+5 x+3 x^2}} \]
-1/87567480*(2*Sqrt[3 + 2*x]*(-666434848 - 6298405666*x - 25296672765*x^2 - 56607962679*x^3 - 76896556902*x^4 - 63978029658*x^5 - 30512259036*x^6 - 6333945660*x^7 + 468822816*x^8 + 315242928*x^9) + 19196233*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] - 4132174*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 = 0.45 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {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.
| \(\displaystyle \int (5-x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{45} \int \frac {(521 x+784) \left (3 x^2+5 x+2\right )^{5/2}}{2 \sqrt {2 x+3}}dx-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \int \frac {(521 x+784) \left (3 x^2+5 x+2\right )^{5/2}}{\sqrt {2 x+3}}dx-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \int \frac {(14847 x+16573) \left (3 x^2+5 x+2\right )^{3/2}}{\sqrt {2 x+3}}dx\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{21} \sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{126} \int -\frac {3 (296393 x+237692) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx\right )\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \int \frac {(296393 x+237692) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx+\frac {1}{21} \sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (-\frac {1}{90} \int -\frac {19196233 x+16240967}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (287729-2667537 x)\right )+\frac {1}{21} \sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \int \frac {19196233 x+16240967}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} (287729-2667537 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {19196233}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {25106765}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{45} (287729-2667537 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {19196233 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {25106765 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (287729-2667537 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {19196233 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {25106765 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (287729-2667537 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {19196233 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {25106765 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (287729-2667537 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{45} \left (\frac {1}{429} \sqrt {2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{858} \left (\frac {1}{42} \left (\frac {1}{90} \left (\frac {19196233 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {25106765 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} (287729-2667537 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {1}{21} \sqrt {2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {2}{45} \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}\) |
(-2*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(7/2))/45 + ((Sqrt[3 + 2*x]*(15467 + 1 7193*x)*(2 + 5*x + 3*x^2)^(5/2))/429 - (5*((Sqrt[3 + 2*x]*(15076 + 34643*x )*(2 + 5*x + 3*x^2)^(3/2))/21 + (-1/45*((287729 - 2667537*x)*Sqrt[3 + 2*x] *Sqrt[2 + 5*x + 3*x^2]) + ((19196233*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcS in[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (2510676 5*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sq rt[3]*Sqrt[2 + 5*x + 3*x^2]))/90)/42))/858)/45
3.26.96.3.1 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 = 166, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(-\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \left (9457287840 x^{9}+14064684480 x^{8}-190018369800 x^{7}-915367771080 x^{6}-1919340889740 x^{5}+8865798 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-19196233 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-2306896707060 x^{4}-1698238880370 x^{3}-760627843920 x^{2}-191831604930 x -21144819420\right )}{1313512200 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(166\) |
| risch | \(-\frac {\left (17513496 x^{6}-29413692 x^{5}-314201916 x^{4}-624522906 x^{3}-552292686 x^{2}-231246315 x -39157073\right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{14594580}-\frac {\left (\frac {16240967 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{437837400 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {2742319 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {E\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{62548200 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(223\) |
| elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {6 x^{6} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5}+\frac {131 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{5}}{65}+\frac {15393 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{715}+\frac {1652177 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{38610}+\frac {2789357 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{73710}+\frac {5138807 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{324324}+\frac {39157073 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{14594580}+\frac {16240967 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{437837400 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {2742319 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{62548200 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(319\) |
-1/1313512200*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(9457287840*x^9+1406468448 0*x^8-190018369800*x^7-915367771080*x^6-1919340889740*x^5+8865798*(-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))-19196233*(-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))-2306896707060*x^4-16982 38880370*x^3-760627843920*x^2-191831604930*x-21144819420)/(6*x^3+19*x^2+19 *x+6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.33 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{14594580} \, {\left (17513496 \, x^{6} - 29413692 \, x^{5} - 314201916 \, x^{4} - 624522906 \, x^{3} - 552292686 \, x^{2} - 231246315 \, x - 39157073\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {72391021}{1576214640} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {2742319}{12509640} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
-1/14594580*(17513496*x^6 - 29413692*x^5 - 314201916*x^4 - 624522906*x^3 - 552292686*x^2 - 231246315*x - 39157073)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3) + 72391021/1576214640*sqrt(6)*weierstrassPInverse(19/27, -28/729, x + 1 9/18) + 2742319/12509640*sqrt(6)*weierstrassZeta(19/27, -28/729, weierstra ssPInverse(19/27, -28/729, x + 19/18))
\[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2} \, dx=- \int \left (- 20 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 96 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 165 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 113 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 15 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 9 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
-Integral(-20*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-96*x*sq rt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-165*x**2*sqrt(2*x + 3)* sqrt(3*x**2 + 5*x + 2), x) - Integral(-113*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-15*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x ) - Integral(9*x**5*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x)
\[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} \sqrt {2 \, x + 3} {\left (x - 5\right )} \,d x } \]
\[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} \sqrt {2 \, x + 3} {\left (x - 5\right )} \,d x } \]
Timed out. \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\int \sqrt {2\,x+3}\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]