Integrand size = 25, antiderivative size = 256 \[ \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 x^{11/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {1521056 \sqrt {x} (2+3 x)}{76545 \sqrt {2+5 x+3 x^2}}-\frac {4 x^{7/2} (1484+1685 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {211144 \sqrt {x} \sqrt {2+5 x+3 x^2}}{5103}-\frac {167336 x^{3/2} \sqrt {2+5 x+3 x^2}}{2835}+\frac {45820}{567} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {1521056 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{76545 \sqrt {2+5 x+3 x^2}}-\frac {211144 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{5103 \sqrt {2+5 x+3 x^2}} \]
2/9*x^(11/2)*(74+95*x)/(3*x^2+5*x+2)^(3/2)-4/27*x^(7/2)*(1484+1685*x)/(3*x ^2+5*x+2)^(1/2)-1521056/76545*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)+1521056/ 76545*(1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1 /2))*2^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)-211144/5103*(1+x)^( 3/2)*(1/(1+x))^(1/2)*EllipticF(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)* ((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)-167336/2835*x^(3/2)*(3*x^2+5*x+2 )^(1/2)+45820/567*x^(5/2)*(3*x^2+5*x+2)^(1/2)+211144/5103*x^(1/2)*(3*x^2+5 *x+2)^(1/2)
Result contains complex when optimal does not.
Time = 21.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.73 \[ \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {-2 \left (3042112+8876240 x+5504080 x^2-2967300 x^3-2106756 x^4+262710 x^5-70956 x^6+18225 x^7\right )-1521056 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-1646104 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{76545 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \]
(-2*(3042112 + 8876240*x + 5504080*x^2 - 2967300*x^3 - 2106756*x^4 + 26271 0*x^5 - 70956*x^6 + 18225*x^7) - (1521056*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x ^(3/2)*(2 + 5*x + 3*x^2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (1 646104*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*EllipticF[ I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(76545*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2) )
Time = 0.51 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {1233, 25, 1233, 27, 1236, 25, 1236, 27, 1236, 25, 1240, 1503, 1413, 1456}
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.
| \(\displaystyle \int \frac {(2-5 x) x^{13/2}}{\left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2}{9} \int -\frac {x^{9/2} (340 x+407)}{\left (3 x^2+5 x+2\right )^{3/2}}dx+\frac {2 (95 x+74) x^{11/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \int \frac {x^{9/2} (340 x+407)}{\left (3 x^2+5 x+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {2}{3} \int -\frac {x^{5/2} (11455 x+10388)}{2 \sqrt {3 x^2+5 x+2}}dx+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {2 x^{7/2} (1685 x+1484)}{3 \sqrt {3 x^2+5 x+2}}-\frac {1}{3} \int \frac {x^{5/2} (11455 x+10388)}{\sqrt {3 x^2+5 x+2}}dx\right )\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (-\frac {2}{21} \int -\frac {x^{3/2} (62751 x+57275)}{\sqrt {3 x^2+5 x+2}}dx-\frac {22910}{21} \sqrt {3 x^2+5 x+2} x^{5/2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {2}{21} \int \frac {x^{3/2} (62751 x+57275)}{\sqrt {3 x^2+5 x+2}}dx-\frac {22910}{21} x^{5/2} \sqrt {3 x^2+5 x+2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {2}{15} \int -\frac {3 \sqrt {x} (131965 x+125502)}{2 \sqrt {3 x^2+5 x+2}}dx+\frac {41834}{5} \sqrt {3 x^2+5 x+2} x^{3/2}\right )-\frac {22910}{21} x^{5/2} \sqrt {3 x^2+5 x+2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {41834}{5} x^{3/2} \sqrt {3 x^2+5 x+2}-\frac {1}{5} \int \frac {\sqrt {x} (131965 x+125502)}{\sqrt {3 x^2+5 x+2}}dx\right )-\frac {22910}{21} x^{5/2} \sqrt {3 x^2+5 x+2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {1}{5} \left (-\frac {2}{9} \int -\frac {95066 x+131965}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {263930}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}\right )+\frac {41834}{5} \sqrt {3 x^2+5 x+2} x^{3/2}\right )-\frac {22910}{21} x^{5/2} \sqrt {3 x^2+5 x+2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {1}{5} \left (\frac {2}{9} \int \frac {95066 x+131965}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {263930}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}\right )+\frac {41834}{5} \sqrt {3 x^2+5 x+2} x^{3/2}\right )-\frac {22910}{21} x^{5/2} \sqrt {3 x^2+5 x+2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {1}{5} \left (\frac {4}{9} \int \frac {95066 x+131965}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {263930}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}\right )+\frac {41834}{5} \sqrt {3 x^2+5 x+2} x^{3/2}\right )-\frac {22910}{21} x^{5/2} \sqrt {3 x^2+5 x+2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {1}{5} \left (\frac {4}{9} \left (131965 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+95066 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {263930}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}\right )+\frac {41834}{5} \sqrt {3 x^2+5 x+2} x^{3/2}\right )-\frac {22910}{21} x^{5/2} \sqrt {3 x^2+5 x+2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {1}{5} \left (\frac {4}{9} \left (95066 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {131965 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )-\frac {263930}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}\right )+\frac {41834}{5} \sqrt {3 x^2+5 x+2} x^{3/2}\right )-\frac {22910}{21} x^{5/2} \sqrt {3 x^2+5 x+2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {2 x^{11/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {1}{5} \left (\frac {4}{9} \left (\frac {131965 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+95066 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {263930}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}\right )+\frac {41834}{5} \sqrt {3 x^2+5 x+2} x^{3/2}\right )-\frac {22910}{21} x^{5/2} \sqrt {3 x^2+5 x+2}\right )+\frac {2 (1685 x+1484) x^{7/2}}{3 \sqrt {3 x^2+5 x+2}}\right )\) |
(2*x^(11/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (2*((2*x^(7/2)*(148 4 + 1685*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + ((-22910*x^(5/2)*Sqrt[2 + 5*x + 3 *x^2])/21 + (2*((41834*x^(3/2)*Sqrt[2 + 5*x + 3*x^2])/5 + ((-263930*Sqrt[x ]*Sqrt[2 + 5*x + 3*x^2])/9 + (4*(95066*((Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5* x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sq rt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (131965*(1 + x)*Sqrt[(2 + 3*x) /(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2] )))/9)/5))/21)/3))/9
3.11.72.3.1 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[((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 + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
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[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q )*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
Time = 0.30 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.06
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (-\frac {22460}{19683}-\frac {32666 x}{19683}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (-\frac {100085}{6561}-\frac {33460 x}{2187}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}-\frac {10 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{189}+\frac {1084 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{2835}-\frac {9286 \sqrt {3 x^{3}+5 x^{2}+2 x}}{5103}-\frac {211144 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{15309 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {760528 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{76545 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(272\) |
| default | \(-\frac {2 \left (1328364 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}+1140792 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}+54675 x^{7}+2213940 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +1901320 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -212868 x^{6}+885576 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+760528 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+788130 x^{5}-26854524 x^{4}-77349420 x^{3}-67906368 x^{2}-19002960 x \right ) \sqrt {3 x^{2}+5 x +2}}{229635 \sqrt {x}\, \left (2+3 x \right )^{2} \left (1+x \right )^{2}}\) | \(312\) |
(x*(3*x^2+5*x+2))^(1/2)/x^(1/2)/(3*x^2+5*x+2)^(1/2)*((-22460/19683-32666/1 9683*x)*(3*x^3+5*x^2+2*x)^(1/2)/(x^2+5/3*x+2/3)^2-2*x*(-100085/6561-33460/ 2187*x)*3^(1/2)/(x*(x^2+5/3*x+2/3))^(1/2)-10/189*x^2*(3*x^3+5*x^2+2*x)^(1/ 2)+1084/2835*x*(3*x^3+5*x^2+2*x)^(1/2)-9286/5103*(3*x^3+5*x^2+2*x)^(1/2)-2 11144/15309*(6*x+4)^(1/2)*(3+3*x)^(1/2)*(-6*x)^(1/2)/(3*x^3+5*x^2+2*x)^(1/ 2)*EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))-760528/76545*(6*x+4)^(1/2)*(3+3* x)^(1/2)*(-6*x)^(1/2)/(3*x^3+5*x^2+2*x)^(1/2)*(1/3*EllipticE(1/2*(6*x+4)^( 1/2),I*2^(1/2))-EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.54 \[ \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (5698840 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 6844752 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) + 27 \, {\left (6075 \, x^{6} - 23652 \, x^{5} + 87570 \, x^{4} - 2983836 \, x^{3} - 8594380 \, x^{2} - 7545152 \, x - 2111440\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{688905 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
-2/688905*(5698840*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstras sPInverse(28/27, 80/729, x + 5/9) - 6844752*sqrt(3)*(9*x^4 + 30*x^3 + 37*x ^2 + 20*x + 4)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 8 0/729, x + 5/9)) + 27*(6075*x^6 - 23652*x^5 + 87570*x^4 - 2983836*x^3 - 85 94380*x^2 - 7545152*x - 2111440)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/(9*x^4 + 3 0*x^3 + 37*x^2 + 20*x + 4)
Timed out. \[ \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {13}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {13}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x^{13/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]