Integrand size = 28, antiderivative size = 249 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}-\frac {23204503328 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}-\frac {697995152 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{194481 \sqrt {33}} \]
-23204503328/6417873*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2)) *33^(1/2)-697995152/6417873*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155 ^(1/2))*33^(1/2)+14/33*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2)+4508/891 *(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+171004/6237*(1-2*x)^(1/2)*(3+5* x)^(1/2)/(2+3*x)^(7/2)+7173272/43659*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^( 5/2)+333930952/305613*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+2320450332 8/2139291*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.40 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.45 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx=\frac {16 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (391506734113+2903435279352 x+8615827181322 x^2+12787628716260 x^3+9492493272732 x^4+2819347154352 x^5\right )}{8 (2+3 x)^{11/2}}+i \sqrt {33} \left (1450281458 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1493906155 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{6417873} \]
(16*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(391506734113 + 2903435279352*x + 8615 827181322*x^2 + 12787628716260*x^3 + 9492493272732*x^4 + 2819347154352*x^5 ))/(8*(2 + 3*x)^(11/2)) + I*Sqrt[33]*(1450281458*EllipticE[I*ArcSinh[Sqrt[ 9 + 15*x]], -2/33] - 1493906155*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33 ])))/6417873
Time = 0.31 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {109, 167, 27, 169, 27, 169, 27, 169, 169, 27, 176, 123, 129}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{13/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{33} \int \frac {(227-223 x) \sqrt {1-2 x}}{(3 x+2)^{11/2} \sqrt {5 x+3}}dx+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {2}{33} \left (\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}-\frac {2}{27} \int -\frac {49835-74876 x}{2 \sqrt {1-2 x} (3 x+2)^{9/2} \sqrt {5 x+3}}dx\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \int \frac {49835-74876 x}{\sqrt {1-2 x} (3 x+2)^{9/2} \sqrt {5 x+3}}dx+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {2}{49} \int \frac {35 (156383-213755 x)}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {10}{7} \int \frac {156383-213755 x}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {10}{7} \left (\frac {2}{35} \int \frac {3 (7936063-8966590 x)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {1793318 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {10}{7} \left (\frac {3}{35} \int \frac {7936063-8966590 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {1793318 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {10}{7} \left (\frac {3}{35} \left (\frac {2}{21} \int \frac {344289256-208706845 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {83482738 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1793318 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {10}{7} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (2900562916 x+1836312083)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2900562916 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {83482738 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1793318 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {10}{7} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {2900562916 x+1836312083}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2900562916 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {83482738 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1793318 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {10}{7} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {479871667}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2900562916}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {2900562916 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {83482738 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1793318 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {10}{7} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {479871667}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2900562916}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {2900562916 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {83482738 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1793318 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{33} \left (\frac {1}{27} \left (\frac {10}{7} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {87249394}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2900562916}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {2900562916 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {83482738 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1793318 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {85502 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{7/2}}\right )+\frac {2254 \sqrt {1-2 x} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}\) |
(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (2*((2254*Sqrt[ 1 - 2*x]*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + ((85502*Sqrt[1 - 2*x]*Sqrt[ 3 + 5*x])/(7*(2 + 3*x)^(7/2)) + (10*((1793318*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) /(35*(2 + 3*x)^(5/2)) + (3*((83482738*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(3/2)) + (2*((2900562916*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3 *x]) + (5*((-2900562916*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5 - (87249394*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2 *x]], 35/33])/5))/7))/21))/35))/7)/27))/33
3.28.95.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ [(b*e - a*f)/b, 0] && PosQ[-b/d] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d *e - c*f)/d, 0] && GtQ[-d/b, 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(( -b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ [((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f /b]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Time = 1.35 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.27
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {171004 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{505197 \left (\frac {2}{3}+x \right )^{4}}+\frac {7173272 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1178793 \left (\frac {2}{3}+x \right )^{3}}+\frac {333930952 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2750517 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {232045033280}{2139291} x^{2}-\frac {23204503328}{2139291} x +\frac {23204503328}{713097}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {29380993328 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{44925111 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {46409006656 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{44925111 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{72171 \left (\frac {2}{3}+x \right )^{6}}+\frac {4256 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{5}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(315\) |
| default | \(\frac {2 \left (2819347154352 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2738201933016 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+9397823847840 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9127339776720 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+12530431797120 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-12169786368960 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+8353621198080 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-8113190912640 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+2784540399360 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2704396970880 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+84580414630560 x^{7}+371272053248 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-360586262784 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+293232839645016 x^{6}+386732216916828 x^{5}+211405262133852 x^{4}-2138118521814 x^{3}-57086936770452 x^{2}-24956397311829 x -3523560607017\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{6417873 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(599\) |
(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*(171004/505197*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+7173272/1178793*(- 30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+333930952/2750517*(-30*x^3-23*x^2+7*x +6)^(1/2)/(2/3+x)^2+23204503328/6417873*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2- 3*x+9))^(1/2)+29380993328/44925111*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x- 9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1 /2))+46409006656/44925111*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/ (-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2) )+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1/2)))+98/72171*(-30*x^3-23*x^2+7 *x+6)^(1/2)/(2/3+x)^6+4256/216513*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (135 \, {\left (2819347154352 \, x^{5} + 9492493272732 \, x^{4} + 12787628716260 \, x^{3} + 8615827181322 \, x^{2} + 2903435279352 \, x + 391506734113\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 197110280804 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 522101324880 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{288804285 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
2/288804285*(135*(2819347154352*x^5 + 9492493272732*x^4 + 12787628716260*x ^3 + 8615827181322*x^2 + 2903435279352*x + 391506734113)*sqrt(5*x + 3)*sqr t(3*x + 2)*sqrt(-2*x + 1) - 197110280804*sqrt(-30)*(729*x^6 + 2916*x^5 + 4 860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 522101324880*sqrt(-30)*(729*x^6 + 2916*x^5 + 486 0*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassZeta(1159/675, 38998/ 91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}\,\sqrt {5\,x+3}} \,d x \]