Integrand size = 28, antiderivative size = 251 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {11171040 \sqrt {1-2 x} \sqrt {2+3 x}}{49 \sqrt {3+5 x}}-\frac {2234208}{49} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {201616}{49} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
2/3*(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2)-201616/147*EllipticF(1/7*21^ (1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2234208/49*EllipticE(1/7*21^ (1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+44/3*(1-2*x)^(1/2)/(2+3*x)^( 5/2)/(3+5*x)^(3/2)+11924/63*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)+2488 904/441*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)-5544440/147*(1-2*x)^(1/2 )*(2+3*x)^(1/2)/(3+5*x)^(3/2)+11171040/49*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5 *x)^(1/2)
Result contains complex when optimal does not.
Time = 8.65 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx=\frac {2}{147} \left (\frac {\sqrt {1-2 x} \left (763335749+5915384456 x+18325125498 x^2+28367736228 x^3+21944379060 x^4+6786406800 x^5\right )}{(2+3 x)^{7/2} (3+5 x)^{3/2}}+8 i \sqrt {33} \left (418914 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-431515 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \]
(2*((Sqrt[1 - 2*x]*(763335749 + 5915384456*x + 18325125498*x^2 + 283677362 28*x^3 + 21944379060*x^4 + 6786406800*x^5))/((2 + 3*x)^(7/2)*(3 + 5*x)^(3/ 2)) + (8*I)*Sqrt[33]*(418914*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 431515*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/147
Time = 0.32 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {109, 27, 167, 27, 169, 169, 27, 169, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{9/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{21} \int \frac {231 \sqrt {1-2 x} (1-x)}{(3 x+2)^{7/2} (5 x+3)^{5/2}}dx+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 22 \int \frac {\sqrt {1-2 x} (1-x)}{(3 x+2)^{7/2} (5 x+3)^{5/2}}dx+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle 22 \left (\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {2}{15} \int -\frac {5 (45-68 x)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}dx\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 22 \left (\frac {1}{3} \int \frac {45-68 x}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}dx+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \int \frac {4911-6775 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {15 (49227-56566 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {56566 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {15}{7} \int \frac {49227-56566 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {56566 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {15}{7} \left (-\frac {2}{33} \int \frac {99 (20367-12601 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {25202 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {56566 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {15}{7} \left (-6 \int \frac {20367-12601 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {25202 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {56566 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {15}{7} \left (-6 \left (-\frac {2}{11} \int \frac {837828 x+530419}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {279276 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {25202 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {56566 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {15}{7} \left (-6 \left (-\frac {1}{11} \int \frac {837828 x+530419}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {279276 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {25202 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {56566 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {15}{7} \left (-6 \left (\frac {1}{11} \left (-\frac {138611}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {837828}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {279276 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {25202 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {56566 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {15}{7} \left (-6 \left (\frac {1}{11} \left (\frac {279276}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {138611}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {279276 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {25202 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {56566 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle 22 \left (\frac {1}{3} \left (\frac {2}{21} \left (\frac {15}{7} \left (-6 \left (\frac {1}{11} \left (\frac {25202}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {279276}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {279276 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {25202 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {56566 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {542 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}\) |
(2*(1 - 2*x)^(3/2))/(3*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)) + 22*((2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + ((542*Sqrt[1 - 2*x])/(21*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (2*((56566*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*( 3 + 5*x)^(3/2)) + (15*((-25202*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3 + 5*x)^(3/2 ) - 6*((-279276*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + ((279276 *Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (25202*Sq rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/11)))/7))/2 1)/3)
3.29.14.3.1 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 = 4.45 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.27
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81 \left (\frac {2}{3}+x \right )^{4}}+\frac {268 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27 \left (\frac {2}{3}+x \right )^{3}}+\frac {28234 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{63 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {41369840}{49} x^{2}-\frac {4136984}{49} x +\frac {12410952}{49}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {8486704 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{1029 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4468416 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{343 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3 \left (x +\frac {3}{5}\right )^{2}}+\frac {-523600 x^{2}-\frac {261800}{3} x +\frac {523600}{3}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(319\) |
| default | \(-\frac {2 \sqrt {1-2 x}\, \left (439405560 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-452427120 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1142454456 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1176310512 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1113160752 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1146148704 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+481718688 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-495994176 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+78116544 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-80431488 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-13572813600 x^{6}-37102351320 x^{5}-34791093396 x^{4}-8282514768 x^{3}+6494356586 x^{2}+4388712958 x +763335749\right )}{147 \left (2+3 x \right )^{\frac {7}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(501\) |
(-(-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 )*(14/81*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+268/27*(-30*x^3-23*x^2+7*x +6)^(1/2)/(2/3+x)^3+28234/63*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+413698 4/147*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+8486704/1029*(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)*Elli pticF((10+15*x)^(1/2),1/35*70^(1/2))+4468416/343*(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)))-242 /3*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x+3/5)^2+52360/3*(-30*x^2-5*x+10)/((x+3/5 )*(-30*x^2-5*x+10))^(1/2))
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 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (5 \, {\left (6786406800 \, x^{5} + 21944379060 \, x^{4} + 28367736228 \, x^{3} + 18325125498 \, x^{2} + 5915384456 \, x + 763335749\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 6326148 \, \sqrt {-30} {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 16756560 \, \sqrt {-30} {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{735 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]
2/735*(5*(6786406800*x^5 + 21944379060*x^4 + 28367736228*x^3 + 18325125498 *x^2 + 5915384456*x + 763335749)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1 ) - 6326148*sqrt(-30)*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224* x^2 + 1344*x + 144)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 16756560*sqrt(-30)*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x ^2 + 1344*x + 144)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInve rse(1159/675, 38998/91125, x + 23/90)))/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]