Integrand size = 28, antiderivative size = 218 \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=-\frac {11346991 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3898125}-\frac {342971 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{866250}-\frac {543 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1508889271 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{7087500 \sqrt {33}}-\frac {11346991 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1771875 \sqrt {33}} \]
-1508889271/233887500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2) )*33^(1/2)-11346991/58471875*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*115 5^(1/2))*33^(1/2)-23/2475*(2+3*x)^(3/2)*(3+5*x)^(5/2)*(1-2*x)^(1/2)+2/55*( 2+3*x)^(5/2)*(3+5*x)^(5/2)*(1-2*x)^(1/2)-342971/866250*(3+5*x)^(3/2)*(1-2* x)^(1/2)*(2+3*x)^(1/2)-543/9625*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)- 11346991/3898125*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.50 \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-27010769+29706255 x+132234750 x^2+156161250 x^3+63787500 x^4\right )+1508889271 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1554277235 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{233887500} \]
(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-27010769 + 29706255*x + 13 2234750*x^2 + 156161250*x^3 + 63787500*x^4) + (1508889271*I)*Sqrt[33]*Elli pticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (1554277235*I)*Sqrt[33]*Elliptic F[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/233887500
Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {112, 27, 171, 27, 171, 25, 171, 27, 171, 25, 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.
| \(\displaystyle \int \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {2}{55} \int -\frac {(3 x+2)^{3/2} (5 x+3)^{3/2} (23 x+27)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \int \frac {(3 x+2)^{3/2} (5 x+3)^{3/2} (23 x+27)}{\sqrt {1-2 x}}dx+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{55} \left (-\frac {1}{45} \int -\frac {\sqrt {3 x+2} (5 x+3)^{3/2} (9774 x+6355)}{2 \sqrt {1-2 x}}dx-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \int \frac {\sqrt {3 x+2} (5 x+3)^{3/2} (9774 x+6355)}{\sqrt {1-2 x}}dx-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \left (-\frac {1}{35} \int -\frac {(5 x+3)^{3/2} (1028913 x+674539)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {9774}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \left (\frac {1}{35} \int \frac {(5 x+3)^{3/2} (1028913 x+674539)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {9774}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \left (\frac {1}{35} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (45387964 x+29496387)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {342971}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9774}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \int \frac {\sqrt {5 x+3} (45387964 x+29496387)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {342971}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9774}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (-\frac {1}{9} \int -\frac {1508889271 x+955260323}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {45387964}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {342971}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9774}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \int \frac {1508889271 x+955260323}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {45387964}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {342971}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9774}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {249633802}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1508889271}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {45387964}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {342971}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9774}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {249633802}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1508889271}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {45387964}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {342971}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9774}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (-\frac {45387964}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1508889271}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {45387964}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {342971}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9774}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {23}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
(2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/55 + ((-23*Sqrt[1 - 2*x] *(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/45 + ((-9774*Sqrt[1 - 2*x]*Sqrt[2 + 3*x] *(3 + 5*x)^(5/2))/35 + ((-342971*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/ 2))/5 + ((-45387964*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-1508 889271*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - ( 45387964*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/ 9)/10)/35)/90)/55
3.27.53.3.1 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[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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 = 1.32 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-57408750000 x^{7}+1465461393 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1508889271 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-184558500000 x^{6}-213367162500 x^{5}-73701994500 x^{4}+59690698650 x^{3}+48677999160 x^{2}-325135590 x -4861938420\right )}{233887500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(160\) |
| risch | \(-\frac {\left (63787500 x^{4}+156161250 x^{3}+132234750 x^{2}+29706255 x -27010769\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{7796250 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {955260323 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{857587500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1508889271 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{857587500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {660139 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{173250}-\frac {27010769 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7796250}+\frac {955260323 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{818606250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1508889271 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{818606250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {58771 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3465}+\frac {90 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}+\frac {661 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{33}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(284\) |
-1/233887500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-57408750000*x^7+1 465461393*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*Ellip ticF((10+15*x)^(1/2),1/35*70^(1/2))-1508889271*5^(1/2)*(2+3*x)^(1/2)*7^(1/ 2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))-1 84558500000*x^6-213367162500*x^5-73701994500*x^4+59690698650*x^3+486779991 60*x^2-325135590*x-4861938420)/(30*x^3+23*x^2-7*x-6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.32 \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {1}{7796250} \, {\left (63787500 \, x^{4} + 156161250 \, x^{3} + 132234750 \, x^{2} + 29706255 \, x - 27010769\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {51268975837}{21049875000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {1508889271}{233887500} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
1/7796250*(63787500*x^4 + 156161250*x^3 + 132234750*x^2 + 29706255*x - 270 10769)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 51268975837/2104987500 0*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 150888 9271/233887500*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstras sPInverse(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\text {Timed out} \]
\[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1} \,d x } \]
\[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1} \,d x } \]
Timed out. \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \]