Integrand size = 28, antiderivative size = 218 \[ \int \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=-\frac {222527 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{51975}-\frac {6691 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6930}-\frac {34}{231} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{55} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {30926081 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{89100 \sqrt {35}}+\frac {222527 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{22275 \sqrt {35}} \] Output:
-222527/51975*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-6691/6930*(1-2*x)^ (1/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)-34/231*(1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5* x)^(3/2)-1/55*(1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2)+2/55*(1-2*x)^(1/2) *(2+3*x)^(3/2)*(3+5*x)^(7/2)-30926081/3118500*EllipticE(1/11*55^(1/2)*(1-2 *x)^(1/2),1/35*1155^(1/2))*35^(1/2)+222527/779625*EllipticF(1/11*55^(1/2)* (1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
Result contains complex when optimal does not.
Time = 6.55 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.50 \[ \int \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-567484+570555 x+2737800 x^2+3354750 x^3+1417500 x^4\right )+30926081 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-31856335 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{3118500} \] Input:
Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2),x]
Output:
(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-567484 + 570555*x + 273780 0*x^2 + 3354750*x^3 + 1417500*x^4) + (30926081*I)*Sqrt[33]*EllipticE[I*Arc Sinh[Sqrt[9 + 15*x]], -2/33] - (31856335*I)*Sqrt[33]*EllipticF[I*ArcSinh[S qrt[9 + 15*x]], -2/33])/3118500
Time = 0.36 (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, 27, 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.
| \(\displaystyle \int \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}-\frac {2}{55} \int -\frac {\sqrt {3 x+2} (5 x+3)^{5/2} (27 x+25)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \int \frac {\sqrt {3 x+2} (5 x+3)^{5/2} (27 x+25)}{\sqrt {1-2 x}}dx+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{55} \left (-\frac {1}{45} \int -\frac {9 (5 x+3)^{5/2} (1062 x+701)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \int \frac {(5 x+3)^{5/2} (1062 x+701)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \left (-\frac {1}{21} \int -\frac {15 (5 x+3)^{3/2} (7031 x+4608)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {354}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \left (\frac {5}{7} \int \frac {(5 x+3)^{3/2} (7031 x+4608)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {354}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \left (\frac {5}{7} \left (-\frac {1}{15} \int -\frac {\sqrt {5 x+3} (930254 x+604557)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {7031}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {354}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \left (\frac {5}{7} \left (\frac {1}{30} \int \frac {\sqrt {5 x+3} (930254 x+604557)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {7031}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {354}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \left (\frac {5}{7} \left (\frac {1}{30} \left (-\frac {1}{9} \int -\frac {30926081 x+19578928}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {930254}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7031}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {354}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \left (\frac {5}{7} \left (\frac {1}{30} \left (\frac {1}{9} \int \frac {30926081 x+19578928}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {930254}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7031}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {354}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \left (\frac {5}{7} \left (\frac {1}{30} \left (\frac {1}{9} \left (\frac {5116397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {30926081}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {930254}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7031}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {354}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \left (\frac {5}{7} \left (\frac {1}{30} \left (\frac {1}{9} \left (\frac {5116397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {30926081}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {930254}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7031}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {354}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{10} \left (\frac {5}{7} \left (\frac {1}{30} \left (\frac {1}{9} \left (-\frac {930254}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {30926081}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {930254}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7031}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {354}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {3}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
Input:
Int[Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2),x]
Output:
(2*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/55 + ((-3*Sqrt[1 - 2*x]* Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/5 + ((-354*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/7 + (5*((-7031*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/1 5 + ((-930254*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-30926081*S qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (930254*S qrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9)/30))/7) /10)/55
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 = 0.42 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-1275750000 x^{7}-3997350000 x^{6}+15349191 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-30926081 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-4481122500 x^{5}-1442934000 x^{4}+1295845650 x^{3}+1004184510 x^{2}-16471740 x -102147120\right )}{3118500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
| risch | \(-\frac {\left (1417500 x^{4}+3354750 x^{3}+2737800 x^{2}+570555 x -567484\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{103950 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {4894732 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2858625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {30926081 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{11434500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {12679 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2310}-\frac {283742 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{51975}+\frac {4894732 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1091475 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {30926081 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{4365900 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2028 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{77}+\frac {355 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11}+\frac {150 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(290\) |
Input:
int((1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/3118500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-1275750000*x^7-3997 350000*x^6+15349191*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ell ipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-30926081*2^(1/2)*(2+3*x)^(1/2)*(- 3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-448 1122500*x^5-1442934000*x^4+1295845650*x^3+1004184510*x^2-16471740*x-102147 120)/(30*x^3+23*x^2-7*x-6)
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.32 \[ \int \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\frac {1}{103950} \, {\left (1417500 \, x^{4} + 3354750 \, x^{3} + 2737800 \, x^{2} + 570555 \, x - 567484\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {1050803657}{280665000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {30926081}{3118500} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x, algorithm="fricas")
Output:
1/103950*(1417500*x^4 + 3354750*x^3 + 2737800*x^2 + 570555*x - 567484)*sqr t(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1050803657/280665000*sqrt(-30)*w eierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 30926081/3118500*sq rt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/67 5, 38998/91125, x + 23/90))
Timed out. \[ \int \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(1/2)*(2+3*x)**(3/2)*(3+5*x)**(5/2),x)
Output:
Timed out
\[ \int \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*sqrt(-2*x + 1), x)
\[ \int \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*sqrt(-2*x + 1), x)
Timed out. \[ \int \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \] Input:
int((1 - 2*x)^(1/2)*(3*x + 2)^(3/2)*(5*x + 3)^(5/2),x)
Output:
int((1 - 2*x)^(1/2)*(3*x + 2)^(3/2)*(5*x + 3)^(5/2), x)
\[ \int \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\frac {150 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{11}+\frac {355 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{11}+\frac {2028 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{77}+\frac {12679 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{2310}-\frac {422447 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{35420}+\frac {30926081 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{106260}-\frac {8274913 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{70840} \] Input:
int((1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x)
Output:
(2898000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 + 6858600*sqrt( 3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 5597280*sqrt(3*x + 2)*sqrt( 5*x + 3)*sqrt( - 2*x + 1)*x**2 + 1166468*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 2534682*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 618 52162*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(30*x**3 + 2 3*x**2 - 7*x - 6),x) - 24824739*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2 *x + 1))/(30*x**3 + 23*x**2 - 7*x - 6),x))/212520