Integrand size = 28, antiderivative size = 246 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {269045681 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{207900}+\frac {4066493 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23100}+\frac {9741}{385} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {419}{66} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {18}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {17888580643 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{189000 \sqrt {33}}+\frac {269045681 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{94500 \sqrt {33}} \]
17888580643/6237000*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))* 33^(1/2)+269045681/3118500*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^ (1/2))*33^(1/2)+(2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2)+419/66*(2+3*x)^( 3/2)*(3+5*x)^(5/2)*(1-2*x)^(1/2)+18/11*(2+3*x)^(5/2)*(3+5*x)^(5/2)*(1-2*x) ^(1/2)+4066493/23100*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)+9741/385*(3 +5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)+269045681/207900*(1-2*x)^(1/2)*(2+ 3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.49 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {-30 \sqrt {2+3 x} \sqrt {3+5 x} \left (-477155552+273928969 x+198895770 x^2+133330950 x^3+60196500 x^4+12757500 x^5\right )-17888580643 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+18426672005 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{6237000 \sqrt {1-2 x}} \]
(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-477155552 + 273928969*x + 198895770*x^2 + 133330950*x^3 + 60196500*x^4 + 12757500*x^5) - (17888580643*I)*Sqrt[33 - 66*x]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (18426672005*I)*Sqrt [33 - 66*x]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(6237000*Sqrt[1 - 2*x])
Time = 0.33 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {108, 27, 171, 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.
| \(\displaystyle \int \frac {(3 x+2)^{7/2} (5 x+3)^{5/2}}{(1-2 x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^{7/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}-\int \frac {(3 x+2)^{5/2} (5 x+3)^{3/2} (180 x+113)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{7/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}-\frac {1}{2} \int \frac {(3 x+2)^{5/2} (5 x+3)^{3/2} (180 x+113)}{\sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{55} \int -\frac {25 (3 x+2)^{3/2} (5 x+3)^{3/2} (1257 x+796)}{\sqrt {1-2 x}}dx+\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \int \frac {(3 x+2)^{3/2} (5 x+3)^{3/2} (1257 x+796)}{\sqrt {1-2 x}}dx\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (-\frac {1}{45} \int -\frac {3 \sqrt {3 x+2} (5 x+3)^{3/2} (116892 x+74995)}{2 \sqrt {1-2 x}}dx-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \int \frac {\sqrt {3 x+2} (5 x+3)^{3/2} (116892 x+74995)}{\sqrt {1-2 x}}dx-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \left (-\frac {1}{35} \int -\frac {(5 x+3)^{3/2} (12199479 x+7996612)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {116892}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \left (\frac {1}{35} \int \frac {(5 x+3)^{3/2} (12199479 x+7996612)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {116892}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \left (\frac {1}{35} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (538091362 x+349693671)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {4066493}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {116892}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \int \frac {\sqrt {5 x+3} (538091362 x+349693671)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {4066493}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {116892}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (-\frac {1}{9} \int -\frac {17888580643 x+11325048884}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {538091362}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {4066493}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {116892}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \int \frac {17888580643 x+11325048884}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {538091362}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {4066493}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {116892}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {2959502491}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {17888580643}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {538091362}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {4066493}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {116892}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {2959502491}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {17888580643}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {538091362}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {4066493}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {116892}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{2} \left (\frac {36}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {5}{11} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (-\frac {538091362}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {17888580643}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {538091362}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {4066493}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {116892}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {419}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + ((36*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/11 - (5*((-419*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*( 3 + 5*x)^(5/2))/15 + ((-116892*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2) )/35 + ((-4066493*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((-5380 91362*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-17888580643*Sqrt[1 1/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (538091362*Sqr t[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9)/10)/35)/3 0))/11)/2
3.30.8.3.1 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 + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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 = 4.58 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.65
| method | result | size |
| default | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (5740875000 x^{7}+17373719319 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-17888580643 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+34360200000 x^{6}+96607282500 x^{5}+176337108000 x^{4}+260638195950 x^{3}-22779247470 x^{2}-222671450220 x -85887999360\right )}{187110000 x^{3}+143451000 x^{2}-43659000 x -37422000}\) | \(160\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {12542447 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18480}+\frac {1660125991 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1663200}-\frac {2831262221 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{5457375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {17888580643 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{21829500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {740527 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1848}+\frac {675 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{22}+\frac {7045 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{44}-\frac {41503 \left (-30 x^{2}-38 x -12\right )}{64 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(312\) |
1/6237000*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(5740875000*x^7+173737 19319*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF ((10+15*x)^(1/2),1/35*70^(1/2))-17888580643*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))+3436 0200000*x^6+96607282500*x^5+176337108000*x^4+260638195950*x^3-22779247470* x^2-222671450220*x-85887999360)/(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 = 93, normalized size of antiderivative = 0.38 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\frac {2700 \, {\left (12757500 \, x^{5} + 60196500 \, x^{4} + 133330950 \, x^{3} + 198895770 \, x^{2} + 273928969 \, x - 477155552\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 607817044771 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 1609972257870 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{561330000 \, {\left (2 \, x - 1\right )}} \]
1/561330000*(2700*(12757500*x^5 + 60196500*x^4 + 133330950*x^3 + 198895770 *x^2 + 273928969*x - 477155552)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 607817044771*sqrt(-30)*(2*x - 1)*weierstrassPInverse(1159/675, 38998/91 125, x + 23/90) - 1609972257870*sqrt(-30)*(2*x - 1)*weierstrassZeta(1159/6 75, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/( 2*x - 1)
Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]