Integrand size = 28, antiderivative size = 249 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=-\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8120161139 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}-\frac {486785077 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{248771250 \sqrt {33}} \]
-8120161139/4104725625*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2 ))*33^(1/2)-486785077/8209451250*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33 *1155^(1/2))*33^(1/2)+326/10725*(1-2*x)^(3/2)*(3+5*x)^(7/2)*(2+3*x)^(1/2)+ 2/65*(1-2*x)^(5/2)*(3+5*x)^(7/2)*(2+3*x)^(1/2)-3872003/30405375*(3+5*x)^(3 /2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-121031/30405375*(3+5*x)^(5/2)*(1-2*x)^(1/2 )*(2+3*x)^(1/2)+2314/111375*(3+5*x)^(7/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-4867 85077/547296750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.45 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (495379991+2923422930 x-1730459250 x^2-7942630500 x^3+2577015000 x^4+8419950000 x^5\right )+16240322278 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-16727107355 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{8209451250} \]
(15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(495379991 + 2923422930*x - 1730459250*x^2 - 7942630500*x^3 + 2577015000*x^4 + 8419950000*x^5) + (1624 0322278*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (1672710 7355*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/8209451250
Time = 0.32 (sec) , antiderivative size = 283, 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 = {112, 27, 171, 27, 171, 27, 171, 27, 171, 27, 171, 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.
| \(\displaystyle \int (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}-\frac {2}{65} \int -\frac {(1-2 x)^{3/2} (5 x+3)^{5/2} (163 x+111)}{2 \sqrt {3 x+2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \int \frac {(1-2 x)^{3/2} (5 x+3)^{5/2} (163 x+111)}{\sqrt {3 x+2}}dx+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {2}{165} \int \frac {\sqrt {1-2 x} (5 x+3)^{5/2} (15041 x+11306)}{2 \sqrt {3 x+2}}dx+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \int \frac {\sqrt {1-2 x} (5 x+3)^{5/2} (15041 x+11306)}{\sqrt {3 x+2}}dx+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {2}{135} \int \frac {(5 x+3)^{5/2} (121031 x+518563)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \int \frac {(5 x+3)^{5/2} (121031 x+518563)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \left (-\frac {1}{21} \int -\frac {5 (5 x+3)^{3/2} (23232018 x+14205479)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {121031}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \left (\frac {5}{42} \int \frac {(5 x+3)^{3/2} (23232018 x+14205479)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {121031}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \left (\frac {5}{42} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (486785077 x+317626266)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {7744006}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {121031}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \left (\frac {5}{42} \left (\frac {1}{5} \int \frac {\sqrt {5 x+3} (486785077 x+317626266)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {7744006}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {121031}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \left (\frac {5}{42} \left (\frac {1}{5} \left (-\frac {1}{9} \int -\frac {32480644556 x+20559313903}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {486785077}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7744006}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {121031}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \int \frac {32480644556 x+20559313903}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {486785077}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7744006}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {121031}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {5354635847}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {32480644556}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {486785077}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7744006}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {121031}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {5354635847}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {32480644556}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {486785077}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7744006}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {121031}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{135} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (-\frac {973570154}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {32480644556}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {486785077}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7744006}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {121031}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {30082}{135} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {326}{165} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
(2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/65 + ((326*(1 - 2*x)^(3/ 2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/165 + ((30082*Sqrt[1 - 2*x]*Sqrt[2 + 3*x ]*(3 + 5*x)^(7/2))/135 + ((-121031*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^( 5/2))/21 + (5*((-7744006*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((-486785077*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-32480644556 *Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (973570 154*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/18)/5 ))/42)/135)/165)/65
3.28.76.3.1 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 = 1.36 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.66
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-3788977500000 x^{8}-4064539500000 x^{7}+15771272649 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-16240322278 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+3569208300000 x^{6}+4547296260000 x^{5}-1320576729750 x^{4}-2128036873050 x^{3}-19688021745 x^{2}+315122962755 x +44584199190\right )}{8209451250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(165\) |
| risch | \(-\frac {\left (8419950000 x^{5}+2577015000 x^{4}-7942630500 x^{3}-1730459250 x^{2}+2923422930 x +495379991\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{547296750 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {20559313903 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{60202642500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {8120161139 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{15050660625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {32482477 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6081075}+\frac {495379991 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{547296750}+\frac {20559313903 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{57466158750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {16240322278 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{28733079375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {59161 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18711}+\frac {2020 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{429}-\frac {168098 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11583}+\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(294\) |
-1/8209451250*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-3788977500000*x^ 8-4064539500000*x^7+15771272649*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))-16240322278*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))+3569208300000*x^6+4547296260000*x^5-1320576729750*x^ 4-2128036873050*x^3-19688021745*x^2+315122962755*x+44584199190)/(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 = 74, normalized size of antiderivative = 0.30 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\frac {1}{547296750} \, {\left (8419950000 \, x^{5} + 2577015000 \, x^{4} - 7942630500 \, x^{3} - 1730459250 \, x^{2} + 2923422930 \, x + 495379991\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {551641713241}{738850612500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {8120161139}{4104725625} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
1/547296750*(8419950000*x^5 + 2577015000*x^4 - 7942630500*x^3 - 1730459250 *x^2 + 2923422930*x + 495379991)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1 ) - 551641713241/738850612500*sqrt(-30)*weierstrassPInverse(1159/675, 3899 8/91125, x + 23/90) + 8120161139/4104725625*sqrt(-30)*weierstrassZeta(1159 /675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\text {Timed out} \]
\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]