Integrand size = 28, antiderivative size = 188 \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {9694}{175} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2511}{350} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1289089 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}+\frac {9694}{875} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
1289089/10500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/ 2)+9694/2625*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2 )+(2+3*x)^(5/2)*(3+5*x)^(3/2)/(1-2*x)^(1/2)+12/7*(2+3*x)^(3/2)*(3+5*x)^(3/ 2)*(1-2*x)^(1/2)+2511/350*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)+9694/1 75*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.59 \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {-30 \sqrt {2+3 x} \sqrt {3+5 x} \left (-34721+17487 x+8460 x^2+2250 x^3\right )-1289089 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+1327865 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{10500 \sqrt {1-2 x}} \]
(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-34721 + 17487*x + 8460*x^2 + 2250*x^3) - (1289089*I)*Sqrt[33 - 66*x]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (1327865*I)*Sqrt[33 - 66*x]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]) /(10500*Sqrt[1 - 2*x])
Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {108, 27, 171, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{(1-2 x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\int \frac {15 (3 x+2)^{3/2} \sqrt {5 x+3} (8 x+5)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \int \frac {(3 x+2)^{3/2} \sqrt {5 x+3} (8 x+5)}{\sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \left (-\frac {1}{35} \int -\frac {\sqrt {3 x+2} \sqrt {5 x+3} (837 x+530)}{\sqrt {1-2 x}}dx-\frac {8}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \left (\frac {1}{35} \int \frac {\sqrt {3 x+2} \sqrt {5 x+3} (837 x+530)}{\sqrt {1-2 x}}dx-\frac {8}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \left (\frac {1}{35} \left (-\frac {1}{25} \int -\frac {\sqrt {5 x+3} (116328 x+75599)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {837}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {8}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \left (\frac {1}{35} \left (\frac {1}{50} \int \frac {\sqrt {5 x+3} (116328 x+75599)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {837}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {8}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \left (\frac {1}{35} \left (\frac {1}{50} \left (-\frac {1}{9} \int -\frac {3 (1289089 x+816107)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {38776}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {837}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {8}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \left (\frac {1}{35} \left (\frac {1}{50} \left (\frac {1}{3} \int \frac {1289089 x+816107}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {38776}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {837}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {8}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \left (\frac {1}{35} \left (\frac {1}{50} \left (\frac {1}{3} \left (\frac {213268}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1289089}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {38776}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {837}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {8}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \left (\frac {1}{35} \left (\frac {1}{50} \left (\frac {1}{3} \left (\frac {213268}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1289089}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {38776}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {837}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {8}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {(3 x+2)^{5/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \left (\frac {1}{35} \left (\frac {1}{50} \left (\frac {1}{3} \left (-\frac {38776}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1289089}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {38776}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {837}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {8}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )\) |
((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] - (15*((-8*Sqrt[1 - 2*x]*( 2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/35 + ((-837*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/25 + ((-38776*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3 + ((-1289089*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]) /5 - (38776*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/ 5)/3)/50)/35))/2
3.29.100.3.1 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 = 1.35 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (1251987 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1289089 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+1012500 x^{5}+5089500 x^{4}+13096350 x^{3}-4134060 x^{2}-16643310 x -6249780\right )}{315000 x^{3}+241500 x^{2}-73500 x -63000}\) | \(150\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {1917 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{140}+\frac {44559 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1400}-\frac {816107 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{36750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1289089 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{36750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {45 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14}-\frac {539 \left (-30 x^{2}-38 x -12\right )}{16 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(268\) |
1/10500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(1251987*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))-1289089*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))+1012500*x^5+5089500*x^4+13096 350*x^3-4134060*x^2-16643310*x-6249780)/(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 = 83, normalized size of antiderivative = 0.44 \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {2700 \, {\left (2250 \, x^{3} + 8460 \, x^{2} + 17487 \, x - 34721\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 43800583 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 116018010 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{945000 \, {\left (2 \, x - 1\right )}} \]
1/945000*(2700*(2250*x^3 + 8460*x^2 + 17487*x - 34721)*sqrt(5*x + 3)*sqrt( 3*x + 2)*sqrt(-2*x + 1) + 43800583*sqrt(-30)*(2*x - 1)*weierstrassPInverse (1159/675, 38998/91125, x + 23/90) - 116018010*sqrt(-30)*(2*x - 1)*weierst rassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(2*x - 1)
Timed out. \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]