Integrand size = 28, antiderivative size = 129 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {335}{63} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}+\frac {4451}{126} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {67}{63} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
4451/378*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+67 /189*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+11/7*( 3+5*x)^(3/2)*(2+3*x)^(1/2)/(1-2*x)^(1/2)+335/63*(1-2*x)^(1/2)*(2+3*x)^(1/2 )*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 5.39 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {6 (632-175 x) \sqrt {2+3 x} \sqrt {3+5 x}-4451 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+4585 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{378 \sqrt {1-2 x}} \]
(6*(632 - 175*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (4451*I)*Sqrt[33 - 66*x]*El lipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (4585*I)*Sqrt[33 - 66*x]*Ellip ticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(378*Sqrt[1 - 2*x])
Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {109, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{3/2} \sqrt {3 x+2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 \sqrt {3 x+2} (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}-\frac {1}{7} \int \frac {5 \sqrt {5 x+3} (134 x+87)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 \sqrt {3 x+2} (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}-\frac {5}{14} \int \frac {\sqrt {5 x+3} (134 x+87)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {11 \sqrt {3 x+2} (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}-\frac {5}{14} \left (-\frac {1}{9} \int -\frac {4451 x+2818}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {134}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {11 \sqrt {3 x+2} (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}-\frac {5}{14} \left (\frac {1}{9} \int \frac {4451 x+2818}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {134}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {11 \sqrt {3 x+2} (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}-\frac {5}{14} \left (\frac {1}{9} \left (\frac {737}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4451}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {134}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {11 \sqrt {3 x+2} (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}-\frac {5}{14} \left (\frac {1}{9} \left (\frac {737}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4451}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {134}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {11 \sqrt {3 x+2} (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}-\frac {5}{14} \left (\frac {1}{9} \left (-\frac {134}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4451}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {134}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )\) |
(11*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]) - (5*((-134*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-4451*Sqrt[11/3]*EllipticE[ArcSin[ Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (134*Sqrt[11/3]*EllipticF[ArcSin[Sqr t[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9))/14
3.30.12.3.1 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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 = 140, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (4323 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4451 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+15750 x^{3}-36930 x^{2}-65748 x -22752\right )}{11340 x^{3}+8694 x^{2}-2646 x -2268}\) | \(140\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {25 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18}-\frac {2818 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{1323 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4451 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{1323 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {121 \left (-30 x^{2}-38 x -12\right )}{28 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(214\) |
1/378*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(4323*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))-4451*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*Elli pticE((10+15*x)^(1/2),1/35*70^(1/2))+15750*x^3-36930*x^2-65748*x-22752)/(3 0*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 = 73, normalized size of antiderivative = 0.57 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {540 \, {\left (175 \, x - 632\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 151247 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 400590 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{34020 \, {\left (2 \, x - 1\right )}} \]
1/34020*(540*(175*x - 632)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 15 1247*sqrt(-30)*(2*x - 1)*weierstrassPInverse(1159/675, 38998/91125, x + 23 /90) - 400590*sqrt(-30)*(2*x - 1)*weierstrassZeta(1159/675, 38998/91125, w eierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(2*x - 1)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {3 x + 2}}\, dx \]
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}} \,d x \]