Integrand size = 28, antiderivative size = 129 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}+\frac {428}{135} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {8314}{675} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {824}{675} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-8314/2025*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+ 824/2025*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+14 /3*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)+428/135*(1-2*x)^(1/2)*(2+3*x) ^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 5.93 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {2 \left (\frac {15 \sqrt {1-2 x} \sqrt {3+5 x} (743+12 x)}{\sqrt {2+3 x}}+4157 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-3745 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{2025} \]
(2*((15*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(743 + 12*x))/Sqrt[2 + 3*x] + (4157*I) *Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (3745*I)*Sqrt[33]* EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/2025
Time = 0.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {109, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{3/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{3} \int \frac {\sqrt {1-2 x} (107 x+62)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {2}{3} \left (\frac {2}{45} \int \frac {4157 x+2041}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \left (\frac {1}{45} \int \frac {4157 x+2041}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{3} \left (\frac {1}{45} \left (\frac {4157}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {2266}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{3} \left (\frac {1}{45} \left (-\frac {2266}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4157}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{3} \left (\frac {1}{45} \left (\frac {412}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4157}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 \sqrt {3 x+2}}\) |
(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*Sqrt[2 + 3*x]) + (2*((214*Sqrt[1 - 2 *x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45 + ((-4157*Sqrt[11/3]*EllipticE[ArcSin[ Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (412*Sqrt[11/3]*EllipticF[ArcSin[Sqr t[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/45))/3
3.28.90.3.1 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 = 1.31 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (3531 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4157 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1800 x^{3}-111630 x^{2}-10605 x +33435\right )}{2025 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(140\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {8 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{135}+\frac {8164 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{14175 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {16628 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{14175 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {980}{9} x^{2}-\frac {98}{9} x +\frac {98}{3}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(214\) |
-2/2025*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(3531*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))-4157*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*El lipticE((10+15*x)^(1/2),1/35*70^(1/2))-1800*x^3-111630*x^2-10605*x+33435)/ (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 = 73, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {1350 \, {\left (12 \, x + 743\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 88079 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 374130 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{91125 \, {\left (3 \, x + 2\right )}} \]
1/91125*(1350*(12*x + 743)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 88 079*sqrt(-30)*(3*x + 2)*weierstrassPInverse(1159/675, 38998/91125, x + 23/ 90) + 374130*sqrt(-30)*(3*x + 2)*weierstrassZeta(1159/675, 38998/91125, we ierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(3*x + 2)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{\frac {3}{2}} \sqrt {5 x + 3}}\, dx \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \]