Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}+\frac {196}{81} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {4418}{405} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {988}{405} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-4418/1215*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+ 988/1215*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/ 9*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+10/3*(1-2*x)^(3/2)*(3+5*x)^(1/ 2)/(2+3*x)^(1/2)+196/81*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.48 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx=\frac {2 \left (\frac {15 \sqrt {1-2 x} \sqrt {3+5 x} \left (653+1077 x+36 x^2\right )}{(2+3 x)^{3/2}}+2209 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1715 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{1215} \]
(2*((15*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(653 + 1077*x + 36*x^2))/(2 + 3*x)^(3/ 2) + (2209*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (1715 *I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/1215
Time = 0.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {108, 27, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} \sqrt {5 x+3}}{(3 x+2)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{9} \int -\frac {5 (1-2 x)^{3/2} (12 x+5)}{2 (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{5/2} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{9} \int \frac {(1-2 x)^{3/2} (12 x+5)}{(3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{9} \left (-\frac {2}{3} \int \frac {3 \sqrt {1-2 x} (49 x+25)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{\sqrt {3 x+2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{9} \left (-2 \int \frac {\sqrt {1-2 x} (49 x+25)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{\sqrt {3 x+2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{9} \left (-2 \left (\frac {2}{45} \int \frac {2209 x+782}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {98}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{\sqrt {3 x+2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{9} \left (-2 \left (\frac {1}{45} \int \frac {2209 x+782}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {98}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{\sqrt {3 x+2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {5}{9} \left (-2 \left (\frac {1}{45} \left (\frac {2209}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {2717}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {98}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{\sqrt {3 x+2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {5}{9} \left (-2 \left (\frac {1}{45} \left (-\frac {2717}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2209}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {98}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{\sqrt {3 x+2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {5}{9} \left (-2 \left (\frac {1}{45} \left (\frac {494}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2209}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {98}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{\sqrt {3 x+2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}\) |
(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(3/2)) - (5*((-6*(1 - 2*x) ^(3/2)*Sqrt[3 + 5*x])/Sqrt[2 + 3*x] - 2*((98*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*S qrt[3 + 5*x])/45 + ((-2209*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (494*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5)/45)))/9
3.28.59.3.1 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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 1.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(-\frac {2 \left (4851 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-6627 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3234 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4418 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5400 x^{4}-162090 x^{3}-112485 x^{2}+38670 x +29385\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{1215 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\) | \(224\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {8 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81}+\frac {3128 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{8505 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8836 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{8505 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {6860}{81} x^{2}-\frac {686}{81} x +\frac {686}{27}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(238\) |
-2/1215*(4851*5^(1/2)*7^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))*x*( 2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)-6627*5^(1/2)*7^(1/2)*EllipticE(( 10+15*x)^(1/2),1/35*70^(1/2))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2) +3234*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))-4418*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))-5400*x^4-16 2090*x^3-112485*x^2+38670*x+29385)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3 )/(2+3*x)^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx=\frac {1350 \, {\left (36 \, x^{2} + 1077 \, x + 653\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 19573 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 198810 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{54675 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/54675*(1350*(36*x^2 + 1077*x + 653)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2* x + 1) - 19573*sqrt(-30)*(9*x^2 + 12*x + 4)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 198810*sqrt(-30)*(9*x^2 + 12*x + 4)*weierstrassZ eta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(9*x^2 + 12*x + 4)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{5/2}} \,d x \]