Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}-\frac {184636 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}-\frac {9124 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{26411 \sqrt {33}} \]
-184636/871563*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1 /2)-9124/871563*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^( 1/2)+4/231*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2)+1072/17787*(3+5*x)^(1 /2)/(2+3*x)^(3/2)/(1-2*x)^(1/2)+974/41503*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3 *x)^(3/2)+184636/290521*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2 \left (\frac {\sqrt {3+5 x} \left (597945-1478206 x-1066908 x^2+3323448 x^3\right )}{(1-2 x)^{3/2} (2+3 x)^{3/2}}+2 i \sqrt {33} \left (46159 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-48440 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{871563} \]
(2*((Sqrt[3 + 5*x]*(597945 - 1478206*x - 1066908*x^2 + 3323448*x^3))/((1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (2*I)*Sqrt[33]*(46159*EllipticE[I*ArcSinh[S qrt[9 + 15*x]], -2/33] - 48440*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] )))/871563
Time = 0.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {115, 27, 169, 27, 169, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^{5/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac {2}{231} \int -\frac {150 x+193}{2 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \int \frac {150 x+193}{(1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1072 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}-\frac {2}{77} \int -\frac {9 (2680 x+1949)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {9}{77} \int \frac {2680 x+1949}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {1072 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {9}{77} \left (\frac {2}{21} \int \frac {13763-2435 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1072 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {9}{77} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (92318 x+60409)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1072 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {9}{77} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {92318 x+60409}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1072 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{231} \left (\frac {9}{77} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {25091}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {92318}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1072 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{231} \left (\frac {9}{77} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {25091}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {92318}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1072 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{231} \left (\frac {9}{77} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {4562}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {92318}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1072 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}\) |
(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + ((1072*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (9*((974*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(3/2)) + (2*((92318*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*S qrt[2 + 3*x]) + (5*((-92318*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (4562*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2 *x]], 35/33])/5))/7))/21))/77)/231
3.30.78.3.1 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*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 = 1.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {107}{29106}-\frac {31 x}{4851}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right )^{2}}-\frac {2 \left (-18-30 x \right ) \left (\frac {135068}{2614689}-\frac {92318 x}{871563}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}+\frac {241636 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6100941 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {369272 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{6100941 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
| default | \(-\frac {2 \sqrt {1-2 x}\, \left (548064 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-553908 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+91344 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-92318 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-182688 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+184636 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-16617240 x^{4}-4635804 x^{3}+10591754 x^{2}+1444893 x -1793835\right )}{871563 \left (2+3 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(311\) |
(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*((107/29106-31/4851*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x^2+1/6*x-1/3)^2-2* (-18-30*x)*(135068/2614689-92318/871563*x)/((x^2+1/6*x-1/3)*(-18-30*x))^(1 /2)+241636/6100941*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^ 3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+369272/6100 941*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6) ^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15 *x)^(1/2),1/35*70^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (45 \, {\left (3323448 \, x^{3} - 1066908 \, x^{2} - 1478206 \, x + 597945\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1656748 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 4154310 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{39220335 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
2/39220335*(45*(3323448*x^3 - 1066908*x^2 - 1478206*x + 597945)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1656748*sqrt(-30)*(36*x^4 + 12*x^3 - 23 *x^2 - 4*x + 4)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 41 54310*sqrt(-30)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*weierstrassZeta(1159/ 675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/ (36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]