Integrand size = 28, antiderivative size = 158 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=-\frac {143 \sqrt {3+5 x}}{49 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {438 \sqrt {1-2 x} \sqrt {3+5 x}}{343 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}-\frac {146}{343} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {17}{343} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-17/1029*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-14 6/343*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+11/21 *(3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^(1/2)-143/49*(3+5*x)^(1/2)/(1-2*x)^(1 /2)/(2+3*x)^(1/2)+438/343*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.59 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\frac {\frac {\sqrt {3+5 x} \left (-72+3445 x+5256 x^2\right )}{(1-2 x)^{3/2} \sqrt {2+3 x}}+i \sqrt {33} \left (438 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-455 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{1029} \]
((Sqrt[3 + 5*x]*(-72 + 3445*x + 5256*x^2))/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + I*Sqrt[33]*(438*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 455*Ellip ticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/1029
Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {109, 27, 167, 25, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{5/2} (3 x+2)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}-\frac {1}{21} \int \frac {3 \sqrt {5 x+3} (120 x+83)}{2 (1-2 x)^{3/2} (3 x+2)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}-\frac {1}{14} \int \frac {\sqrt {5 x+3} (120 x+83)}{(1-2 x)^{3/2} (3 x+2)^{3/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{14} \left (-\frac {1}{7} \int -\frac {116-45 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {286 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \int \frac {116-45 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {286 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (\frac {2}{7} \int \frac {5 (876 x+563)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {286 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (\frac {5}{7} \int \frac {876 x+563}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {286 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (\frac {5}{7} \left (\frac {187}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {286 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (\frac {5}{7} \left (\frac {187}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {292}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {286 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (\frac {5}{7} \left (-\frac {34}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {292}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {286 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\) |
(11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + ((-286*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) + ((876*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) /(7*Sqrt[2 + 3*x]) + (5*((-292*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (34*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2* x]], 35/33])/5))/7)/7)/14
3.30.67.3.1 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[(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[(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 = 1.40 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.44
| method | result | size |
| default | \(-\frac {\left (858 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-876 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-429 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+438 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-26280 x^{3}-32993 x^{2}-9975 x +216\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}}{1029 \left (15 x^{2}+19 x +6\right ) \left (-1+2 x \right )^{2}}\) | \(228\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-3 x +9\right )}{1029 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {563 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{7203 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {292 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2401 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {4400}{343} x^{2}-\frac {16720}{1029} x -\frac {1760}{343}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{588 \left (x -\frac {1}{2}\right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
-1/1029*(858*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)-876*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)-4 29*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF((1 0+15*x)^(1/2),1/35*70^(1/2))+438*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))-26280*x^3-32993 *x^2-9975*x+216)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(15*x^2+19*x+6) /(-1+2*x)^2
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\frac {30 \, {\left (5256 \, x^{2} + 3445 \, x - 72\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 5087 \, \sqrt {-30} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 13140 \, \sqrt {-30} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{30870 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
1/30870*(30*(5256*x^2 + 3445*x - 72)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 5087*sqrt(-30)*(12*x^3 - 4*x^2 - 5*x + 2)*weierstrassPInverse(1159 /675, 38998/91125, x + 23/90) + 13140*sqrt(-30)*(12*x^3 - 4*x^2 - 5*x + 2) *weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 3899 8/91125, x + 23/90)))/(12*x^3 - 4*x^2 - 5*x + 2)
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}} \,d x \]