Integrand size = 28, antiderivative size = 160 \[ \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {2645}{378} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {20}{21} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {1}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {17587}{378} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {529}{378} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-17587/1134*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2) -529/1134*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2 0/21*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-1/7*(3+5*x)^(5/2)*(1-2*x)^( 1/2)*(2+3*x)^(1/2)-2645/378*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 3.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\frac {-3 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (4211+3420 x+1350 x^2\right )+17587 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-18116 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{1134} \]
(-3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(4211 + 3420*x + 1350*x^2) + (17587*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (18116*I )*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/1134
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 = {112, 27, 171, 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.
| \(\displaystyle \int \frac {\sqrt {3 x+2} (5 x+3)^{5/2}}{\sqrt {1-2 x}} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {1}{7} \int \frac {(5 x+3)^{3/2} (200 x+131)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {(5 x+3)^{3/2} (200 x+131)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{14} \left (-\frac {1}{15} \int -\frac {5 \sqrt {5 x+3} (2645 x+1719)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {40}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{3} \int \frac {\sqrt {5 x+3} (2645 x+1719)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {40}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{3} \left (-\frac {1}{9} \int -\frac {175870 x+111341}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2645}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {40}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{3} \left (\frac {1}{18} \int \frac {175870 x+111341}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2645}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {40}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{3} \left (\frac {1}{18} \left (5819 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+35174 \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {2645}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {40}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{3} \left (\frac {1}{18} \left (5819 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-35174 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {2645}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {40}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{3} \left (\frac {1}{18} \left (-1058 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-35174 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {2645}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {40}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
-1/7*(Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2)) + ((-40*Sqrt[1 - 2*x]*S qrt[2 + 3*x]*(3 + 5*x)^(3/2))/3 + ((-2645*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt [3 + 5*x])/9 + (-35174*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] ], 35/33] - 1058*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/ 33])/18)/3)/14
3.29.33.3.1 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*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 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[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 = 150, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(-\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (85404 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-87935 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+607500 x^{5}+2004750 x^{4}+2933100 x^{3}+972195 x^{2}-749955 x -378990\right )}{5670 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(150\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {190 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21}-\frac {4211 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{378}+\frac {111341 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{39690 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {17587 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{3969 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {25 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
| risch | \(\frac {\left (1350 x^{2}+3420 x +4211\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{378 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {111341 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{41580 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {17587 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{4158 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(251\) |
-1/5670*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(85404*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*7 0^(1/2))-87935*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))+607500*x^5+2004750*x^4+2933100*x^ 3+972195*x^2-749955*x-378990)/(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 = 59, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{378} \, {\left (1350 \, x^{2} + 3420 \, x + 4211\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {149392}{25515} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {17587}{1134} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
-1/378*(1350*x^2 + 3420*x + 4211)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 149392/25515*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 17587/1134*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weie rstrassPInverse(1159/675, 38998/91125, x + 23/90))
\[ \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int \frac {\sqrt {3 x + 2} \left (5 x + 3\right )^{\frac {5}{2}}}{\sqrt {1 - 2 x}}\, dx \]
\[ \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2}}{\sqrt {-2 \, x + 1}} \,d x } \]
\[ \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2}}{\sqrt {-2 \, x + 1}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int \frac {\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]