Integrand size = 28, antiderivative size = 158 \[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=-\frac {4839 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1750}-\frac {104}{175} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {1}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {56041 \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8750}-\frac {5057 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{8750} \]
-5057/26250*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2) -56041/8750*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2) -104/175*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-1/7*(2+3*x)^(5/2)*(1-2* x)^(1/2)*(3+5*x)^(1/2)-4839/1750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 3.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\frac {168123 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5 \left (3 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (7919+6120 x+2250 x^2\right )+34636 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{26250} \]
((168123*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 5*(3*Sq rt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(7919 + 6120*x + 2250*x^2) + (3463 6*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/26250
Time = 0.25 (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 = {112, 27, 171, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^{5/2} \sqrt {5 x+3}}{\sqrt {1-2 x}} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {1}{7} \int \frac {(3 x+2)^{3/2} (208 x+127)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {(3 x+2)^{3/2} (208 x+127)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{14} \left (-\frac {1}{25} \int -\frac {\sqrt {3 x+2} (14517 x+8950)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {208}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{25} \int \frac {\sqrt {3 x+2} (14517 x+8950)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {208}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{25} \left (-\frac {1}{15} \int -\frac {3 (336246 x+212873)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4839}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {208}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{25} \left (\frac {1}{10} \int \frac {336246 x+212873}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4839}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {208}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{25} \left (\frac {1}{10} \left (\frac {55627}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {336246}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {4839}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {208}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{25} \left (\frac {1}{10} \left (\frac {55627}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {112082}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {4839}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {208}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{25} \left (\frac {1}{10} \left (-\frac {10114}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {112082}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {4839}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {208}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\) |
-1/7*(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + ((-208*Sqrt[1 - 2*x]* (2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/25 + ((-4839*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sq rt[3 + 5*x])/5 + ((-112082*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2* x]], 35/33])/5 - (10114*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5)/10)/25)/14
3.29.15.3.1 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 = 1.36 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (54428 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-56041 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+337500 x^{5}+1176750 x^{4}+1812900 x^{3}+628985 x^{2}-460765 x -237570\right )}{8750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(150\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {612 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{175}-\frac {7919 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1750}+\frac {212873 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{183750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {56041 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{30625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {9 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
| risch | \(\frac {\left (2250 x^{2}+6120 x +7919\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{1750 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {212873 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{192500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {168123 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{96250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(251\) |
-1/8750*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(54428*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))-56041*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))+337500*x^5+1176750*x^4+1812900*x^ 3+628985*x^2-460765*x-237570)/(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 \[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=-\frac {1}{1750} \, {\left (2250 \, x^{2} + 6120 \, x + 7919\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {476038}{196875} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {56041}{8750} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
-1/1750*(2250*x^2 + 6120*x + 7919)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 476038/196875*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 56041/8750*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, we ierstrassPInverse(1159/675, 38998/91125, x + 23/90))
\[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\int \frac {\left (3 x + 2\right )^{\frac {5}{2}} \sqrt {5 x + 3}}{\sqrt {1 - 2 x}}\, dx \]
\[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \]
\[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}}{\sqrt {1-2\,x}} \,d x \]