Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx=\frac {30922 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{2625}+\frac {2}{35} (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {408311 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}-\frac {132824 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{590625} \]
-408311/1771875*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^( 1/2)-132824/1771875*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))* 33^(1/2)+326/2625*(1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+2/35*(1-2*x)^( 5/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+30922/118125*(1-2*x)^(1/2)*(2+3*x)^(1/2)* (3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 3.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (26171-28170 x+13500 x^2\right )+408311 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-541135 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{1771875} \]
(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(26171 - 28170*x + 13500*x^2 ) + (408311*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (541 135*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/1771875
Time = 0.24 (sec) , antiderivative size = 175, 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, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} \sqrt {3 x+2}}{\sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{35} (1-2 x)^{5/2} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {2}{35} \int -\frac {(1-2 x)^{3/2} (163 x+111)}{2 \sqrt {3 x+2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \int \frac {(1-2 x)^{3/2} (163 x+111)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2}{35} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{75} \int \frac {\sqrt {1-2 x} (15461 x+11096)}{2 \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {326}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{35} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \left (\frac {1}{75} \int \frac {\sqrt {1-2 x} (15461 x+11096)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {326}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{35} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{35} \left (\frac {1}{75} \left (\frac {2}{45} \int \frac {408311 x+391093}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {30922}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {326}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{35} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \left (\frac {1}{75} \left (\frac {1}{45} \int \frac {408311 x+391093}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {30922}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {326}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{35} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{35} \left (\frac {1}{75} \left (\frac {1}{45} \left (\frac {730532}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {408311}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {30922}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {326}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{35} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{35} \left (\frac {1}{75} \left (\frac {1}{45} \left (\frac {730532}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {408311}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {30922}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {326}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{35} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{35} \left (\frac {1}{75} \left (\frac {1}{45} \left (-\frac {132824}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {408311}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {30922}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {326}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{35} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
(2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/35 + ((326*(1 - 2*x)^(3/2) *Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/75 + ((30922*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqr t[3 + 5*x])/45 + ((-408311*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (132824*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/45)/75)/35
3.28.88.3.1 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 = 0.94
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (510213 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-408311 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-12150000 x^{5}+16038000 x^{4}-1281600 x^{3}-21543690 x^{2}+425310 x +4710780\right )}{1771875 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(150\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {1252 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2625}+\frac {52342 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{118125}+\frac {782186 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{12403125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {816622 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{12403125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
| risch | \(-\frac {2 \left (13500 x^{2}-28170 x +26171\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{118125 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {391093 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{6496875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {408311 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{6496875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(252\) |
-1/1771875*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(510213*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))-408311*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))-12150000*x^5+16038000*x^4-12 81600*x^3-21543690*x^2+425310*x+4710780)/(30*x^3+23*x^2-7*x-6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.37 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx=\frac {2}{118125} \, {\left (13500 \, x^{2} - 28170 \, x + 26171\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {25807217}{159468750} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {408311}{1771875} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
2/118125*(13500*x^2 - 28170*x + 26171)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2 *x + 1) - 25807217/159468750*sqrt(-30)*weierstrassPInverse(1159/675, 38998 /91125, x + 23/90) + 408311/1771875*sqrt(-30)*weierstrassZeta(1159/675, 38 998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90))
\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {3 x + 2}}{\sqrt {5 x + 3}}\, dx \]
\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx=\int { \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx=\int { \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}}{\sqrt {5\,x+3}} \,d x \]