Integrand size = 35, antiderivative size = 227 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}-\frac {8953 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1390350 \sqrt {5-2 x}}+\frac {397 \sqrt {\frac {3}{22}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{89125 \sqrt {-5+2 x}}-\frac {14832503 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{287339000 \sqrt {11} \sqrt {-5+2 x}} \]
397/1960750*EllipticF(1/11*33^(1/2)*(1+4*x)^(1/2),1/3*3^(1/2))*66^(1/2)*(5 -2*x)^(1/2)/(-5+2*x)^(1/2)-14832503/3160729000*EllipticPi(2/11*(2-3*x)^(1/ 2)*11^(1/2),55/124,1/2*I*2^(1/2))*(5-2*x)^(1/2)*11^(1/2)/(-5+2*x)^(1/2)-89 53/1390350*EllipticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))*11^(1/2)*( -5+2*x)^(1/2)/(5-2*x)^(1/2)-1/10*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2 )/(7+5*x)^2+8953/556140*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)
Time = 5.77 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=\frac {\sqrt {-5+2 x} \left (\frac {17050 \sqrt {2-3 x} \sqrt {1+4 x} (7057+44765 x)}{(7+5 x)^2}+\frac {\sqrt {11} \left (-61059460 E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )+5759676 \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+44497509 \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{\sqrt {5-2 x}}\right )}{9482187000} \]
(Sqrt[-5 + 2*x]*((17050*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(7057 + 44765*x))/(7 + 5*x)^2 + (Sqrt[11]*(-61059460*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11] ], -1/2] + 5759676*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 4 4497509*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/Sqr t[5 - 2*x]))/9482187000
Time = 0.62 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {178, 25, 2107, 2110, 176, 124, 123, 131, 27, 129, 186, 27, 413, 27, 412}
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.
| \(\displaystyle \int \frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{(5 x+7)^3} \, dx\) |
| \(\Big \downarrow \) 178 |
| \(\displaystyle \frac {1}{20} \int -\frac {72 x^2-140 x+21}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2}dx-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{20} \int \frac {72 x^2-140 x+21}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2}dx-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 2107 |
| \(\displaystyle \frac {1}{20} \left (\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}-\frac {\int \frac {-214872 x^2+199200 x+106729}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{55614}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 2110 |
| \(\displaystyle \frac {1}{20} \left (\frac {\frac {14832503}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx-\int \frac {\frac {2500104}{25}-\frac {214872 x}{5}}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{20} \left (\frac {\frac {185796}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {107436}{5} \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx+\frac {14832503}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {1}{20} \left (\frac {\frac {107436 \sqrt {2 x-5} \int \frac {\sqrt {5-2 x}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx}{5 \sqrt {5-2 x}}+\frac {185796}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {14832503}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{20} \left (\frac {\frac {185796}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {14832503}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx+\frac {17906 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {1}{20} \left (\frac {\frac {185796 \sqrt {\frac {2}{11}} \sqrt {5-2 x} \int \frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{25 \sqrt {2 x-5}}+\frac {14832503}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx+\frac {17906 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{20} \left (\frac {\frac {185796 \sqrt {5-2 x} \int \frac {1}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{25 \sqrt {2 x-5}}+\frac {14832503}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx+\frac {17906 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{20} \left (\frac {\frac {14832503}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx+\frac {61932 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}+\frac {17906 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {1}{20} \left (\frac {-\frac {29665006}{25} \int \frac {3}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}+\frac {61932 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}+\frac {17906 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{20} \left (\frac {-\frac {88995018}{25} \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}+\frac {61932 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}+\frac {17906 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {1}{20} \left (\frac {-\frac {88995018 \sqrt {2 (2-3 x)+11} \int \frac {\sqrt {11}}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{25 \sqrt {11} \sqrt {-2 (2-3 x)-11}}+\frac {61932 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}+\frac {17906 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{20} \left (\frac {-\frac {88995018 \sqrt {2 (2-3 x)+11} \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{25 \sqrt {-2 (2-3 x)-11}}+\frac {61932 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}+\frac {17906 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {1}{20} \left (\frac {\frac {61932 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}+\frac {17906 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}-\frac {44497509 \sqrt {2 (2-3 x)+11} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{775 \sqrt {11} \sqrt {-2 (2-3 x)-11}}}{55614}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{27807 (5 x+7)}\right )-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2}\) |
-1/10*(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(7 + 5*x)^2 + ((8953*Sq rt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(27807*(7 + 5*x)) + ((17906*Sqrt [66]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(5*S qrt[5 - 2*x]) + (61932*Sqrt[6/11]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11 ]*Sqrt[1 + 4*x]], 1/3])/(25*Sqrt[-5 + 2*x]) - (44497509*Sqrt[11 + 2*(2 - 3 *x)]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(775*Sq rt[11]*Sqrt[-11 - 2*(2 - 3*x)]))/55614)/20
3.1.41.3.1 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[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[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x] Int[1/(Sq rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && Simpler Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
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]
Int[((a_.) + (b_.)*(x_))^(m_)*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*( x_)]*Sqrt[(g_.) + (h_.)*(x_)], x_] :> Simp[(a + b*x)^(m + 1)*Sqrt[c + d*x]* Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*(m + 1))), x] - Simp[1/(2*b*(m + 1)) Int[ ((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[d*e*g + c*f*g + c*e*h + 2*(d*f*g + d*e*h + c*f*h)*x + 3*d*f*h*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && LtQ[m, -1]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[( c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sy mbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[ e + f*x]*(Sqrt[g + h*x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] - Simp[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)) Int[((a + b*x)^( m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - (b*B - a*C)*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a ^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B + a^2*C)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2*m] && LtQ[m, -1]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.)*((g_.) + (h_.)*(x_))^(q_.), x_Symbol] :> Simp[PolynomialRem ainder[Px, a + b*x, x] Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^ q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d *x)^n*(e + f*x)^p*(g + h*x)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p , q}, x] && PolyQ[Px, x] && EqQ[m, -1]
Time = 1.64 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.20
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {\sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{10 \left (7+5 x \right )^{2}}+\frac {8953 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{556140 \left (7+5 x \right )}-\frac {104171 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{140193625 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {8953 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{28038725 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {14832503 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{19346720250 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(273\) |
| risch | \(-\frac {\left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (7057+44765 x \right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{556140 \left (7+5 x \right )^{2} \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {104171 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{420580875 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {8953 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{84116175 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {14832503 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \Pi \left (\frac {2 \sqrt {22-33 x}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )}{104304057000 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(311\) |
| default | \(\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (512860900 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}+283138625 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-741625150 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x^{2}+1436010520 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x +792788150 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -2076550420 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x +1005207364 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+554951705 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-1453585294 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )+6795327000 x^{4}-18748451150 x^{3}+2821424375 x^{2}+3768732275 x +446355250\right )}{3517585500 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (7+5 x \right )^{2}}\) | \(434\) |
(-(-2+3*x)*(-5+2*x)*(1+4*x))^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1 /2)*(-1/10/(7+5*x)^2*(-24*x^3+70*x^2-21*x-10)^(1/2)+8953/556140/(7+5*x)*(- 24*x^3+70*x^2-21*x-10)^(1/2)-104171/140193625*(11+44*x)^(1/2)*(22-33*x)^(1 /2)*(110-44*x)^(1/2)/(-24*x^3+70*x^2-21*x-10)^(1/2)*EllipticF(1/11*(11+44* x)^(1/2),3^(1/2))+8953/28038725*(11+44*x)^(1/2)*(22-33*x)^(1/2)*(110-44*x) ^(1/2)/(-24*x^3+70*x^2-21*x-10)^(1/2)*(-11/12*EllipticE(1/11*(11+44*x)^(1/ 2),3^(1/2))+2/3*EllipticF(1/11*(11+44*x)^(1/2),3^(1/2)))+14832503/19346720 250*(11+44*x)^(1/2)*(22-33*x)^(1/2)*(110-44*x)^(1/2)/(-24*x^3+70*x^2-21*x- 10)^(1/2)*EllipticPi(1/11*(11+44*x)^(1/2),-55/23,3^(1/2)))
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3}} \,d x } \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}{\left (5 x + 7\right )^{3}}\, dx \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3}} \,d x } \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^3} \,d x \]