Integrand size = 35, antiderivative size = 281 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3 \, dx=-\frac {1182926269 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1603800}-\frac {12243139 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)}{356400}-\frac {17561 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2}{8910}-\frac {427 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3}{2970}+\frac {2}{55} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^4-\frac {6489123157 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{699840 \sqrt {5-2 x}}+\frac {522167393 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{23328 \sqrt {-5+2 x}} \]
522167393/139968*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)-6489123157/699840*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)-1182926 269/1603800*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)-12243139/356400*(7+ 5*x)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)-17561/8910*(7+5*x)^2*(2-3* x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)-427/2970*(7+5*x)^3*(2-3*x)^(1/2)*(-5 +2*x)^(1/2)*(1+4*x)^(1/2)+2/55*(7+5*x)^4*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4 *x)^(1/2)
Time = 5.06 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.48 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3 \, dx=\frac {24 \sqrt {2-3 x} \sqrt {1+4 x} \left (3325071575-797747975 x-670058262 x^2-167736600 x^3+67338000 x^4+29160000 x^5\right )-71380354727 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )+57438413230 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{15396480 \sqrt {-5+2 x}} \]
(24*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(3325071575 - 797747975*x - 670058262*x^2 - 167736600*x^3 + 67338000*x^4 + 29160000*x^5) - 71380354727*Sqrt[66]*Sqrt [5 - 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3] + 57438413230*S qrt[66]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(1 5396480*Sqrt[-5 + 2*x])
Time = 0.87 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.457, Rules used = {179, 25, 2103, 27, 2103, 27, 2103, 27, 2118, 27, 176, 124, 123, 131, 27, 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.
| \(\displaystyle \int \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3 \, dx\) |
| \(\Big \downarrow \) 179 |
| \(\displaystyle \frac {1}{55} \int -\frac {(5 x+7)^3 \left (-854 x^2+1190 x+3\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4-\frac {1}{55} \int \frac {(5 x+7)^3 \left (-854 x^2+1190 x+3\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\) |
| \(\Big \downarrow \) 2103 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{216} \int -\frac {2 (5 x+7)^2 \left (-983416 x^2+796645 x+193137\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (-\frac {1}{108} \int \frac {(5 x+7)^2 \left (-983416 x^2+796645 x+193137\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 2103 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{168} \int -\frac {56 (5 x+7) \left (-36729417 x^2+11636345 x+10149544\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (-\frac {1}{3} \int \frac {(5 x+7) \left (-36729417 x^2+11636345 x+10149544\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 2103 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (\frac {1}{120} \int -\frac {3 \left (-18926820304 x^2-2853602035 x+5865927653\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (-\frac {1}{40} \int \frac {-18926820304 x^2-2853602035 x+5865927653}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 2118 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (\frac {1}{40} \left (-\frac {1}{108} \int \frac {79860 (15398385-53629117 x)}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {4731705076}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (\frac {1}{40} \left (-\frac {6655}{9} \int \frac {15398385-53629117 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {4731705076}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (\frac {1}{40} \left (-\frac {6655}{9} \left (-\frac {237348815}{2} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {53629117}{2} \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx\right )-\frac {4731705076}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (\frac {1}{40} \left (-\frac {6655}{9} \left (-\frac {53629117 \sqrt {2 x-5} \int \frac {\sqrt {5-2 x}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx}{2 \sqrt {5-2 x}}-\frac {237348815}{2} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\right )-\frac {4731705076}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (\frac {1}{40} \left (-\frac {6655}{9} \left (-\frac {237348815}{2} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {53629117 \sqrt {\frac {11}{6}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{2 \sqrt {5-2 x}}\right )-\frac {4731705076}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (\frac {1}{40} \left (-\frac {6655}{9} \left (-\frac {21577165 \sqrt {\frac {11}{2}} \sqrt {5-2 x} \int \frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{\sqrt {2 x-5}}-\frac {53629117 \sqrt {\frac {11}{6}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{2 \sqrt {5-2 x}}\right )-\frac {4731705076}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (\frac {1}{40} \left (-\frac {6655}{9} \left (-\frac {237348815 \sqrt {5-2 x} \int \frac {1}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{2 \sqrt {2 x-5}}-\frac {53629117 \sqrt {\frac {11}{6}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{2 \sqrt {5-2 x}}\right )-\frac {4731705076}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{55} \left (\frac {1}{108} \left (\frac {1}{3} \left (\frac {1}{40} \left (-\frac {6655}{9} \left (-\frac {21577165 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {53629117 \sqrt {\frac {11}{6}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{2 \sqrt {5-2 x}}\right )-\frac {4731705076}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )-\frac {12243139}{20} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )-\frac {35122}{3} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )-\frac {427}{54} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\right )+\frac {2}{55} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^4\) |
(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^4)/55 + ((-427*Sqr t[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^3)/54 + ((-35122*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^2)/3 + ((-12243139*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x))/20 + ((-4731705076*Sqrt[2 - 3 *x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/9 - (6655*((-53629117*Sqrt[11/6]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(2*Sqrt[5 - 2*x] ) - (21577165*Sqrt[11/6]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/Sqrt[-5 + 2*x]))/9)/40)/3)/108)/55
3.1.35.3.1 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[2*(a + b*x)^(m + 1)*Sqrt[c + d*x ]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*(2*m + 5))), x] + Simp[1/(b*(2*m + 5)) Int[((a + b*x)^m/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[3*b*c*e* g - a*(d*e*g + c*f*g + c*e*h) + 2*(b*(d*e*g + c*f*g + c*e*h) - a*(d*f*g + d *e*h + c*f*h))*x - (3*a*d*f*h - b*(d*f*g + d*e*h + c*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[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[ (c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S ymbol] :> Simp[2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/( d*f*h*(2*m + 3))), x] + Simp[1/(d*f*h*(2*m + 3)) Int[((a + b*x)^(m - 1)/( Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a *(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + ((A*b + a*B)*d*f*h*(2*m + 3) - C* (2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + (b *B*d*f*h*(2*m + 3) + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h)))*x ^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2 *m] && GtQ[m, 0]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]
Time = 1.76 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.55
| method | result | size |
| default | \(-\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (-8398080000 x^{7}-15894144000 x^{6}+57788380800 x^{5}+29554530236 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-71380354727 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+176080611456 x^{4}+141293068560 x^{3}-1085513167176 x^{2}+360716686200 x +159603435600\right )}{15396480 \left (24 x^{3}-70 x^{2}+21 x +10\right )}\) | \(154\) |
| risch | \(-\frac {\left (14580000 x^{4}+70119000 x^{3}+91429200 x^{2}-106456131 x -665014315\right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{641520 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {1026559 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{23328 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {53629117 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{349920 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(262\) |
| elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {11828459 x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{71280}-\frac {133002863 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{128304}-\frac {1026559 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{7776 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {53629117 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{116640 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {126985 x^{2} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{891}+\frac {250 x^{4} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{11}+\frac {64925 x^{3} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{594}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(272\) |
-1/15396480*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(-8398080000*x^7-15 894144000*x^6+57788380800*x^5+29554530236*(1+4*x)^(1/2)*(2-3*x)^(1/2)*22^( 1/2)*(5-2*x)^(1/2)*EllipticF(1/11*(11+44*x)^(1/2),3^(1/2))-71380354727*(1+ 4*x)^(1/2)*(2-3*x)^(1/2)*22^(1/2)*(5-2*x)^(1/2)*EllipticE(1/11*(11+44*x)^( 1/2),3^(1/2))+176080611456*x^4+141293068560*x^3-1085513167176*x^2+36071668 6200*x+159603435600)/(24*x^3-70*x^2+21*x+10)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.25 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3 \, dx=\frac {1}{641520} \, {\left (14580000 \, x^{4} + 70119000 \, x^{3} + 91429200 \, x^{2} - 106456131 \, x - 665014315\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} - \frac {32008789087}{5038848} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) + \frac {6489123157}{699840} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]
1/641520*(14580000*x^4 + 70119000*x^3 + 91429200*x^2 - 106456131*x - 66501 4315)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2) - 32008789087/5038848*sqr t(-6)*weierstrassPInverse(847/108, 6655/2916, x - 35/36) + 6489123157/6998 40*sqrt(-6)*weierstrassZeta(847/108, 6655/2916, weierstrassPInverse(847/10 8, 6655/2916, x - 35/36))
\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3 \, dx=\int \sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \left (5 x + 7\right )^{3}\, dx \]
\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3 \, dx=\int { {\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]
\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3 \, dx=\int { {\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]
Timed out. \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3 \, dx=\int \sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^3 \,d x \]