Integrand size = 28, antiderivative size = 185 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}+\frac {5440 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-1088 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-120 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-120/11*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-108 8/3*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+14/9*(1 -2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)+404/9*(1-2*x)^(1/2)/(3+5*x)^(3/2)/ (2+3*x)^(1/2)-300*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+5440/3*(1-2*x) ^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \sqrt {1-2 x} \left (30977+147122 x+232590 x^2+122400 x^3\right )}{3 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {8 i \left (1496 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1541 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{\sqrt {33}} \]
(2*Sqrt[1 - 2*x]*(30977 + 147122*x + 232590*x^2 + 122400*x^3))/(3*(2 + 3*x )^(3/2)*(3 + 5*x)^(3/2)) + ((8*I)*(1496*EllipticE[I*ArcSinh[Sqrt[9 + 15*x] ], -2/33] - 1541*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/Sqrt[33]
Time = 0.27 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {109, 169, 27, 169, 27, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^{5/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{9} \int \frac {123-169 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{9} \left (\frac {2}{7} \int \frac {21 (879-1010 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {202 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \left (3 \int \frac {879-1010 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {202 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{9} \left (3 \left (-\frac {2}{33} \int \frac {33 (1091-675 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {450 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {202 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \left (3 \left (-2 \int \frac {1091-675 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {450 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {202 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{9} \left (3 \left (-2 \left (-\frac {2}{11} \int \frac {33 (1360 x+861)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1360 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )-\frac {450 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {202 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \left (3 \left (-2 \left (-3 \int \frac {1360 x+861}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1360 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )-\frac {450 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {202 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{9} \left (3 \left (-2 \left (-3 \left (45 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+272 \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {1360 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )-\frac {450 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {202 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{9} \left (3 \left (-2 \left (-3 \left (45 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-272 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1360 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )-\frac {450 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {202 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{9} \left (3 \left (-2 \left (-3 \left (-30 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-272 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1360 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )-\frac {450 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {202 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
(14*Sqrt[1 - 2*x])/(9*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (2*((202*Sqrt[1 - 2*x])/(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + 3*((-450*Sqrt[1 - 2*x]*Sqrt[2 + 3 *x])/(3 + 5*x)^(3/2) - 2*((-1360*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x ] - 3*(-272*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] - 30*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])))))/9
3.28.52.3.1 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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 = 1.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.23
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {86}{675}-\frac {136 x}{675}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right )^{2}}-\frac {2 \left (15-30 x \right ) \left (-\frac {1034}{9}-\frac {544 x}{3}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {328 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{5 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2176 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
| default | \(-\frac {2 \sqrt {1-2 x}\, \left (277380 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-285600 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+351348 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-361760 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+110952 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-114240 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-8568000 x^{4}-11997300 x^{3}-2157890 x^{2}+2980880 x +1084195\right )}{105 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(311\) |
(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*((-86/675-136/675*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x^2+19/15*x+2/5)^2-2* (15-30*x)*(-1034/9-544/3*x)/((x^2+19/15*x+2/5)*(15-30*x))^(1/2)+328/5*(10+ 15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*E llipticF((10+15*x)^(1/2),1/35*70^(1/2))+2176/21*(10+15*x)^(1/2)*(21-42*x)^ (1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15* x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (122400 \, x^{3} + 232590 \, x^{2} + 147122 \, x + 30977\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 9242 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 24480 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{135 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
2/135*(45*(122400*x^3 + 232590*x^2 + 147122*x + 30977)*sqrt(5*x + 3)*sqrt( 3*x + 2)*sqrt(-2*x + 1) - 9242*sqrt(-30)*(225*x^4 + 570*x^3 + 541*x^2 + 22 8*x + 36)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 24480*sq rt(-30)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*weierstrassZeta(1159/67 5, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(2 25*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]