Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {1232 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {35948 \sqrt {1-2 x}}{45 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16016 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {96808 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {96808}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2912}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
14/15*(1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)-96808/15*EllipticE(1/7*21^ (1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2912/15*EllipticF(1/7*21^(1/ 2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+1232/45*(1-2*x)^(1/2)/(2+3*x)^( 3/2)/(3+5*x)^(3/2)+35948/45*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)-1601 6/3*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+96808/3*(1-2*x)^(1/2)*(2+3*x )^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.41 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.46 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {2}{15} \left (\frac {\sqrt {1-2 x} \left (5512543+34450018 x+80662602 x^2+83867940 x^3+32672700 x^4\right )}{(2+3 x)^{5/2} (3+5 x)^{3/2}}+4 i \sqrt {33} \left (12101 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-12465 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \]
(2*((Sqrt[1 - 2*x]*(5512543 + 34450018*x + 80662602*x^2 + 83867940*x^3 + 3 2672700*x^4))/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (4*I)*Sqrt[33]*(12101*El lipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 12465*EllipticF[I*ArcSinh[Sqrt [9 + 15*x]], -2/33])))/15
Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {109, 27, 167, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{7/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{15} \int \frac {33 (6-5 x) \sqrt {1-2 x}}{(3 x+2)^{5/2} (5 x+3)^{5/2}}dx+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{5} \int \frac {(6-5 x) \sqrt {1-2 x}}{(3 x+2)^{5/2} (5 x+3)^{5/2}}dx+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {22}{5} \left (\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac {2}{9} \int -\frac {993-1370 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \int \frac {993-1370 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {105 (711-817 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {1634 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \left (30 \int \frac {711-817 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {1634 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \left (30 \left (-\frac {2}{33} \int \frac {33 (1765-1092 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {364 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {1634 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \left (30 \left (-\int \frac {1765-1092 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {364 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {1634 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \left (30 \left (\frac {2}{11} \int \frac {3 (12101 x+7661)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {24202 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}-\frac {364 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {1634 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \left (30 \left (\frac {6}{11} \int \frac {12101 x+7661}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {24202 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}-\frac {364 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {1634 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \left (30 \left (\frac {6}{11} \left (\frac {2002}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {12101}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {24202 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}-\frac {364 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {1634 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \left (30 \left (\frac {6}{11} \left (\frac {2002}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {12101}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {24202 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}-\frac {364 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {1634 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {22}{5} \left (\frac {1}{9} \left (30 \left (\frac {6}{11} \left (-\frac {364}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {12101}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {24202 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}-\frac {364 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {1634 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {56 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (22*((56*Sqrt[ 1 - 2*x])/(9*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + ((1634*Sqrt[1 - 2*x])/(Sqr t[2 + 3*x]*(3 + 5*x)^(3/2)) + 30*((-364*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3 + 5*x)^(3/2) + (24202*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + (6*( (-12101*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (364*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/11) )/9))/5
3.29.13.3.1 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*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 1.33 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{135 \left (\frac {2}{3}+x \right )^{3}}+\frac {728 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15 \left (\frac {2}{3}+x \right )^{2}}+\frac {-110676 x^{2}-\frac {55338}{5} x +\frac {166014}{5}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {122576 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{105 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {193616 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{105 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15 \left (x +\frac {3}{5}\right )^{2}}+\frac {-82940 x^{2}-\frac {41470}{3} x +\frac {82940}{3}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(295\) |
| default | \(\frac {2 \sqrt {1-2 x}\, \left (15247260 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-14808420 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+29478036 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-28629612 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+18974368 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-18428256 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+4065936 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-3948912 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+457417800 x^{5}+945442260 x^{4}+542200848 x^{3}-82337962 x^{2}-163974524 x -38587801\right )}{105 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(406\) |
(-(-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 )*(98/135*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+728/15*(-30*x^3-23*x^2+7* x+6)^(1/2)/(2/3+x)^2+18446/5*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/ 2)+122576/105*(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)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+193616/105*(10+1 5*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)))-242/15*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x+3/5)^2+8294/3*(-30 *x^2-5*x+10)/((x+3/5)*(-30*x^2-5*x+10))^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (32672700 \, x^{4} + 83867940 \, x^{3} + 80662602 \, x^{2} + 34450018 \, x + 5512543\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 822334 \, \sqrt {-30} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2178180 \, \sqrt {-30} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{675 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
2/675*(45*(32672700*x^4 + 83867940*x^3 + 80662602*x^2 + 34450018*x + 55125 43)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 822334*sqrt(-30)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*weierstrassPInverse(1159/6 75, 38998/91125, x + 23/90) + 2178180*sqrt(-30)*(675*x^5 + 2160*x^4 + 2763 *x^3 + 1766*x^2 + 564*x + 72)*weierstrassZeta(1159/675, 38998/91125, weier strassPInverse(1159/675, 38998/91125, x + 23/90)))/(675*x^5 + 2160*x^4 + 2 763*x^3 + 1766*x^2 + 564*x + 72)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]