Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}+\frac {301304 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}-\frac {301304}{35} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {33232}{35} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-33232/385*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)- 301304/105*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+ 14/15*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)+536/45*(1-2*x)^(1/2)/(2+3* x)^(3/2)/(3+5*x)^(3/2)+111884/315*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2 )-16616/7*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+301304/21*(1-2*x)^(1/2 )*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.76 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {2 \sqrt {1-2 x} \left (17157169+107221804 x+251053266 x^2+261029520 x^3+101690100 x^4\right )}{105 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {8 i \left (414293 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-426755 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{35 \sqrt {33}} \]
(2*Sqrt[1 - 2*x]*(17157169 + 107221804*x + 251053266*x^2 + 261029520*x^3 + 101690100*x^4))/(105*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (((8*I)/35)*(4142 93*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 426755*EllipticF[I*ArcSin h[Sqrt[9 + 15*x]], -2/33]))/Sqrt[33]
Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {109, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^{7/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{15} \int \frac {156-235 x}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}dx+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{15} \left (\frac {2}{21} \int \frac {7 (4857-6700 x)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \int \frac {4857-6700 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {2}{7} \int \frac {15 (24342-27971 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {55942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {30}{7} \int \frac {24342-27971 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {55942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {30}{7} \left (-\frac {2}{33} \int \frac {33 (60427-37386 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {12462 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {55942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {30}{7} \left (-\int \frac {60427-37386 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {12462 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {55942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {30}{7} \left (\frac {2}{11} \int \frac {33 (37663 x+23844)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {75326 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}-\frac {12462 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {55942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {30}{7} \left (6 \int \frac {37663 x+23844}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {75326 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}-\frac {12462 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {55942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {30}{7} \left (6 \left (\frac {6231}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {37663}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {75326 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}-\frac {12462 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {55942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {30}{7} \left (6 \left (\frac {6231}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {37663}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {75326 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}-\frac {12462 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {55942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {30}{7} \left (6 \left (-\frac {4154}{5} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {37663}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {75326 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}-\frac {12462 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}\right )+\frac {55942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {268 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
(14*Sqrt[1 - 2*x])/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (2*((268*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + ((55942*Sqrt[1 - 2*x])/(7*Sq rt[2 + 3*x]*(3 + 5*x)^(3/2)) + (30*((-12462*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/( 3 + 5*x)^(3/2) + (75326*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] + 6*((- 37663*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (4 154*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)))/7)/ 3))/15
3.28.53.3.1 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 = 295, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45 \left (\frac {2}{3}+x \right )^{3}}+\frac {956 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {342908}{7} x^{2}-\frac {171454}{35} x +\frac {514362}{35}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {127168 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{245 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {602608 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{735 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3 \left (x +\frac {3}{5}\right )^{2}}+\frac {-37100 x^{2}-\frac {18550}{3} x +\frac {37100}{3}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(295\) |
| default | \(-\frac {2 \sqrt {1-2 x}\, \left (6584220 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-6779340 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+12729492 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-13106724 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+8193696 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-8436512 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1755792 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1807824 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-203380200 x^{5}-420368940 x^{4}-241077012 x^{3}+36609658 x^{2}+72907466 x +17157169\right )}{105 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(406\) |
(-(-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 )*(14/45*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+956/45*(-30*x^3-23*x^2+7*x +6)^(1/2)/(2/3+x)^2+171454/105*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^( 1/2)+127168/245*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-2 3*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+602608/735*(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)))-22/3*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x+3/5)^2+3710/3*(-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 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (101690100 \, x^{4} + 261029520 \, x^{3} + 251053266 \, x^{2} + 107221804 \, x + 17157169\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 2559422 \, \sqrt {-30} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 6779340 \, \sqrt {-30} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{4725 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
2/4725*(45*(101690100*x^4 + 261029520*x^3 + 251053266*x^2 + 107221804*x + 17157169)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 2559422*sqrt(-30)*( 675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*weierstrassPInverse (1159/675, 38998/91125, x + 23/90) + 6779340*sqrt(-30)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*weierstrassZeta(1159/675, 38998/91125 , weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(675*x^5 + 2160* x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]