Integrand size = 28, antiderivative size = 249 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {19548 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {4115652 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {181551856 \sqrt {1-2 x} \sqrt {2+3 x}}{871563 (3+5 x)^{3/2}}+\frac {12071114168 \sqrt {1-2 x} \sqrt {2+3 x}}{9587193 \sqrt {3+5 x}}-\frac {12071114168 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1452605 \sqrt {33}}-\frac {363103712 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1452605 \sqrt {33}} \]
-12071114168/47935965*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2) )*33^(1/2)-363103712/47935965*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*11 55^(1/2))*33^(1/2)+4/77/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)+138/2695 *(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)+19548/18865*(1-2*x)^(1/2)/(2+3* x)^(3/2)/(3+5*x)^(3/2)+4115652/132055*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^ (1/2)-181551856/871563*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+120711141 68/9587193*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.77 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {687365548973+2920885694212 x+1466692421066 x^2-9658241620704 x^3-16841199826980 x^4-8148002063400 x^5}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+4 i \sqrt {33} \left (1508889271 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1554277235 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{47935965} \]
(2*((687365548973 + 2920885694212*x + 1466692421066*x^2 - 9658241620704*x^ 3 - 16841199826980*x^4 - 8148002063400*x^5)/(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2) *(3 + 5*x)^(3/2)) + (4*I)*Sqrt[33]*(1508889271*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 1554277235*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]) ))/47935965
Time = 0.32 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {115, 27, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^{7/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {2}{77} \int -\frac {270 x+203}{2 \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \int \frac {270 x+203}{\sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}}dx+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \int \frac {3277-2415 x}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}dx+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {2}{21} \int \frac {3 (180071-244350 x)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \int \frac {180071-244350 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {2}{7} \int \frac {5 (2686753-3086739 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {2057826 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \int \frac {2686753-3086739 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {2057826 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {2}{33} \int \frac {220079519-136163892 x}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {45387964 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {2057826 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {1}{33} \int \frac {220079519-136163892 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {45387964 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {2057826 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {3 (1508889271 x+955260323)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {3017778542 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {45387964 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {2057826 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \int \frac {1508889271 x+955260323}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {3017778542 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {45387964 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {2057826 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (\frac {249633802}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1508889271}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {3017778542 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {45387964 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {2057826 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (\frac {249633802}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1508889271}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {3017778542 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {45387964 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {2057826 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (-\frac {45387964}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1508889271}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {3017778542 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {45387964 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {2057826 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {9774 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + ((138*Sqrt[1 - 2*x] )/(35*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (2*((9774*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + ((2057826*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (10*((-45387964*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(33*(3 + 5*x )^(3/2)) + ((3017778542*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + (6*((-1508889271*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/ 33])/5 - (45387964*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3 5/33])/5))/11)/33))/7)/7))/35)/77
3.30.44.3.1 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*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
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 = 4.63 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.30
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{245 \left (\frac {2}{3}+x \right )^{3}}+\frac {3048 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {10178676}{2401} x^{2}-\frac {5089338}{12005} x +\frac {15268014}{12005}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {15284165168 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{335551755 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {24142228336 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{335551755 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{363 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {4412500}{1331} x^{2}-\frac {2206250}{3993} x +\frac {4412500}{3993}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {64 \left (-30 x^{2}-38 x -12\right )}{3195731 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(323\) |
| default | \(\frac {2 \sqrt {1-2 x}\, \left (271600068780 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-263783050740 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+525093466308 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-509980564764 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+337991196704 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-328263352032 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+72426685008 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-70342146864 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+8148002063400 x^{5}+16841199826980 x^{4}+9658241620704 x^{3}-1466692421066 x^{2}-2920885694212 x -687365548973\right )}{47935965 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(406\) |
(-(-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 )*(6/245*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+3048/1715*(-30*x^3-23*x^2+ 7*x+6)^(1/2)/(2/3+x)^2+1696446/12005*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x +9))^(1/2)+15284165168/335551755*(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 ))+24142228336/335551755*(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)))-250/363*(-30*x^3-23*x^2+7*x +6)^(1/2)/(x+3/5)^2+441250/3993*(-30*x^2-5*x+10)/((x+3/5)*(-30*x^2-5*x+10) )^(1/2)-64/3195731*(-30*x^2-38*x-12)/((x-1/2)*(-30*x^2-38*x-12))^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (8148002063400 \, x^{5} + 16841199826980 \, x^{4} + 9658241620704 \, x^{3} - 1466692421066 \, x^{2} - 2920885694212 \, x - 687365548973\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 102537951674 \, \sqrt {-30} {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 271600068780 \, \sqrt {-30} {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{2157118425 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )}} \]
2/2157118425*(45*(8148002063400*x^5 + 16841199826980*x^4 + 9658241620704*x ^3 - 1466692421066*x^2 - 2920885694212*x - 687365548973)*sqrt(5*x + 3)*sqr t(3*x + 2)*sqrt(-2*x + 1) - 102537951674*sqrt(-30)*(1350*x^6 + 3645*x^5 + 3366*x^4 + 769*x^3 - 638*x^2 - 420*x - 72)*weierstrassPInverse(1159/675, 3 8998/91125, x + 23/90) + 271600068780*sqrt(-30)*(1350*x^6 + 3645*x^5 + 336 6*x^4 + 769*x^3 - 638*x^2 - 420*x - 72)*weierstrassZeta(1159/675, 38998/91 125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(1350*x^6 + 3 645*x^5 + 3366*x^4 + 769*x^3 - 638*x^2 - 420*x - 72)
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]