Integrand size = 28, antiderivative size = 222 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}+\frac {1400888 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{132055} \]
46585232/1452605*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^ (1/2)+1400888/1452605*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2) )*33^(1/2)+4/77/(2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)+138/2695*(1-2*x) ^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2)+14928/18865*(1-2*x)^(1/2)/(2+3*x)^(3/2) /(3+5*x)^(1/2)+2101332/132055*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-46 585232/290521*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.52 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.46 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (\frac {-884250959-2283681406 x+1919527182 x^2+9225477612 x^3+6289006320 x^4}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-4 i \sqrt {33} \left (5823154 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5998265 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{1452605} \]
(2*((-884250959 - 2283681406*x + 1919527182*x^2 + 9225477612*x^3 + 6289006 320*x^4)/(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) - (4*I)*Sqrt[33]*(5 823154*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 5998265*EllipticF[I*A rcSinh[Sqrt[9 + 15*x]], -2/33])))/1452605
Time = 0.30 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {115, 27, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^{7/2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}-\frac {2}{77} \int -\frac {210 x+163}{2 \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \int \frac {210 x+163}{\sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \int \frac {2582-1725 x}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}dx+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {2}{21} \int \frac {3 (100471-111960 x)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {7464 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \int \frac {100471-111960 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {7464 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {2}{7} \int \frac {5 (849431-525333 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {1050666 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {7464 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \int \frac {849431-525333 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {1050666 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {7464 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {2}{11} \int \frac {3 (11646308 x+7373029)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {11646308 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {1050666 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {7464 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \int \frac {11646308 x+7373029}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {11646308 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {1050666 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {7464 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \left (\frac {1926221}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {11646308}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {11646308 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {1050666 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {7464 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \left (\frac {1926221}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {11646308}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {11646308 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {1050666 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {7464 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \left (-\frac {350222}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {11646308}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {11646308 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {1050666 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {7464 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + ((138*Sqrt[1 - 2*x])/ (35*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (2*((7464*Sqrt[1 - 2*x])/(7*(2 + 3*x) ^(3/2)*Sqrt[3 + 5*x]) + ((1050666*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (10*((-11646308*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) - (3*((-11646308*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3 3])/5 - (350222*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3 3])/5))/11))/7)/7))/35)/77
3.30.34.3.1 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 = 1.37 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (-\frac {7503029}{2905210}+\frac {1500641 x}{290521}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{245 \left (\frac {2}{3}+x \right )^{3}}-\frac {666 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{2}}-\frac {260982 \left (-30 x^{2}-3 x +9\right )}{12005 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {58984232 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{10168235 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {93170464 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{10168235 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(271\) |
| default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (209633544 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-203598252 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+279511392 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-271464336 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+93170464 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-90488112 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+6289006320 x^{4}+9225477612 x^{3}+1919527182 x^{2}-2283681406 x -884250959\right )}{1452605 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\) | \(314\) |
(-(-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 )*(-2*(-20-30*x)*(-7503029/2905210+1500641/290521*x)/((-3/10+x^2+1/10*x)*( -20-30*x))^(1/2)-2/245*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3-666/1715*(-3 0*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2-260982/12005*(-30*x^2-3*x+9)/((2/3+x)* (-30*x^2-3*x+9))^(1/2)-58984232/10168235*(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))-93170464/10168235*(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 = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (45 \, {\left (6289006320 \, x^{4} + 9225477612 \, x^{3} + 1919527182 \, x^{2} - 2283681406 \, x - 884250959\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 395707526 \, \sqrt {-30} {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 1048167720 \, \sqrt {-30} {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{65367225 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} \]
-2/65367225*(45*(6289006320*x^4 + 9225477612*x^3 + 1919527182*x^2 - 228368 1406*x - 884250959)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 395707526 *sqrt(-30)*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*weierstrass PInverse(1159/675, 38998/91125, x + 23/90) + 1048167720*sqrt(-30)*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*weierstrassZeta(1159/675, 3899 8/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]