Integrand size = 28, antiderivative size = 218 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {456}{5929 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {5034 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1523260 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 (3+5 x)^{3/2}}+\frac {99425780 \sqrt {1-2 x} \sqrt {2+3 x}}{15065589 \sqrt {3+5 x}}-\frac {19885156 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{456533 \sqrt {33}}-\frac {609304 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{456533 \sqrt {33}} \]
-19885156/15065589*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*3 3^(1/2)-609304/15065589*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/ 2))*33^(1/2)+4/231/(1-2*x)^(3/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)+456/5929/(3+5 *x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)+5034/41503*(1-2*x)^(1/2)/(3+5*x)^(3/ 2)/(2+3*x)^(1/2)-1523260/1369599*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2) +99425780/15065589*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.70 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.47 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {283144937-211488180 x-1802210526 x^2+694871080 x^3+2982773400 x^4}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+2 i \sqrt {33} \left (4971289 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5123615 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{15065589} \]
(2*((283144937 - 211488180*x - 1802210526*x^2 + 694871080*x^3 + 2982773400 *x^4)/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (2*I)*Sqrt[33]*(49 71289*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 5123615*EllipticF[I*Ar cSinh[Sqrt[9 + 15*x]], -2/33])))/15065589
Time = 0.30 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {115, 27, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {2}{231} \int -\frac {3 (70 x+79)}{2 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \int \frac {70 x+79}{(1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {2}{77} \int -\frac {17100 x+12239}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \int \frac {17100 x+12239}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{7} \int \frac {5 (10702-7551 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {5034 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {10}{7} \int \frac {10702-7551 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {5034 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {10}{7} \left (-\frac {2}{33} \int \frac {720071-456978 x}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {152326 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {5034 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {10}{7} \left (-\frac {1}{33} \int \frac {720071-456978 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {152326 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {5034 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {3 (4971289 x+3150332)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {9942578 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {152326 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {5034 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \int \frac {4971289 x+3150332}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {9942578 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {152326 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {5034 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (\frac {837793}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4971289}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {9942578 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {152326 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {5034 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (\frac {837793}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4971289}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {9942578 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {152326 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {5034 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (-\frac {152326}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4971289}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {9942578 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {152326 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {5034 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {456}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
4/(231*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (456/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + ((5034*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3* x]*(3 + 5*x)^(3/2)) + (10*((-152326*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(33*(3 + 5*x)^(3/2)) + ((9942578*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + (6*((-4971289*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] )/5 - (152326*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] )/5))/11)/33))/7)/77)/77
3.30.98.3.1 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 = 256, normalized size of antiderivative = 1.17
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {6101}{889350}+\frac {1229 x}{88935}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (\frac {6580681}{30131178}-\frac {6384815 x}{15065589}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}+\frac {-\frac {4860}{343} x^{2}-\frac {486}{343} x +\frac {1458}{343}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {25202656 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{105459123 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {39770312 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{105459123 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
| default | \(\frac {2 \sqrt {1-2 x}\, \left (99425780 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-96616740 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+9942578 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9661674 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-29827734 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+28985022 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+2982773400 x^{4}+694871080 x^{3}-1802210526 x^{2}-211488180 x +283144937\right )}{15065589 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {2+3 x}}\) | \(311\) |
(-(-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 )*((-6101/889350+1229/88935*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/(-3/10+x^2+1/1 0*x)^2-2*(-20-30*x)*(6580681/30131178-6384815/15065589*x)/((-3/10+x^2+1/10 *x)*(-20-30*x))^(1/2)+162/343*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1 /2)+25202656/105459123*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-3 0*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+3977031 2/105459123*(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*Elliptic F((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 = 148, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (2982773400 \, x^{4} + 694871080 \, x^{3} - 1802210526 \, x^{2} - 211488180 \, x + 283144937\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 169190233 \, \sqrt {-30} {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 447416010 \, \sqrt {-30} {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{677951505 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )}} \]
2/677951505*(45*(2982773400*x^4 + 694871080*x^3 - 1802210526*x^2 - 2114881 80*x + 283144937)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 169190233*s qrt(-30)*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*weierstrassPI nverse(1159/675, 38998/91125, x + 23/90) + 447416010*sqrt(-30)*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*weierstrassZeta(1159/675, 38998/9 1125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(300*x^5 + 2 60*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]