Integrand size = 24, antiderivative size = 152 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx=-\frac {667615}{195657 (1-2 x)^{3/2}}-\frac {7554245}{5021863 \sqrt {1-2 x}}-\frac {505}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {32765}{1694 (1-2 x)^{3/2} (3+5 x)}+\frac {17820}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {738625 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]
-667615/195657/(1-2*x)^(3/2)-505/154/(1-2*x)^(3/2)/(3+5*x)^2+3/7/(1-2*x)^( 3/2)/(2+3*x)/(3+5*x)^2+32765/1694/(1-2*x)^(3/2)/(3+5*x)+17820/2401*arctanh (1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-738625/161051*arctanh(1/11*55^(1/2)* (1-2*x)^(1/2))*55^(1/2)-7554245/5021863/(1-2*x)^(1/2)
Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {645558882-479695050 x-4110847595 x^2+1580768100 x^3+6798820500 x^4}{30131178 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {17820}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {738625 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]
(645558882 - 479695050*x - 4110847595*x^2 + 1580768100*x^3 + 6798820500*x^ 4)/(30131178*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^2) + (17820*Sqrt[3/7]*Arc Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (738625*Sqrt[5/11]*ArcTanh[Sqrt[5/11] *Sqrt[1 - 2*x]])/14641
Time = 0.26 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {114, 27, 168, 168, 25, 169, 27, 169, 27, 174, 73, 219}
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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{7} \int \frac {5 (4-27 x)}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^3}dx+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \int \frac {4-27 x}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^3}dx+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{7} \left (-\frac {1}{22} \int \frac {38-2121 x}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2}dx-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{22} \left (\frac {1}{11} \int -\frac {98295 x+17614}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx+\frac {6553}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{22} \left (\frac {6553}{11 (1-2 x)^{3/2} (5 x+3)}-\frac {1}{11} \int \frac {98295 x+17614}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx\right )-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{231} \int \frac {3 (245998-2002845 x)}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {267046}{231 (1-2 x)^{3/2}}\right )+\frac {6553}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \int \frac {245998-2002845 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {267046}{231 (1-2 x)^{3/2}}\right )+\frac {6553}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (-\frac {2}{77} \int -\frac {37072034-22662735 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {3021698}{77 \sqrt {1-2 x}}\right )-\frac {267046}{231 (1-2 x)^{3/2}}\right )+\frac {6553}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \int \frac {37072034-22662735 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {3021698}{77 \sqrt {1-2 x}}\right )-\frac {267046}{231 (1-2 x)^{3/2}}\right )+\frac {6553}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (253348375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-156541572 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {3021698}{77 \sqrt {1-2 x}}\right )-\frac {267046}{231 (1-2 x)^{3/2}}\right )+\frac {6553}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (156541572 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-253348375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {3021698}{77 \sqrt {1-2 x}}\right )-\frac {267046}{231 (1-2 x)^{3/2}}\right )+\frac {6553}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (104361048 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-101339350 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {3021698}{77 \sqrt {1-2 x}}\right )-\frac {267046}{231 (1-2 x)^{3/2}}\right )+\frac {6553}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {101}{22 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\) |
3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^2) + (5*(-101/(22*(1 - 2*x)^(3/2) *(3 + 5*x)^2) + (6553/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) + (-267046/(231*(1 - 2*x)^(3/2)) + (-3021698/(77*Sqrt[1 - 2*x]) + (104361048*Sqrt[3/7]*ArcTanh[ Sqrt[3/7]*Sqrt[1 - 2*x]] - 101339350*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77)/77)/11)/22))/7
3.22.98.3.1 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_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
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] && ILtQ[m, -1]
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[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 5.95 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(-\frac {6798820500 x^{4}+1580768100 x^{3}-4110847595 x^{2}-479695050 x +645558882}{30131178 \left (-1+2 x \right ) \sqrt {1-2 x}\, \left (3+5 x \right )^{2} \left (2+3 x \right )}+\frac {17820 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}-\frac {738625 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}\) | \(88\) |
| derivativedivides | \(\frac {-\frac {3125 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {74375 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {738625 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}-\frac {162 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}+\frac {17820 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {32}{195657 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5472}{5021863 \sqrt {1-2 x}}\) | \(100\) |
| default | \(\frac {-\frac {3125 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {74375 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {738625 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}-\frac {162 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}+\frac {17820 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {32}{195657 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5472}{5021863 \sqrt {1-2 x}}\) | \(100\) |
| pseudoelliptic | \(-\frac {445500 \left (-\frac {753152029}{6522565500}+\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{2} \sqrt {21}}{25}-\frac {14187509 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{2} \sqrt {55}}{573985764}-\frac {10575943 x^{4}}{8696754}-\frac {12294863 x^{3}}{43483770}+\frac {5755186633 x^{2}}{7827078600}+\frac {7461923 x}{86967540}\right )}{2401 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right ) \left (3+5 x \right )^{2}}\) | \(124\) |
| trager | \(\frac {\left (6798820500 x^{4}+1580768100 x^{3}-4110847595 x^{2}-479695050 x +645558882\right ) \sqrt {1-2 x}}{30131178 \left (10 x^{2}+x -3\right )^{2} \left (2+3 x \right )}-\frac {8910 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{2401}+\frac {2375 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5319655\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5319655\right ) x +17105 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5319655\right )}{3+5 x}\right )}{322102}\) | \(136\) |
-1/30131178*(6798820500*x^4+1580768100*x^3-4110847595*x^2-479695050*x+6455 58882)/(-1+2*x)/(1-2*x)^(1/2)/(3+5*x)^2/(2+3*x)+17820/2401*arctanh(1/7*21^ (1/2)*(1-2*x)^(1/2))*21^(1/2)-738625/161051*arctanh(1/11*55^(1/2)*(1-2*x)^ (1/2))*55^(1/2)
Time = 0.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {5320315875 \, \sqrt {11} \sqrt {5} {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 8609786460 \, \sqrt {7} \sqrt {3} {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (6798820500 \, x^{4} + 1580768100 \, x^{3} - 4110847595 \, x^{2} - 479695050 \, x + 645558882\right )} \sqrt {-2 \, x + 1}}{2320100706 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )}} \]
1/2320100706*(5320315875*sqrt(11)*sqrt(5)*(300*x^5 + 260*x^4 - 137*x^3 - 1 36*x^2 + 15*x + 18)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 8609786460*sqrt(7)*sqrt(3)*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 7 7*(6798820500*x^4 + 1580768100*x^3 - 4110847595*x^2 - 479695050*x + 645558 882)*sqrt(-2*x + 1))/(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)
Result contains complex when optimal does not.
Time = 12.75 (sec) , antiderivative size = 2966, normalized size of antiderivative = 19.51 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx=\text {Too large to display} \]
376926608520000*sqrt(2)*I*(x - 1/2)**(17/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705 295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592 *(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 2135605689756000*sqrt(2)* I*(x - 1/2)**(15/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 10592025 35612298*(x - 1/2)**3) + 4838544255837600*sqrt(2)*I*(x - 1/2)**(13/2)/(501 141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 96536606255812 80*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 5479255948116720*sqrt(2)*I*(x - 1/2)**(11/2)/(501141752496000*(x - 1/2)** 9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 145827 05295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 56123978510365 92*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 3100767153386980*sqrt(2 )*I*(x - 1/2)**(9/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202 535612298*(x - 1/2)**3) + 700998571871598*sqrt(2)*I*(x - 1/2)**(7/2)/(5011 41752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 965366062558...
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {738625}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8910}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1699705125 \, {\left (2 \, x - 1\right )}^{4} + 7589204550 \, {\left (2 \, x - 1\right )}^{3} + 8458535305 \, {\left (2 \, x - 1\right )}^{2} - 22225280 \, x + 13199648}{15065589 \, {\left (75 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 505 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 1133 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 847 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
738625/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sq rt(-2*x + 1))) - 8910/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sq rt(21) + 3*sqrt(-2*x + 1))) - 1/15065589*(1699705125*(2*x - 1)^4 + 7589204 550*(2*x - 1)^3 + 8458535305*(2*x - 1)^2 - 22225280*x + 13199648)/(75*(-2* x + 1)^(9/2) - 505*(-2*x + 1)^(7/2) + 1133*(-2*x + 1)^(5/2) - 847*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {738625}{322102} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {8910}{2401} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {64 \, {\left (513 \, x - 295\right )}}{15065589 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {243 \, \sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 2\right )}} - \frac {625 \, {\left (55 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 119 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \]
738625/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt( 55) + 5*sqrt(-2*x + 1))) - 8910/2401*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6* sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/15065589*(513*x - 295) /((2*x - 1)*sqrt(-2*x + 1)) + 243/343*sqrt(-2*x + 1)/(3*x + 2) - 625/5324* (55*(-2*x + 1)^(3/2) - 119*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 1.43 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {\frac {153791551\,{\left (2\,x-1\right )}^2}{20543985}-\frac {5248\,x}{266805}+\frac {33729798\,{\left (2\,x-1\right )}^3}{5021863}+\frac {7554245\,{\left (2\,x-1\right )}^4}{5021863}+\frac {15584}{1334025}}{\frac {847\,{\left (1-2\,x\right )}^{3/2}}{75}-\frac {1133\,{\left (1-2\,x\right )}^{5/2}}{75}+\frac {101\,{\left (1-2\,x\right )}^{7/2}}{15}-{\left (1-2\,x\right )}^{9/2}}+\frac {17820\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {738625\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051} \]
((153791551*(2*x - 1)^2)/20543985 - (5248*x)/266805 + (33729798*(2*x - 1)^ 3)/5021863 + (7554245*(2*x - 1)^4)/5021863 + 15584/1334025)/((847*(1 - 2*x )^(3/2))/75 - (1133*(1 - 2*x)^(5/2))/75 + (101*(1 - 2*x)^(7/2))/15 - (1 - 2*x)^(9/2)) + (17820*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (738625*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/161051