Integrand size = 24, antiderivative size = 166 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx=-\frac {15987390}{456533 \sqrt {1-2 x}}-\frac {35825}{1078 \sqrt {1-2 x} (3+5 x)^2}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac {435}{98 \sqrt {1-2 x} (2+3 x) (3+5 x)^2}+\frac {1176400}{5929 \sqrt {1-2 x} (3+5 x)}+\frac {414315}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1561125 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
414315/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1561125/14641*arc tanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-15987390/456533/(1-2*x)^(1/2)-3 5825/1078/(3+5*x)^2/(1-2*x)^(1/2)+3/14/(2+3*x)^2/(3+5*x)^2/(1-2*x)^(1/2)+4 35/98/(2+3*x)/(3+5*x)^2/(1-2*x)^(1/2)+1176400/5929/(3+5*x)/(1-2*x)^(1/2)
Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {909821467+2503057145 x-1810042755 x^2-10073172600 x^3-7194325500 x^4}{913066 \sqrt {1-2 x} \left (6+19 x+15 x^2\right )^2}+\frac {414315}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1561125 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
(909821467 + 2503057145*x - 1810042755*x^2 - 10073172600*x^3 - 7194325500* x^4)/(913066*Sqrt[1 - 2*x]*(6 + 19*x + 15*x^2)^2) + (414315*Sqrt[3/7]*ArcT anh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1561125*Sqrt[5/11]*ArcTanh[Sqrt[5/11] *Sqrt[1 - 2*x]])/1331
Time = 0.27 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13, 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, 27, 168, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{14} \int \frac {5 (11-27 x)}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^3}dx+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \int \frac {11-27 x}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^3}dx+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \int \frac {1039-3045 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^3}dx+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (-\frac {1}{22} \int \frac {2 (29627-107475 x)}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}dx-\frac {7165}{11 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (-\frac {1}{11} \int \frac {29627-107475 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}dx-\frac {7165}{11 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (\frac {1}{11} \int \frac {3 (172927-1411680 x)}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {470560}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {7165}{11 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (\frac {3}{11} \int \frac {172927-1411680 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {470560}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {7165}{11 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (\frac {3}{11} \left (-\frac {2}{77} \int -\frac {26105291-15987390 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {2131652}{77 \sqrt {1-2 x}}\right )+\frac {470560}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {7165}{11 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (\frac {3}{11} \left (\frac {1}{77} \int \frac {26105291-15987390 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {2131652}{77 \sqrt {1-2 x}}\right )+\frac {470560}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {7165}{11 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (\frac {3}{11} \left (\frac {1}{77} \left (178488625 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-110290653 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {2131652}{77 \sqrt {1-2 x}}\right )+\frac {470560}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {7165}{11 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (\frac {3}{11} \left (\frac {1}{77} \left (110290653 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-178488625 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {2131652}{77 \sqrt {1-2 x}}\right )+\frac {470560}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {7165}{11 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (\frac {3}{11} \left (\frac {1}{77} \left (73527102 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-71395450 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {2131652}{77 \sqrt {1-2 x}}\right )+\frac {470560}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {7165}{11 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {87}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}\) |
3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^2) + (5*(87/(7*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^2) + (-7165/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (470560/(11* Sqrt[1 - 2*x]*(3 + 5*x)) + (3*(-2131652/(77*Sqrt[1 - 2*x]) + (73527102*Sqr t[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 71395450*Sqrt[5/11]*ArcTanh[Sqrt [5/11]*Sqrt[1 - 2*x]])/77))/11)/11)/7))/14
3.22.34.3.1 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_), 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 = 1.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(-\frac {7194325500 x^{4}+10073172600 x^{3}+1810042755 x^{2}-2503057145 x -909821467}{913066 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}}+\frac {414315 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}-\frac {1561125 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}\) | \(79\) |
| derivativedivides | \(\frac {-\frac {596875 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {118125 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {1561125 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}-\frac {8748 \left (\frac {217 \left (1-2 x \right )^{\frac {3}{2}}}{36}-\frac {511 \sqrt {1-2 x}}{36}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {414315 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {64}{456533 \sqrt {1-2 x}}\) | \(103\) |
| default | \(\frac {-\frac {596875 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {118125 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {1561125 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}-\frac {8748 \left (\frac {217 \left (1-2 x \right )^{\frac {3}{2}}}{36}-\frac {511 \sqrt {1-2 x}}{36}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {414315 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {64}{456533 \sqrt {1-2 x}}\) | \(103\) |
| pseudoelliptic | \(-\frac {14050125 \left (-\frac {10008036137}{9638385750}-\frac {44933229 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {21}}{249884075}+\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {55}}{9}+\frac {11724086 x^{4}}{1427909}+\frac {246233108 x^{3}}{21418635}+\frac {1327364687 x^{2}}{642559050}-\frac {30423899 x}{10650150}\right )}{14641 \sqrt {1-2 x}\, \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(118\) |
| trager | \(\frac {\left (7194325500 x^{4}+10073172600 x^{3}+1810042755 x^{2}-2503057145 x -909821467\right ) \sqrt {1-2 x}}{913066 \left (15 x^{2}+19 x +6\right )^{2} \left (-1+2 x \right )}-\frac {8625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1801855\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1801855\right ) x +9955 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1801855\right )}{3+5 x}\right )}{29282}-\frac {414315 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{4802}\) | \(138\) |
-1/913066*(7194325500*x^4+10073172600*x^3+1810042755*x^2-2503057145*x-9098 21467)/(15*x^2+19*x+6)^2/(1-2*x)^(1/2)+414315/2401*arctanh(1/7*21^(1/2)*(1 -2*x)^(1/2))*21^(1/2)-1561125/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*5 5^(1/2)
Time = 0.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {3748261125 \, \sqrt {11} \sqrt {5} {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 6065985915 \, \sqrt {7} \sqrt {3} {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (7194325500 \, x^{4} + 10073172600 \, x^{3} + 1810042755 \, x^{2} - 2503057145 \, x - 909821467\right )} \sqrt {-2 \, x + 1}}{70306082 \, {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )}} \]
1/70306082*(3748261125*sqrt(11)*sqrt(5)*(450*x^5 + 915*x^4 + 512*x^3 - 85* x^2 - 156*x - 36)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3 )) + 6065985915*sqrt(7)*sqrt(3)*(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 15 6*x - 36)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77* (7194325500*x^4 + 10073172600*x^3 + 1810042755*x^2 - 2503057145*x - 909821 467)*sqrt(-2*x + 1))/(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)
Result contains complex when optimal does not.
Time = 13.98 (sec) , antiderivative size = 5216, normalized size of antiderivative = 31.42 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx=\text {Too large to display} \]
-97154928360000000*sqrt(55)*I*(x - 1/2)**(21/2)*atan(sqrt(110)*sqrt(x - 1/ 2)/11)/(911166822720000*(x - 1/2)**(21/2) + 8261245859328000*(x - 1/2)**(1 9/2) + 32765558945011200*(x - 1/2)**(17/2) + 74250241951226880*(x - 1/2)** (15/2) + 105148838074841792*(x - 1/2)**(13/2) + 95287810504074496*(x - 1/2 )**(11/2) + 53963055273603168*(x - 1/2)**(9/2) + 17460793314336064*(x - 1/ 2)**(7/2) + 2471472583095362*(x - 1/2)**(5/2)) + 157230354916800000*sqrt(2 1)*I*(x - 1/2)**(21/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(911166822720000*(x - 1/2)**(21/2) + 8261245859328000*(x - 1/2)**(19/2) + 32765558945011200*(x - 1/2)**(17/2) + 74250241951226880*(x - 1/2)**(15/2) + 105148838074841792 *(x - 1/2)**(13/2) + 95287810504074496*(x - 1/2)**(11/2) + 539630552736031 68*(x - 1/2)**(9/2) + 17460793314336064*(x - 1/2)**(7/2) + 247147258309536 2*(x - 1/2)**(5/2)) - 78615177458400000*sqrt(21)*I*pi*(x - 1/2)**(21/2)/(9 11166822720000*(x - 1/2)**(21/2) + 8261245859328000*(x - 1/2)**(19/2) + 32 765558945011200*(x - 1/2)**(17/2) + 74250241951226880*(x - 1/2)**(15/2) + 105148838074841792*(x - 1/2)**(13/2) + 95287810504074496*(x - 1/2)**(11/2) + 53963055273603168*(x - 1/2)**(9/2) + 17460793314336064*(x - 1/2)**(7/2) + 2471472583095362*(x - 1/2)**(5/2)) + 48577464180000000*sqrt(55)*I*pi*(x - 1/2)**(21/2)/(911166822720000*(x - 1/2)**(21/2) + 8261245859328000*(x - 1/2)**(19/2) + 32765558945011200*(x - 1/2)**(17/2) + 74250241951226880*(x - 1/2)**(15/2) + 105148838074841792*(x - 1/2)**(13/2) + 95287810504074...
Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {1561125}{29282} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {414315}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (1798581375 \, {\left (2 \, x - 1\right )}^{4} + 12230911800 \, {\left (2 \, x - 1\right )}^{3} + 27711289905 \, {\left (2 \, x - 1\right )}^{2} + 41836111240 \, x - 20918245348\right )}}{456533 \, {\left (225 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 2040 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 6934 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 10472 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 5929 \, \sqrt {-2 \, x + 1}\right )}} \]
1561125/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sq rt(-2*x + 1))) - 414315/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/( sqrt(21) + 3*sqrt(-2*x + 1))) - 2/456533*(1798581375*(2*x - 1)^4 + 1223091 1800*(2*x - 1)^3 + 27711289905*(2*x - 1)^2 + 41836111240*x - 20918245348)/ (225*(-2*x + 1)^(9/2) - 2040*(-2*x + 1)^(7/2) + 6934*(-2*x + 1)^(5/2) - 10 472*(-2*x + 1)^(3/2) + 5929*sqrt(-2*x + 1))
Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {1561125}{29282} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {414315}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {64}{456533 \, \sqrt {-2 \, x + 1}} + \frac {2 \, {\left (256941225 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 1747282440 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 3958787399 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2988341532 \, \sqrt {-2 \, x + 1}\right )}}{65219 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]
1561125/29282*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt( 55) + 5*sqrt(-2*x + 1))) - 414315/4802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/456533/sqrt(-2*x + 1 ) + 2/65219*(256941225*(2*x - 1)^3*sqrt(-2*x + 1) + 1747282440*(2*x - 1)^2 *sqrt(-2*x + 1) - 3958787399*(-2*x + 1)^(3/2) + 2988341532*sqrt(-2*x + 1)) /(15*(2*x - 1)^2 + 136*x + 9)^2
Time = 1.62 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {414315\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {1561125\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641}-\frac {\frac {217330448\,x}{266805}+\frac {3694838654\,{\left (2\,x-1\right )}^2}{6847995}+\frac {108719216\,{\left (2\,x-1\right )}^3}{456533}+\frac {15987390\,{\left (2\,x-1\right )}^4}{456533}-\frac {543331048}{1334025}}{\frac {5929\,\sqrt {1-2\,x}}{225}-\frac {10472\,{\left (1-2\,x\right )}^{3/2}}{225}+\frac {6934\,{\left (1-2\,x\right )}^{5/2}}{225}-\frac {136\,{\left (1-2\,x\right )}^{7/2}}{15}+{\left (1-2\,x\right )}^{9/2}} \]
(414315*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (1561125*55^( 1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641 - ((217330448*x)/266805 + (3694838654*(2*x - 1)^2)/6847995 + (108719216*(2*x - 1)^3)/456533 + (1598 7390*(2*x - 1)^4)/456533 - 543331048/1334025)/((5929*(1 - 2*x)^(1/2))/225 - (10472*(1 - 2*x)^(3/2))/225 + (6934*(1 - 2*x)^(5/2))/225 - (136*(1 - 2*x )^(7/2))/15 + (1 - 2*x)^(9/2))