Integrand size = 24, antiderivative size = 139 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx=-\frac {224967}{65219 \sqrt {1-2 x}}-\frac {505}{154 \sqrt {1-2 x} (3+5 x)^2}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^2}+\frac {33115}{1694 \sqrt {1-2 x} (3+5 x)}+\frac {5832}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {153825 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
5832/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-153825/14641*arctanh (1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-224967/65219/(1-2*x)^(1/2)-505/154/ (3+5*x)^2/(1-2*x)^(1/2)+3/7/(2+3*x)/(3+5*x)^2/(1-2*x)^(1/2)+33115/1694/(3+ 5*x)/(1-2*x)^(1/2)
Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {5832}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {\frac {11 \left (6400750+8019782 x-24742935 x^2-33745050 x^3\right )}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2}-15074850 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1434818} \]
(5832*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + ((11*(6400750 + 801 9782*x - 24742935*x^2 - 33745050*x^3))/(Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^ 2) - 15074850*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1434818
Time = 0.24 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {114, 168, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{7} \int \frac {38-105 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^3}dx+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{7} \left (-\frac {1}{22} \int \frac {2078-7575 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}dx-\frac {505}{22 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{22} \left (\frac {1}{11} \int \frac {3 (12178-99345 x)}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {33115}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {505}{22 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{22} \left (\frac {3}{11} \int \frac {12178-99345 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {33115}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {505}{22 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{22} \left (\frac {3}{11} \left (-\frac {2}{77} \int -\frac {1837574-1124835 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {149978}{77 \sqrt {1-2 x}}\right )+\frac {33115}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {505}{22 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{22} \left (\frac {3}{11} \left (\frac {1}{77} \int \frac {1837574-1124835 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {149978}{77 \sqrt {1-2 x}}\right )+\frac {33115}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {505}{22 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{22} \left (\frac {3}{11} \left (\frac {1}{77} \left (12562375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-7762392 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {149978}{77 \sqrt {1-2 x}}\right )+\frac {33115}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {505}{22 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{22} \left (\frac {3}{11} \left (\frac {1}{77} \left (7762392 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-12562375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {149978}{77 \sqrt {1-2 x}}\right )+\frac {33115}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {505}{22 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{22} \left (\frac {3}{11} \left (\frac {1}{77} \left (5174928 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-5024950 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {149978}{77 \sqrt {1-2 x}}\right )+\frac {33115}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {505}{22 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}\) |
3/(7*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^2) + (-505/(22*Sqrt[1 - 2*x]*(3 + 5 *x)^2) + (33115/(11*Sqrt[1 - 2*x]*(3 + 5*x)) + (3*(-149978/(77*Sqrt[1 - 2* x]) + (5174928*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 5024950*Sqrt[5 /11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77))/11)/22)/7
3.22.33.3.1 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 = 3.55 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.55
| method | result | size |
| risch | \(-\frac {33745050 x^{3}+24742935 x^{2}-8019782 x -6400750}{130438 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (2+3 x \right )}+\frac {5832 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}-\frac {153825 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}\) | \(76\) |
| derivativedivides | \(\frac {-\frac {78125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {15375 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {153825 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}-\frac {54 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}+\frac {5832 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {32}{65219 \sqrt {1-2 x}}\) | \(91\) |
| default | \(\frac {-\frac {78125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {15375 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {153825 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}-\frac {54 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}+\frac {5832 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {32}{65219 \sqrt {1-2 x}}\) | \(91\) |
| pseudoelliptic | \(\frac {\frac {3200375}{65219}+\frac {5832 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {21}}{343}-\frac {153825 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {55}}{14641}-\frac {16872525 x^{3}}{65219}-\frac {3534705 x^{2}}{18634}+\frac {4009891 x}{65219}}{\left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {1-2 x}}\) | \(113\) |
| trager | \(\frac {\left (33745050 x^{3}+24742935 x^{2}-8019782 x -6400750\right ) \sqrt {1-2 x}}{130438 \left (3+5 x \right )^{2} \left (6 x^{2}+x -2\right )}-\frac {2916 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{343}-\frac {525 \operatorname {RootOf}\left (\textit {\_Z}^{2}-4721695\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-4721695\right ) x +16115 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-4721695\right )}{3+5 x}\right )}{29282}\) | \(131\) |
-1/130438*(33745050*x^3+24742935*x^2-8019782*x-6400750)/(3+5*x)^2/(1-2*x)^ (1/2)/(2+3*x)+5832/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-153825 /14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {52761975 \, \sqrt {11} \sqrt {5} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 85386312 \, \sqrt {7} \sqrt {3} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (33745050 \, x^{3} + 24742935 \, x^{2} - 8019782 \, x - 6400750\right )} \sqrt {-2 \, x + 1}}{10043726 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \]
1/10043726*(52761975*sqrt(11)*sqrt(5)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 85386312 *sqrt(7)*sqrt(3)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*log(-(sqrt(7)*sq rt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(33745050*x^3 + 24742935*x ^2 - 8019782*x - 6400750)*sqrt(-2*x + 1))/(150*x^4 + 205*x^3 + 34*x^2 - 51 *x - 18)
Result contains complex when optimal does not.
Time = 11.04 (sec) , antiderivative size = 2222, normalized size of antiderivative = 15.99 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx=\text {Too large to display} \]
1039347540000*sqrt(2)*I*(x - 1/2)**(13/2)/(602623560000*(x - 1/2)**7 + 335 4604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 46076 68296000*sqrt(2)*I*(x - 1/2)**(11/2)/(602623560000*(x - 1/2)**7 + 33546044 84000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)* *4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 7658622448 400*sqrt(2)*I*(x - 1/2)**(9/2)/(602623560000*(x - 1/2)**7 + 3354604484000* (x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4 625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 5656411074160*sq rt(2)*I*(x - 1/2)**(7/2)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1 /2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396 959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 1565987216794*sqrt(2)* I*(x - 1/2)**(5/2)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876 *(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) - 252527968*sqrt(2)*I*(x - 1/2 )**(3/2)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514 653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2) **3 + 1029351346562*(x - 1/2)**2) - 6331437000000*sqrt(55)*I*(x - 1/2)**7* atan(sqrt(110)*sqrt(x - 1/2)/11)/(602623560000*(x - 1/2)**7 + 335460448400 0*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**...
Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {153825}{29282} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2916}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {16872525 \, {\left (2 \, x - 1\right )}^{3} + 75360510 \, {\left (2 \, x - 1\right )}^{2} + 168127762 \, x - 84090985}{65219 \, {\left (75 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 505 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 1133 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 847 \, \sqrt {-2 \, x + 1}\right )}} \]
153825/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr t(-2*x + 1))) - 2916/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt (21) + 3*sqrt(-2*x + 1))) + 1/65219*(16872525*(2*x - 1)^3 + 75360510*(2*x - 1)^2 + 168127762*x - 84090985)/(75*(-2*x + 1)^(7/2) - 505*(-2*x + 1)^(5/ 2) + 1133*(-2*x + 1)^(3/2) - 847*sqrt(-2*x + 1))
Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {153825}{29282} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2916}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (215526 \, x - 107875\right )}}{65219 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} - \frac {125 \, {\left (625 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1353 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \]
153825/29282*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(5 5) + 5*sqrt(-2*x + 1))) - 2916/343*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sq rt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/65219*(215526*x - 107875) /(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1)) - 125/5324*(625*(-2*x + 1)^(3/2) - 1353*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {5832\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {15284342\,x}{444675}+\frac {5024034\,{\left (2\,x-1\right )}^2}{326095}+\frac {224967\,{\left (2\,x-1\right )}^3}{65219}-\frac {1528927}{88935}}{\frac {847\,\sqrt {1-2\,x}}{75}-\frac {1133\,{\left (1-2\,x\right )}^{3/2}}{75}+\frac {101\,{\left (1-2\,x\right )}^{5/2}}{15}-{\left (1-2\,x\right )}^{7/2}}-\frac {153825\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641} \]
(5832*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - ((15284342*x)/44 4675 + (5024034*(2*x - 1)^2)/326095 + (224967*(2*x - 1)^3)/65219 - 1528927 /88935)/((847*(1 - 2*x)^(1/2))/75 - (1133*(1 - 2*x)^(3/2))/75 + (101*(1 - 2*x)^(5/2))/15 - (1 - 2*x)^(7/2)) - (153825*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641