Integrand size = 24, antiderivative size = 201 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx=-\frac {8836825 \sqrt {1-2 x}}{378 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac {1393 \sqrt {1-2 x}}{108 (2+3 x)^3 (3+5 x)^2}+\frac {11243 \sqrt {1-2 x}}{72 (2+3 x)^2 (3+5 x)^2}+\frac {522385 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^2}+\frac {23680975 \sqrt {1-2 x}}{168 (3+5 x)}+\frac {163363895 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28 \sqrt {21}}-171675 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/12*(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^2+163363895/588*arctanh(1/7*21^(1/2)* (1-2*x)^(1/2))*21^(1/2)-171675*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/ 2)-8836825/378*(1-2*x)^(1/2)/(3+5*x)^2+1393/108*(1-2*x)^(1/2)/(2+3*x)^3/(3 +5*x)^2+11243/72*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2+522385/168*(1-2*x)^(1/2 )/(2+3*x)/(3+5*x)^2+23680975/168*(1-2*x)^(1/2)/(3+5*x)
Time = 0.42 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (359378534+2785562634 x+8630749831 x^2+13362164665 x^3+10337268075 x^4+3196931625 x^5\right )}{56 (2+3 x)^4 (3+5 x)^2}+\frac {163363895 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28 \sqrt {21}}-171675 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(359378534 + 2785562634*x + 8630749831*x^2 + 13362164665*x^ 3 + 10337268075*x^4 + 3196931625*x^5))/(56*(2 + 3*x)^4*(3 + 5*x)^2) + (163 363895*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(28*Sqrt[21]) - 171675*Sqrt[55]*A rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.30 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {109, 166, 25, 168, 27, 168, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^5 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{12} \int \frac {(265-299 x) \sqrt {1-2 x}}{(3 x+2)^4 (5 x+3)^3}dx+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{12} \left (\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}-\frac {1}{9} \int -\frac {38107-60891 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^3}dx\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \int \frac {38107-60891 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^3}dx+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {1}{14} \int \frac {35 (156029-236103 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \int \frac {156029-236103 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {1}{7} \int \frac {17001229-23507325 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {940293 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {1}{7} \left (-\frac {1}{22} \int \frac {198 (6177953-7069460 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {7069460 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {940293 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {1}{7} \left (-9 \int \frac {6177953-7069460 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {7069460 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {940293 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {1}{7} \left (-9 \left (-\frac {1}{11} \int \frac {33 (7733463-4736195 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {4736195 \sqrt {1-2 x}}{5 x+3}\right )-\frac {7069460 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {940293 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {1}{7} \left (-9 \left (-3 \int \frac {7733463-4736195 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {4736195 \sqrt {1-2 x}}{5 x+3}\right )-\frac {7069460 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {940293 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {1}{7} \left (-9 \left (-3 \left (52875900 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-32672779 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {4736195 \sqrt {1-2 x}}{5 x+3}\right )-\frac {7069460 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {940293 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {1}{7} \left (-9 \left (-3 \left (32672779 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-52875900 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {4736195 \sqrt {1-2 x}}{5 x+3}\right )-\frac {7069460 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {940293 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {1}{7} \left (-9 \left (-3 \left (\frac {65345558 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-1922760 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {4736195 \sqrt {1-2 x}}{5 x+3}\right )-\frac {7069460 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {940293 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {33729 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\right )+\frac {1393 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}\) |
(7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*(3 + 5*x)^2) + ((1393*Sqrt[1 - 2*x])/( 9*(2 + 3*x)^3*(3 + 5*x)^2) + ((33729*Sqrt[1 - 2*x])/(2*(2 + 3*x)^2*(3 + 5* x)^2) + (5*((940293*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^2) + ((-7069460* Sqrt[1 - 2*x])/(3 + 5*x)^2 - 9*((-4736195*Sqrt[1 - 2*x])/(3 + 5*x) - 3*((6 5345558*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 1922760*Sqrt[55]*ArcT anh[Sqrt[5/11]*Sqrt[1 - 2*x]])))/7))/2)/9)/12
3.20.100.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 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[(((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 = 1.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(-\frac {6393863250 x^{6}+17477604525 x^{5}+16387061255 x^{4}+3899334997 x^{3}-3059624563 x^{2}-2066805566 x -359378534}{56 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}+\frac {163363895 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{588}-171675 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(91\) |
| derivativedivides | \(\frac {-466125 \left (1-2 x \right )^{\frac {3}{2}}+1019425 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-171675 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {3170015 \left (1-2 x \right )^{\frac {7}{2}}}{168}-\frac {28695733 \left (1-2 x \right )^{\frac {5}{2}}}{216}+\frac {202051885 \left (1-2 x \right )^{\frac {3}{2}}}{648}-\frac {52696315 \sqrt {1-2 x}}{216}\right )}{\left (-4-6 x \right )^{4}}+\frac {163363895 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{588}\) | \(112\) |
| default | \(\frac {-466125 \left (1-2 x \right )^{\frac {3}{2}}+1019425 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-171675 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {3170015 \left (1-2 x \right )^{\frac {7}{2}}}{168}-\frac {28695733 \left (1-2 x \right )^{\frac {5}{2}}}{216}+\frac {202051885 \left (1-2 x \right )^{\frac {3}{2}}}{648}-\frac {52696315 \sqrt {1-2 x}}{216}\right )}{\left (-4-6 x \right )^{4}}+\frac {163363895 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{588}\) | \(112\) |
| pseudoelliptic | \(\frac {326727790 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \left (3+5 x \right )^{2} \sqrt {21}-201889800 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{4} \left (3+5 x \right )^{2} \sqrt {55}+21 \sqrt {1-2 x}\, \left (3196931625 x^{5}+10337268075 x^{4}+13362164665 x^{3}+8630749831 x^{2}+2785562634 x +359378534\right )}{1176 \left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}\) | \(116\) |
| trager | \(\frac {\left (3196931625 x^{5}+10337268075 x^{4}+13362164665 x^{3}+8630749831 x^{2}+2785562634 x +359378534\right ) \sqrt {1-2 x}}{56 \left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-\frac {163363895 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{1176}+\frac {1575 \operatorname {RootOf}\left (\textit {\_Z}^{2}-653455\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-653455\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-653455\right )+5995 \sqrt {1-2 x}}{3+5 x}\right )}{2}\) | \(138\) |
-1/56*(6393863250*x^6+17477604525*x^5+16387061255*x^4+3899334997*x^3-30596 24563*x^2-2066805566*x-359378534)/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)^2+163363 895/588*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-171675*arctanh(1/11*5 5^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {100944900 \, \sqrt {55} {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 163363895 \, \sqrt {21} {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (3196931625 \, x^{5} + 10337268075 \, x^{4} + 13362164665 \, x^{3} + 8630749831 \, x^{2} + 2785562634 \, x + 359378534\right )} \sqrt {-2 \, x + 1}}{1176 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]
1/1176*(100944900*sqrt(55)*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3) ) + 163363895*sqrt(21)*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224 *x^2 + 1344*x + 144)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(3196931625*x^5 + 10337268075*x^4 + 13362164665*x^3 + 8630749831*x^2 + 2785562634*x + 359378534)*sqrt(-2*x + 1))/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {171675}{2} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {163363895}{1176} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3196931625 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 36659194275 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 168116119510 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 385408507778 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 441689778145 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 202435240315 \, \sqrt {-2 \, x + 1}}{28 \, {\left (2025 \, {\left (2 \, x - 1\right )}^{6} + 27810 \, {\left (2 \, x - 1\right )}^{5} + 159111 \, {\left (2 \, x - 1\right )}^{4} + 485436 \, {\left (2 \, x - 1\right )}^{3} + 832951 \, {\left (2 \, x - 1\right )}^{2} + 1524292 \, x - 471625\right )}} \]
171675/2*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 *x + 1))) - 163363895/1176*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sq rt(21) + 3*sqrt(-2*x + 1))) - 1/28*(3196931625*(-2*x + 1)^(11/2) - 3665919 4275*(-2*x + 1)^(9/2) + 168116119510*(-2*x + 1)^(7/2) - 385408507778*(-2*x + 1)^(5/2) + 441689778145*(-2*x + 1)^(3/2) - 202435240315*sqrt(-2*x + 1)) /(2025*(2*x - 1)^6 + 27810*(2*x - 1)^5 + 159111*(2*x - 1)^4 + 485436*(2*x - 1)^3 + 832951*(2*x - 1)^2 + 1524292*x - 471625)
Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {171675}{2} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {163363895}{1176} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {275 \, {\left (1695 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3707 \, \sqrt {-2 \, x + 1}\right )}}{4 \, {\left (5 \, x + 3\right )}^{2}} + \frac {85590405 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 602610393 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1414363195 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1106622615 \, \sqrt {-2 \, x + 1}}{448 \, {\left (3 \, x + 2\right )}^{4}} \]
171675/2*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 163363895/1176*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6* sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 275/4*(1695*(-2*x + 1)^(3 /2) - 3707*sqrt(-2*x + 1))/(5*x + 3)^2 + 1/448*(85590405*(2*x - 1)^3*sqrt( -2*x + 1) + 602610393*(2*x - 1)^2*sqrt(-2*x + 1) - 1414363195*(-2*x + 1)^( 3/2) + 1106622615*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.50 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {\frac {5783864009\,\sqrt {1-2\,x}}{1620}-\frac {12619707947\,{\left (1-2\,x\right )}^{3/2}}{1620}+\frac {27529179127\,{\left (1-2\,x\right )}^{5/2}}{4050}-\frac {16811611951\,{\left (1-2\,x\right )}^{7/2}}{5670}+\frac {488789257\,{\left (1-2\,x\right )}^{9/2}}{756}-\frac {4736195\,{\left (1-2\,x\right )}^{11/2}}{84}}{\frac {1524292\,x}{2025}+\frac {832951\,{\left (2\,x-1\right )}^2}{2025}+\frac {161812\,{\left (2\,x-1\right )}^3}{675}+\frac {5893\,{\left (2\,x-1\right )}^4}{75}+\frac {206\,{\left (2\,x-1\right )}^5}{15}+{\left (2\,x-1\right )}^6-\frac {18865}{81}}+\frac {163363895\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{588}-171675\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right ) \]
((5783864009*(1 - 2*x)^(1/2))/1620 - (12619707947*(1 - 2*x)^(3/2))/1620 + (27529179127*(1 - 2*x)^(5/2))/4050 - (16811611951*(1 - 2*x)^(7/2))/5670 + (488789257*(1 - 2*x)^(9/2))/756 - (4736195*(1 - 2*x)^(11/2))/84)/((1524292 *x)/2025 + (832951*(2*x - 1)^2)/2025 + (161812*(2*x - 1)^3)/675 + (5893*(2 *x - 1)^4)/75 + (206*(2*x - 1)^5)/15 + (2*x - 1)^6 - 18865/81) + (16336389 5*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/588 - 171675*55^(1/2)*atan h((55^(1/2)*(1 - 2*x)^(1/2))/11)