Integrand size = 24, antiderivative size = 181 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {7738475 \sqrt {1-2 x}}{504 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac {287 \sqrt {1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac {22109 \sqrt {1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac {288770 \sqrt {1-2 x}}{189 (2+3 x) (3+5 x)}-\frac {53384095 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}+18700 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/12*(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)-53384095/1764*arctanh(1/7*21^(1/2)*(1 -2*x)^(1/2))*21^(1/2)+18700*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)- 7738475/504*(1-2*x)^(1/2)/(3+5*x)+287/27*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)+2 2109/216*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)+288770/189*(1-2*x)^(1/2)/(2+3*x)/ (3+5*x)
Time = 0.34 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (39145938+238179048 x+543154477 x^2+550239720 x^3+208938825 x^4\right )}{168 (2+3 x)^4 (3+5 x)}-\frac {53384095 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}+18700 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-1/168*(Sqrt[1 - 2*x]*(39145938 + 238179048*x + 543154477*x^2 + 550239720* x^3 + 208938825*x^4))/((2 + 3*x)^4*(3 + 5*x)) - (53384095*ArcTanh[Sqrt[3/7 ]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) + 18700*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {109, 166, 25, 168, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^5 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{12} \int \frac {(230-229 x) \sqrt {1-2 x}}{(3 x+2)^4 (5 x+3)^2}dx+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{12} \left (\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}-\frac {1}{9} \int -\frac {25717-38806 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \int \frac {25717-38806 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {1}{14} \int \frac {35 (80314-110545 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx+\frac {22109 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)}\right )+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \int \frac {80314-110545 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx+\frac {22109 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)}\right )+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {1}{7} \int \frac {3 (2018833-2310160 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {462032 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {22109 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)}\right )+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {3}{7} \int \frac {2018833-2310160 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {462032 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {22109 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)}\right )+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {3}{7} \left (-\frac {1}{11} \int \frac {33 (2527143-1547695 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {1547695 \sqrt {1-2 x}}{5 x+3}\right )+\frac {462032 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {22109 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)}\right )+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {3}{7} \left (-3 \int \frac {2527143-1547695 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {1547695 \sqrt {1-2 x}}{5 x+3}\right )+\frac {462032 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {22109 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)}\right )+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {3}{7} \left (-3 \left (17278800 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-10676819 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {1547695 \sqrt {1-2 x}}{5 x+3}\right )+\frac {462032 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {22109 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)}\right )+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {3}{7} \left (-3 \left (10676819 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-17278800 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1547695 \sqrt {1-2 x}}{5 x+3}\right )+\frac {462032 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {22109 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)}\right )+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{9} \left (\frac {5}{2} \left (\frac {3}{7} \left (-3 \left (\frac {21353638 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-628320 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {1547695 \sqrt {1-2 x}}{5 x+3}\right )+\frac {462032 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {22109 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)}\right )+\frac {1148 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}\) |
(7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*(3 + 5*x)) + ((1148*Sqrt[1 - 2*x])/(9* (2 + 3*x)^3*(3 + 5*x)) + ((22109*Sqrt[1 - 2*x])/(2*(2 + 3*x)^2*(3 + 5*x)) + (5*((462032*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) + (3*((-1547695*Sqrt[ 1 - 2*x])/(3 + 5*x) - 3*((21353638*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[ 21] - 628320*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])))/7))/2)/9)/12
3.20.89.3.1 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 = 1.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {417877650 x^{5}+891540615 x^{4}+536069234 x^{3}-66796381 x^{2}-159887172 x -39145938}{168 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \left (3+5 x \right )}-\frac {53384095 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1764}+18700 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(86\) |
| derivativedivides | \(\frac {1210 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+18700 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {\frac {11184975 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {11266013 \left (1-2 x \right )^{\frac {5}{2}}}{4}+\frac {79444085 \left (1-2 x \right )^{\frac {3}{2}}}{12}-\frac {62254745 \sqrt {1-2 x}}{12}}{\left (-4-6 x \right )^{4}}-\frac {53384095 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1764}\) | \(100\) |
| default | \(\frac {1210 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+18700 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {\frac {11184975 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {11266013 \left (1-2 x \right )^{\frac {5}{2}}}{4}+\frac {79444085 \left (1-2 x \right )^{\frac {3}{2}}}{12}-\frac {62254745 \sqrt {1-2 x}}{12}}{\left (-4-6 x \right )^{4}}-\frac {53384095 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1764}\) | \(100\) |
| pseudoelliptic | \(\frac {-106768190 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \left (3+5 x \right ) \sqrt {21}+65973600 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{4} \left (3+5 x \right ) \sqrt {55}-21 \sqrt {1-2 x}\, \left (208938825 x^{4}+550239720 x^{3}+543154477 x^{2}+238179048 x +39145938\right )}{3528 \left (2+3 x \right )^{4} \left (3+5 x \right )}\) | \(107\) |
| trager | \(-\frac {\left (208938825 x^{4}+550239720 x^{3}+543154477 x^{2}+238179048 x +39145938\right ) \sqrt {1-2 x}}{168 \left (2+3 x \right )^{4} \left (3+5 x \right )}+\frac {53384095 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{3528}-9350 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )\) | \(133\) |
1/168*(417877650*x^5+891540615*x^4+536069234*x^3-66796381*x^2-159887172*x- 39145938)/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)-53384095/1764*arctanh(1/7*21^(1/ 2)*(1-2*x)^(1/2))*21^(1/2)+18700*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^( 1/2)
Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=\frac {32986800 \, \sqrt {55} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 53384095 \, \sqrt {21} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (208938825 \, x^{4} + 550239720 \, x^{3} + 543154477 \, x^{2} + 238179048 \, x + 39145938\right )} \sqrt {-2 \, x + 1}}{3528 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]
1/3528*(32986800*sqrt(55)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368* x + 48)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 53384095*sqrt (21)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log((3*x + sq rt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(208938825*x^4 + 550239720*x^3 + 543154477*x^2 + 238179048*x + 39145938)*sqrt(-2*x + 1))/(405*x^5 + 1323* x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)
Time = 135.50 (sec) , antiderivative size = 955, normalized size of antiderivative = 5.28 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=\text {Too large to display} \]
103455*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqr t(21)/3))/7 - 9405*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5)) - 200376*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt (1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2 *x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3))) + 60984*Piecewise((sqr t(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2* x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqr t(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqr t(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqr t(1 - 2*x) < sqrt(21)/3))) - 50176*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sq rt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sq rt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/3 + 10976*Piecewise((sqrt(21)*(35*log(sqrt(21)*sqrt(1 - 2*x )/7 - 1)/256 - 35*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/256 + 35/(256*(sqrt(21 )*sqrt(1 - 2*x)/7 + 1)) + 15/(256*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 5/( 192*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) + 1/(128*(sqrt(21)*sqrt(1 - 2*x)...
Time = 0.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=-9350 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {53384095}{3528} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {208938825 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 1936234740 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 6727689178 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 10387861820 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 6013803565 \, \sqrt {-2 \, x + 1}}{84 \, {\left (405 \, {\left (2 \, x - 1\right )}^{5} + 4671 \, {\left (2 \, x - 1\right )}^{4} + 21546 \, {\left (2 \, x - 1\right )}^{3} + 49686 \, {\left (2 \, x - 1\right )}^{2} + 114562 \, x - 30870\right )}} \]
-9350*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 53384095/3528*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) - 1/84*(208938825*(-2*x + 1)^(9/2) - 1936234740*(- 2*x + 1)^(7/2) + 6727689178*(-2*x + 1)^(5/2) - 10387861820*(-2*x + 1)^(3/2 ) + 6013803565*sqrt(-2*x + 1))/(405*(2*x - 1)^5 + 4671*(2*x - 1)^4 + 21546 *(2*x - 1)^3 + 49686*(2*x - 1)^2 + 114562*x - 30870)
Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=-9350 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {53384095}{3528} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3025 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {33554925 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 236586273 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 556108595 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 435783215 \, \sqrt {-2 \, x + 1}}{1344 \, {\left (3 \, x + 2\right )}^{4}} \]
-9350*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5* sqrt(-2*x + 1))) + 53384095/3528*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt (-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 3025*sqrt(-2*x + 1)/(5*x + 3) - 1/1344*(33554925*(2*x - 1)^3*sqrt(-2*x + 1) + 236586273*(2*x - 1)^2*sqr t(-2*x + 1) - 556108595*(-2*x + 1)^(3/2) + 435783215*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.50 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=18700\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {53384095\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1764}-\frac {\frac {171822959\,\sqrt {1-2\,x}}{972}-\frac {74199013\,{\left (1-2\,x\right )}^{3/2}}{243}+\frac {480549227\,{\left (1-2\,x\right )}^{5/2}}{2430}-\frac {32270579\,{\left (1-2\,x\right )}^{7/2}}{567}+\frac {1547695\,{\left (1-2\,x\right )}^{9/2}}{252}}{\frac {114562\,x}{405}+\frac {16562\,{\left (2\,x-1\right )}^2}{135}+\frac {266\,{\left (2\,x-1\right )}^3}{5}+\frac {173\,{\left (2\,x-1\right )}^4}{15}+{\left (2\,x-1\right )}^5-\frac {686}{9}} \]
18700*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) - (53384095*21^(1/2)*a tanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1764 - ((171822959*(1 - 2*x)^(1/2))/97 2 - (74199013*(1 - 2*x)^(3/2))/243 + (480549227*(1 - 2*x)^(5/2))/2430 - (3 2270579*(1 - 2*x)^(7/2))/567 + (1547695*(1 - 2*x)^(9/2))/252)/((114562*x)/ 405 + (16562*(2*x - 1)^2)/135 + (266*(2*x - 1)^3)/5 + (173*(2*x - 1)^4)/15 + (2*x - 1)^5 - 686/9)