Integrand size = 24, antiderivative size = 208 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx=-\frac {323422735 \sqrt {1-2 x}}{3528 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac {1379 \sqrt {1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac {16549 \sqrt {1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac {924025 \sqrt {1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac {12068887 \sqrt {1-2 x}}{1323 (2+3 x) (3+5 x)}-\frac {2231141147 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}+111650 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/15*(1-2*x)^(3/2)/(2+3*x)^5/(3+5*x)-2231141147/12348*arctanh(1/7*21^(1/2) *(1-2*x)^(1/2))*21^(1/2)+111650*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1 /2)-323422735/3528*(1-2*x)^(1/2)/(3+5*x)+1379/180*(1-2*x)^(1/2)/(2+3*x)^4/ (3+5*x)+16549/270*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)+924025/1512*(1-2*x)^(1/2 )/(2+3*x)^2/(3+5*x)+12068887/1323*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)
Time = 0.44 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (16360698684+124085884254 x+376323861626 x^2+570477768855 x^3+432275892930 x^4+130986207675 x^5\right )}{5880 (2+3 x)^5 (3+5 x)}-\frac {2231141147 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}}+111650 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-1/5880*(Sqrt[1 - 2*x]*(16360698684 + 124085884254*x + 376323861626*x^2 + 570477768855*x^3 + 432275892930*x^4 + 130986207675*x^5))/((2 + 3*x)^5*(3 + 5*x)) - (2231141147*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) + 11 1650*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.12, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^6 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{15} \int \frac {(263-295 x) \sqrt {1-2 x}}{(3 x+2)^5 (5 x+3)^2}dx+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{15} \left (\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}-\frac {1}{12} \int -\frac {37432-59695 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)^2}dx\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \int \frac {37432-59695 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)^2}dx+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {1}{21} \int \frac {35 (153551-231686 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {5}{3} \int \frac {153551-231686 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {5}{3} \left (\frac {1}{14} \int \frac {16783282-23100625 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx+\frac {924025 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {5}{3} \left (\frac {1}{14} \left (\frac {1}{7} \int \frac {3 (421876729-482755480 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {96551096 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {924025 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {5}{3} \left (\frac {1}{14} \left (\frac {3}{7} \int \frac {421876729-482755480 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {96551096 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {924025 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {5}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-\frac {1}{11} \int \frac {33 (528098559-323422735 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {323422735 \sqrt {1-2 x}}{5 x+3}\right )+\frac {96551096 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {924025 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {5}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-3 \int \frac {528098559-323422735 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {323422735 \sqrt {1-2 x}}{5 x+3}\right )+\frac {96551096 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {924025 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {5}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-3 \left (3610761000 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-2231141147 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {323422735 \sqrt {1-2 x}}{5 x+3}\right )+\frac {96551096 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {924025 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {5}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-3 \left (2231141147 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-3610761000 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {323422735 \sqrt {1-2 x}}{5 x+3}\right )+\frac {96551096 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {924025 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{12} \left (\frac {5}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-3 \left (\frac {4462282294 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-131300400 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {323422735 \sqrt {1-2 x}}{5 x+3}\right )+\frac {96551096 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {924025 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {33098 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {1379 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}\) |
(7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5*(3 + 5*x)) + ((1379*Sqrt[1 - 2*x])/(12 *(2 + 3*x)^4*(3 + 5*x)) + ((33098*Sqrt[1 - 2*x])/(3*(2 + 3*x)^3*(3 + 5*x)) + (5*((924025*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)) + ((96551096*Sqrt [1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) + (3*((-323422735*Sqrt[1 - 2*x])/(3 + 5 *x) - 3*((4462282294*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 13130040 0*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])))/7)/14))/3)/12)/15
3.20.90.3.1 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 = 1.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {261972415350 x^{6}+733565578185 x^{5}+708679644780 x^{4}+182169954397 x^{3}-128152093118 x^{2}-91364486886 x -16360698684}{5880 \left (2+3 x \right )^{5} \sqrt {1-2 x}\, \left (3+5 x \right )}-\frac {2231141147 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}+111650 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(91\) |
| derivativedivides | \(\frac {6050 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+111650 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {\frac {1458333369 \left (1-2 x \right )^{\frac {9}{2}}}{196}-\frac {139690761 \left (1-2 x \right )^{\frac {7}{2}}}{2}+\frac {1229445796 \left (1-2 x \right )^{\frac {5}{2}}}{5}-\frac {2308578797 \left (1-2 x \right )^{\frac {3}{2}}}{6}+\frac {2709545797 \sqrt {1-2 x}}{12}}{\left (-4-6 x \right )^{5}}-\frac {2231141147 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\) | \(109\) |
| default | \(\frac {6050 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+111650 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {\frac {1458333369 \left (1-2 x \right )^{\frac {9}{2}}}{196}-\frac {139690761 \left (1-2 x \right )^{\frac {7}{2}}}{2}+\frac {1229445796 \left (1-2 x \right )^{\frac {5}{2}}}{5}-\frac {2308578797 \left (1-2 x \right )^{\frac {3}{2}}}{6}+\frac {2709545797 \sqrt {1-2 x}}{12}}{\left (-4-6 x \right )^{5}}-\frac {2231141147 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\) | \(109\) |
| pseudoelliptic | \(\frac {-22311411470 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \left (3+5 x \right ) \sqrt {21}+13786542000 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{5} \left (3+5 x \right ) \sqrt {55}-21 \sqrt {1-2 x}\, \left (130986207675 x^{5}+432275892930 x^{4}+570477768855 x^{3}+376323861626 x^{2}+124085884254 x +16360698684\right )}{123480 \left (2+3 x \right )^{5} \left (3+5 x \right )}\) | \(112\) |
| trager | \(-\frac {\left (130986207675 x^{5}+432275892930 x^{4}+570477768855 x^{3}+376323861626 x^{2}+124085884254 x +16360698684\right ) \sqrt {1-2 x}}{5880 \left (2+3 x \right )^{5} \left (3+5 x \right )}-\frac {2231141147 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{24696}-55825 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )\) | \(138\) |
1/5880*(261972415350*x^6+733565578185*x^5+708679644780*x^4+182169954397*x^ 3-128152093118*x^2-91364486886*x-16360698684)/(2+3*x)^5/(1-2*x)^(1/2)/(3+5 *x)-2231141147/12348*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+111650*a rctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.24 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx=\frac {6893271000 \, \sqrt {55} {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11155705735 \, \sqrt {21} {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (130986207675 \, x^{5} + 432275892930 \, x^{4} + 570477768855 \, x^{3} + 376323861626 \, x^{2} + 124085884254 \, x + 16360698684\right )} \sqrt {-2 \, x + 1}}{123480 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \]
1/123480*(6893271000*sqrt(55)*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 11155705735*sqrt(21)*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360* x^2 + 880*x + 96)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21* (130986207675*x^5 + 432275892930*x^4 + 570477768855*x^3 + 376323861626*x^2 + 124085884254*x + 16360698684)*sqrt(-2*x + 1))/(1215*x^6 + 4779*x^5 + 78 30*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx=-55825 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2231141147}{24696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {130986207675 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 1519482824235 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 7049980295610 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 16353496911178 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 18965427342155 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 8796956467915 \, \sqrt {-2 \, x + 1}}{2940 \, {\left (1215 \, {\left (2 \, x - 1\right )}^{6} + 16848 \, {\left (2 \, x - 1\right )}^{5} + 97335 \, {\left (2 \, x - 1\right )}^{4} + 299880 \, {\left (2 \, x - 1\right )}^{3} + 519645 \, {\left (2 \, x - 1\right )}^{2} + 960400 \, x - 295323\right )}} \]
-55825*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2231141147/24696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sq rt(21) + 3*sqrt(-2*x + 1))) + 1/2940*(130986207675*(-2*x + 1)^(11/2) - 151 9482824235*(-2*x + 1)^(9/2) + 7049980295610*(-2*x + 1)^(7/2) - 16353496911 178*(-2*x + 1)^(5/2) + 18965427342155*(-2*x + 1)^(3/2) - 8796956467915*sqr t(-2*x + 1))/(1215*(2*x - 1)^6 + 16848*(2*x - 1)^5 + 97335*(2*x - 1)^4 + 2 99880*(2*x - 1)^3 + 519645*(2*x - 1)^2 + 960400*x - 295323)
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx=-55825 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2231141147}{24696} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {15125 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {21875000535 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 205345418670 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 722914128048 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1131203610530 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 663838720265 \, \sqrt {-2 \, x + 1}}{94080 \, {\left (3 \, x + 2\right )}^{5}} \]
-55825*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5 *sqrt(-2*x + 1))) + 2231141147/24696*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6* sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 15125*sqrt(-2*x + 1)/(5*x + 3) - 1/94080*(21875000535*(2*x - 1)^4*sqrt(-2*x + 1) + 205345418670*(2* x - 1)^3*sqrt(-2*x + 1) + 722914128048*(2*x - 1)^2*sqrt(-2*x + 1) - 113120 3610530*(-2*x + 1)^(3/2) + 663838720265*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx=111650\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {2231141147\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12348}-\frac {\frac {35905944767\,\sqrt {1-2\,x}}{14580}-\frac {77409907519\,{\left (1-2\,x\right )}^{3/2}}{14580}+\frac {166872417461\,{\left (1-2\,x\right )}^{5/2}}{36450}-\frac {4795904963\,{\left (1-2\,x\right )}^{7/2}}{2430}+\frac {33766284983\,{\left (1-2\,x\right )}^{9/2}}{79380}-\frac {64684547\,{\left (1-2\,x\right )}^{11/2}}{1764}}{\frac {192080\,x}{243}+\frac {34643\,{\left (2\,x-1\right )}^2}{81}+\frac {6664\,{\left (2\,x-1\right )}^3}{27}+\frac {721\,{\left (2\,x-1\right )}^4}{9}+\frac {208\,{\left (2\,x-1\right )}^5}{15}+{\left (2\,x-1\right )}^6-\frac {98441}{405}} \]
111650*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) - (2231141147*21^(1/2 )*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/12348 - ((35905944767*(1 - 2*x)^(1/ 2))/14580 - (77409907519*(1 - 2*x)^(3/2))/14580 + (166872417461*(1 - 2*x)^ (5/2))/36450 - (4795904963*(1 - 2*x)^(7/2))/2430 + (33766284983*(1 - 2*x)^ (9/2))/79380 - (64684547*(1 - 2*x)^(11/2))/1764)/((192080*x)/243 + (34643* (2*x - 1)^2)/81 + (6664*(2*x - 1)^3)/27 + (721*(2*x - 1)^4)/9 + (208*(2*x - 1)^5)/15 + (2*x - 1)^6 - 98441/405)