Integrand size = 24, antiderivative size = 153 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {7 \sqrt {1-2 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {5732 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {120077 \sqrt {1-2 x}}{882 (2+3 x)^2}+\frac {2788127 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {96169877 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}-2750 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
96169877/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2750*arctanh(1 /11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+7/15*(1-2*x)^(1/2)/(2+3*x)^5+41/15*(1 -2*x)^(1/2)/(2+3*x)^4+5732/315*(1-2*x)^(1/2)/(2+3*x)^3+120077/882*(1-2*x)^ (1/2)/(2+3*x)^2+2788127/2058*(1-2*x)^(1/2)/(2+3*x)
Time = 0.55 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (235067382+1391064622 x+3088510878 x^2+3049001415 x^3+1129191435 x^4\right )}{10290 (2+3 x)^5}+\frac {96169877 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}-2750 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(235067382 + 1391064622*x + 3088510878*x^2 + 3049001415*x^3 + 1129191435*x^4))/(10290*(2 + 3*x)^5) + (96169877*ArcTanh[Sqrt[3/7]*Sqrt [1 - 2*x]])/(1029*Sqrt[21]) - 2750*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2* x]]
Time = 0.25 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {109, 168, 27, 168, 27, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^6 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{15} \int \frac {186-295 x}{\sqrt {1-2 x} (3 x+2)^5 (5 x+3)}dx+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{28} \int \frac {28 (954-1435 x)}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\int \frac {954-1435 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{21} \int \frac {5 (20919-28660 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {5732 \sqrt {1-2 x}}{21 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {5}{21} \int \frac {20919-28660 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {5732 \sqrt {1-2 x}}{21 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{15} \left (\frac {5}{21} \left (\frac {1}{14} \int \frac {3 (529119-600385 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {120077 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {5732 \sqrt {1-2 x}}{21 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {5}{21} \left (\frac {3}{14} \int \frac {529119-600385 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {120077 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {5732 \sqrt {1-2 x}}{21 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{15} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {22762869-13940635 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {2788127 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {120077 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {5732 \sqrt {1-2 x}}{21 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{15} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \left (155636250 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-96169877 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {2788127 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {120077 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {5732 \sqrt {1-2 x}}{21 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{15} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \left (96169877 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-155636250 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2788127 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {120077 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {5732 \sqrt {1-2 x}}{21 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{15} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \left (\frac {192339754 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-5659500 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {2788127 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {120077 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {5732 \sqrt {1-2 x}}{21 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{(3 x+2)^4}\right )+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}\) |
(7*Sqrt[1 - 2*x])/(15*(2 + 3*x)^5) + ((41*Sqrt[1 - 2*x])/(2 + 3*x)^4 + (57 32*Sqrt[1 - 2*x])/(21*(2 + 3*x)^3) + (5*((120077*Sqrt[1 - 2*x])/(14*(2 + 3 *x)^2) + (3*((2788127*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + ((192339754*ArcTanh[S qrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 5659500*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sq rt[1 - 2*x]])/7))/14))/21)/15
3.20.7.3.1 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 + 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 = 1.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {2258382870 x^{5}+4968811395 x^{4}+3128020341 x^{3}-306381634 x^{2}-920929858 x -235067382}{10290 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {96169877 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}-2750 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(79\) |
| pseudoelliptic | \(\frac {961698770 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}-594247500 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{5} \sqrt {55}+21 \sqrt {1-2 x}\, \left (1129191435 x^{4}+3049001415 x^{3}+3088510878 x^{2}+1391064622 x +235067382\right )}{216090 \left (2+3 x \right )^{5}}\) | \(90\) |
| derivativedivides | \(-2750 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {486 \left (\frac {2788127 \left (1-2 x \right )^{\frac {9}{2}}}{6174}-\frac {2406977 \left (1-2 x \right )^{\frac {7}{2}}}{567}+\frac {127289798 \left (1-2 x \right )^{\frac {5}{2}}}{8505}-\frac {17098361 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {20099611 \sqrt {1-2 x}}{1458}\right )}{\left (-4-6 x \right )^{5}}+\frac {96169877 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(93\) |
| default | \(-2750 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {486 \left (\frac {2788127 \left (1-2 x \right )^{\frac {9}{2}}}{6174}-\frac {2406977 \left (1-2 x \right )^{\frac {7}{2}}}{567}+\frac {127289798 \left (1-2 x \right )^{\frac {5}{2}}}{8505}-\frac {17098361 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {20099611 \sqrt {1-2 x}}{1458}\right )}{\left (-4-6 x \right )^{5}}+\frac {96169877 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(93\) |
| trager | \(\frac {\left (1129191435 x^{4}+3049001415 x^{3}+3088510878 x^{2}+1391064622 x +235067382\right ) \sqrt {1-2 x}}{10290 \left (2+3 x \right )^{5}}-\frac {96169877 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{43218}+1375 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )\) | \(126\) |
-1/10290*(2258382870*x^5+4968811395*x^4+3128020341*x^3-306381634*x^2-92092 9858*x-235067382)/(2+3*x)^5/(1-2*x)^(1/2)+96169877/21609*arctanh(1/7*21^(1 /2)*(1-2*x)^(1/2))*21^(1/2)-2750*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 = 1.11 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {297123750 \, \sqrt {55} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 480849385 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (1129191435 \, x^{4} + 3049001415 \, x^{3} + 3088510878 \, x^{2} + 1391064622 \, x + 235067382\right )} \sqrt {-2 \, x + 1}}{216090 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/216090*(297123750*sqrt(55)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240 *x + 32)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 480849385*sq rt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x - sq rt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(1129191435*x^4 + 3049001415*x^ 3 + 3088510878*x^2 + 1391064622*x + 235067382)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx=1375 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {96169877}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1129191435 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 10614768570 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 37423200612 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 58647378230 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 34470832865 \, \sqrt {-2 \, x + 1}}{5145 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
1375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 96169877/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) + 1/5145*(1129191435*(-2*x + 1)^(9/2) - 1061476857 0*(-2*x + 1)^(7/2) + 37423200612*(-2*x + 1)^(5/2) - 58647378230*(-2*x + 1) ^(3/2) + 34470832865*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)
Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx=1375 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {96169877}{43218} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1129191435 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 10614768570 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 37423200612 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 58647378230 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 34470832865 \, \sqrt {-2 \, x + 1}}{164640 \, {\left (3 \, x + 2\right )}^{5}} \]
1375*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s qrt(-2*x + 1))) - 96169877/43218*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt (-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/164640*(1129191435*(2*x - 1 )^4*sqrt(-2*x + 1) + 10614768570*(2*x - 1)^3*sqrt(-2*x + 1) + 37423200612* (2*x - 1)^2*sqrt(-2*x + 1) - 58647378230*(-2*x + 1)^(3/2) + 34470832865*sq rt(-2*x + 1))/(3*x + 2)^5
Time = 1.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {96169877\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}-2750\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {20099611\,\sqrt {1-2\,x}}{729}-\frac {34196722\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {254579596\,{\left (1-2\,x\right )}^{5/2}}{8505}-\frac {4813954\,{\left (1-2\,x\right )}^{7/2}}{567}+\frac {2788127\,{\left (1-2\,x\right )}^{9/2}}{3087}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
(96169877*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/21609 - 2750*55^(1 /2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((20099611*(1 - 2*x)^(1/2))/729 - (34196722*(1 - 2*x)^(3/2))/729 + (254579596*(1 - 2*x)^(5/2))/8505 - (48 13954*(1 - 2*x)^(7/2))/567 + (2788127*(1 - 2*x)^(9/2))/3087)/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)