Integrand size = 24, antiderivative size = 145 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=-\frac {6899 \sqrt {1-2 x}}{18 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {931 \sqrt {1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac {2311 \sqrt {1-2 x}}{3+5 x}+4555 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-14073 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/6*(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^2-14073/5*arctanh(1/11*55^(1/2)*(1-2*x )^(1/2))*55^(1/2)+4555*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-6899/1 8*(1-2*x)^(1/2)/(3+5*x)^2+931/18*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+2311*(1-2 *x)^(1/2)/(3+5*x)
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (52607+249939 x+395215 x^2+207990 x^3\right )}{2 \left (6+19 x+15 x^2\right )^2}+4555 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-14073 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(52607 + 249939*x + 395215*x^2 + 207990*x^3))/(2*(6 + 19*x + 15*x^2)^2) + 4555*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 14073*Sqrt [11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {109, 166, 25, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^3 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{6} \int \frac {(199-167 x) \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^3}dx+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {931 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}-\frac {1}{3} \int -\frac {16591-22941 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \int \frac {16591-22941 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {931 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-\frac {1}{22} \int \frac {198 (6029-6899 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {6899 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {931 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-9 \int \frac {6029-6899 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {6899 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {931 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-9 \left (-\frac {1}{11} \int \frac {33 (7547-4622 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {4622 \sqrt {1-2 x}}{5 x+3}\right )-\frac {6899 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {931 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-9 \left (-3 \int \frac {7547-4622 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {4622 \sqrt {1-2 x}}{5 x+3}\right )-\frac {6899 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {931 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-9 \left (-3 \left (51601 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-31885 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {4622 \sqrt {1-2 x}}{5 x+3}\right )-\frac {6899 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {931 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-9 \left (-3 \left (31885 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-51601 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {4622 \sqrt {1-2 x}}{5 x+3}\right )-\frac {6899 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {931 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-9 \left (-3 \left (9110 \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-9382 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {4622 \sqrt {1-2 x}}{5 x+3}\right )-\frac {6899 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {931 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}\) |
(7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 + 5*x)^2) + ((931*Sqrt[1 - 2*x])/(3* (2 + 3*x)*(3 + 5*x)^2) + ((-6899*Sqrt[1 - 2*x])/(3 + 5*x)^2 - 9*((-4622*Sq rt[1 - 2*x])/(3 + 5*x) - 3*(9110*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x] ] - 9382*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])))/3)/6
3.20.98.3.1 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 = 1.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \left (207990 x^{3}+395215 x^{2}+249939 x +52607\right )}{2 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}}+4555 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}-\frac {14073 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}\) | \(79\) |
| derivativedivides | \(\frac {-11385 \left (1-2 x \right )^{\frac {3}{2}}+24805 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {14073 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}-\frac {252 \left (\frac {67 \left (1-2 x \right )^{\frac {3}{2}}}{4}-\frac {1421 \sqrt {1-2 x}}{36}\right )}{\left (-4-6 x \right )^{2}}+4555 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}\) | \(94\) |
| default | \(\frac {-11385 \left (1-2 x \right )^{\frac {3}{2}}+24805 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {14073 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}-\frac {252 \left (\frac {67 \left (1-2 x \right )^{\frac {3}{2}}}{4}-\frac {1421 \sqrt {1-2 x}}{36}\right )}{\left (-4-6 x \right )^{2}}+4555 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}\) | \(94\) |
| pseudoelliptic | \(\frac {45550 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {21}-28146 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {55}+5 \sqrt {1-2 x}\, \left (207990 x^{3}+395215 x^{2}+249939 x +52607\right )}{10 \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(102\) |
| trager | \(\frac {\left (207990 x^{3}+395215 x^{2}+249939 x +52607\right ) \sqrt {1-2 x}}{2 \left (15 x^{2}+19 x +6\right )^{2}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-17428341\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-17428341\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-17428341\right )-19131 \sqrt {1-2 x}}{2+3 x}\right )}{2}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1210301455\right ) \ln \left (-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1210301455\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1210301455\right )-258005 \sqrt {1-2 x}}{3+5 x}\right )}{10}\) | \(128\) |
-1/2*(-1+2*x)*(207990*x^3+395215*x^2+249939*x+52607)/(15*x^2+19*x+6)^2/(1- 2*x)^(1/2)+4555*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-14073/5*arcta nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.08 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {14073 \, \sqrt {11} \sqrt {5} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 22775 \, \sqrt {21} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 5 \, {\left (207990 \, x^{3} + 395215 \, x^{2} + 249939 \, x + 52607\right )} \sqrt {-2 \, x + 1}}{10 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
1/10*(14073*sqrt(11)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*lo g((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 22775*sqrt(21)* (225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 5*(207990*x^3 + 395215*x^2 + 249939*x + 52607)*sqrt( -2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
Time = 83.24 (sec) , antiderivative size = 651, normalized size of antiderivative = 4.49 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=- 2244 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right ) + 1428 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right ) + 19404 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 2744 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) + 50820 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right ) + 10648 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right ) \]
-2244*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3)) + 1428*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2 *x) + sqrt(55)/5)) + 19404*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))) - 2744*Piecewise((sqrt(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)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21) *sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3))) + 50820*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqr t(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5))) + 10648*Piecewise(( sqrt(55)*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqrt(5 5)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 1/ (16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)**2))/6655, (sqrt(1 - 2*x) > -sqrt(55)/ 5) & (sqrt(1 - 2*x) < sqrt(55)/5)))
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {14073}{10} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4555}{2} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (103995 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 707200 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 1602293 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1209516 \, \sqrt {-2 \, x + 1}\right )}}{225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543} \]
14073/10*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 *x + 1))) - 4555/2*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2*(103995*(-2*x + 1)^(7/2) - 707200*(-2*x + 1)^(5/2) + 1602293*(-2*x + 1)^(3/2) - 1209516*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2 040*(2*x - 1)^3 + 6934*(2*x - 1)^2 + 20944*x - 4543)
Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {14073}{10} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {4555}{2} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (103995 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 707200 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1602293 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1209516 \, \sqrt {-2 \, x + 1}\right )}}{{\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]
14073/10*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4555/2*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2* x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2*(103995*(2*x - 1)^3*sqrt(-2*x + 1) + 707200*(2*x - 1)^2*sqrt(-2*x + 1) - 1602293*(-2*x + 1)^(3/2) + 12095 16*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=4555\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )-\frac {14073\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{5}+\frac {\frac {806344\,\sqrt {1-2\,x}}{75}-\frac {3204586\,{\left (1-2\,x\right )}^{3/2}}{225}+\frac {56576\,{\left (1-2\,x\right )}^{5/2}}{9}-\frac {4622\,{\left (1-2\,x\right )}^{7/2}}{5}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \]
4555*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7) - (14073*55^(1/2)*atanh( (55^(1/2)*(1 - 2*x)^(1/2))/11))/5 + ((806344*(1 - 2*x)^(1/2))/75 - (320458 6*(1 - 2*x)^(3/2))/225 + (56576*(1 - 2*x)^(5/2))/9 - (4622*(1 - 2*x)^(7/2) )/5)/((20944*x)/225 + (6934*(2*x - 1)^2)/225 + (136*(2*x - 1)^3)/15 + (2*x - 1)^4 - 4543/225)