Integrand size = 24, antiderivative size = 121 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx=-\frac {15 \sqrt {1-2 x}}{2 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^2}+\frac {995 \sqrt {1-2 x}}{22 (3+5 x)}+624 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {6665}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
624/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-6665/121*arctanh(1/11*5 5^(1/2)*(1-2*x)^(1/2))*55^(1/2)-15/2*(1-2*x)^(1/2)/(3+5*x)^2+(1-2*x)^(1/2) /(2+3*x)/(3+5*x)^2+995/22*(1-2*x)^(1/2)/(3+5*x)
Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (5662+18410 x+14925 x^2\right )}{22 (2+3 x) (3+5 x)^2}+624 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {6665}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(5662 + 18410*x + 14925*x^2))/(22*(2 + 3*x)*(3 + 5*x)^2) + 624*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (6665*Sqrt[5/11]*ArcTanh[ Sqrt[5/11]*Sqrt[1 - 2*x]])/11
Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {110, 25, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}-\int -\frac {18-25 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {18-25 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {1}{22} \int \frac {11 (118-135 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {15 \sqrt {1-2 x}}{2 (5 x+3)^2}+\frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} \int \frac {118-135 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {15 \sqrt {1-2 x}}{2 (5 x+3)^2}+\frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{11} \int \frac {4874-2985 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {995 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {15 \sqrt {1-2 x}}{2 (5 x+3)^2}+\frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{11} \left (33325 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-20592 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {995 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {15 \sqrt {1-2 x}}{2 (5 x+3)^2}+\frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{11} \left (20592 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-33325 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {995 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {15 \sqrt {1-2 x}}{2 (5 x+3)^2}+\frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{11} \left (13728 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13330 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {995 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {15 \sqrt {1-2 x}}{2 (5 x+3)^2}+\frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\) |
(-15*Sqrt[1 - 2*x])/(2*(3 + 5*x)^2) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)^2 ) + ((995*Sqrt[1 - 2*x])/(11*(3 + 5*x)) + (13728*Sqrt[3/7]*ArcTanh[Sqrt[3/ 7]*Sqrt[1 - 2*x]] - 13330*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11 )/2
3.19.54.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (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 = 3.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.63
| method | result | size |
| risch | \(-\frac {29850 x^{3}+21895 x^{2}-7086 x -5662}{22 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (2+3 x \right )}+\frac {624 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}-\frac {6665 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\) | \(76\) |
| derivativedivides | \(\frac {-\frac {3325 \left (1-2 x \right )^{\frac {3}{2}}}{11}+655 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {6665 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \sqrt {1-2 x}}{-\frac {4}{3}-2 x}+\frac {624 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(82\) |
| default | \(\frac {-\frac {3325 \left (1-2 x \right )^{\frac {3}{2}}}{11}+655 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {6665 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \sqrt {1-2 x}}{-\frac {4}{3}-2 x}+\frac {624 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(82\) |
| pseudoelliptic | \(\frac {151008 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {21}-93310 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {55}+77 \sqrt {1-2 x}\, \left (14925 x^{2}+18410 x +5662\right )}{1694 \left (2+3 x \right ) \left (3+5 x \right )^{2}}\) | \(97\) |
| trager | \(\frac {\left (14925 x^{2}+18410 x +5662\right ) \sqrt {1-2 x}}{22 \left (3+5 x \right )^{2} \left (2+3 x \right )}-\frac {6665 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{242}-\frac {312 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{7}\) | \(123\) |
-1/22*(29850*x^3+21895*x^2-7086*x-5662)/(3+5*x)^2/(1-2*x)^(1/2)/(2+3*x)+62 4/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-6665/121*arctanh(1/11*55^ (1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {46655 \, \sqrt {11} \sqrt {5} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 75504 \, \sqrt {7} \sqrt {3} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (14925 \, x^{2} + 18410 \, x + 5662\right )} \sqrt {-2 \, x + 1}}{1694 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]
1/1694*(46655*sqrt(11)*sqrt(5)*(75*x^3 + 140*x^2 + 87*x + 18)*log((sqrt(11 )*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 75504*sqrt(7)*sqrt(3)*(75 *x^3 + 140*x^2 + 87*x + 18)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5 )/(3*x + 2)) + 77*(14925*x^2 + 18410*x + 5662)*sqrt(-2*x + 1))/(75*x^3 + 1 40*x^2 + 87*x + 18)
Time = 52.50 (sec) , antiderivative size = 488, normalized size of antiderivative = 4.03 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx=- \frac {309 \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 )}{7} + \frac {309 \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 )}{11} + 252 \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 ) + 1360 \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 ) + 440 \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 ) \]
-309*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 21)/3))/7 + 309*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2 *x) + sqrt(55)/5))/11 + 252*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))) + 1360*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)*sqrt(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))) + 440*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(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqr t(1 - 2*x)/11 - 1)) - 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)**2))/6655, (sq rt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {6665}{242} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {312}{7} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {14925 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 66670 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 74393 \, \sqrt {-2 \, x + 1}}{11 \, {\left (75 \, {\left (2 \, x - 1\right )}^{3} + 505 \, {\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \]
6665/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 *x + 1))) - 312/7*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/11*(14925*(-2*x + 1)^(5/2) - 66670*(-2*x + 1)^(3/2) + 74393*sqrt(-2*x + 1))/(75*(2*x - 1)^3 + 505*(2*x - 1)^2 + 2266*x - 286)
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {6665}{242} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {312}{7} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {9 \, \sqrt {-2 \, x + 1}}{3 \, x + 2} - \frac {5 \, {\left (665 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1441 \, \sqrt {-2 \, x + 1}\right )}}{44 \, {\left (5 \, x + 3\right )}^{2}} \]
6665/242*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 312/7*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 9*sqrt(-2*x + 1)/(3*x + 2) - 5/44* (665*(-2*x + 1)^(3/2) - 1441*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {624\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {6665\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}+\frac {\frac {6763\,\sqrt {1-2\,x}}{75}-\frac {13334\,{\left (1-2\,x\right )}^{3/2}}{165}+\frac {199\,{\left (1-2\,x\right )}^{5/2}}{11}}{\frac {2266\,x}{75}+\frac {101\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {286}{75}} \]