Integrand size = 24, antiderivative size = 113 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3}+\frac {52 \sqrt {1-2 x}}{21 (2+3 x)^2}+\frac {1207 \sqrt {1-2 x}}{49 (2+3 x)}+\frac {83264 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}-50 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
83264/1029*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50*arctanh(1/11*55 ^(1/2)*(1-2*x)^(1/2))*55^(1/2)+1/3*(1-2*x)^(1/2)/(2+3*x)^3+52/21*(1-2*x)^( 1/2)/(2+3*x)^2+1207/49*(1-2*x)^(1/2)/(2+3*x)
Time = 0.36 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (5087+14848 x+10863 x^2\right )}{49 (2+3 x)^3}+\frac {83264 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}-50 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(5087 + 14848*x + 10863*x^2))/(49*(2 + 3*x)^3) + (83264*Arc Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[21]) - 50*Sqrt[55]*ArcTanh[Sqrt[5/ 11]*Sqrt[1 - 2*x]]
Time = 0.22 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^4 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x}}{3 (3 x+2)^3}-\frac {1}{3} \int -\frac {18-25 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \int \frac {18-25 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \int \frac {6 (229-260 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{7} \int \frac {229-260 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{7} \left (\frac {1}{7} \int \frac {9854-6035 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {1207 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{7} \left (\frac {1}{7} \left (67375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-41632 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {1207 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{7} \left (\frac {1}{7} \left (41632 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-67375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {1207 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{7} \left (\frac {1}{7} \left (\frac {83264 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-2450 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {1207 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\) |
Sqrt[1 - 2*x]/(3*(2 + 3*x)^3) + ((52*Sqrt[1 - 2*x])/(7*(2 + 3*x)^2) + (3*( (1207*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + ((83264*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2* x]])/Sqrt[21] - 2450*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/7))/7)/3
3.19.36.3.1 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 = 3.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(-\frac {21726 x^{3}+18833 x^{2}-4674 x -5087}{49 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {83264 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}-50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(69\) |
| derivativedivides | \(-50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1207 \left (1-2 x \right )^{\frac {5}{2}}}{147}-\frac {7346 \left (1-2 x \right )^{\frac {3}{2}}}{189}+\frac {1243 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {83264 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\) | \(75\) |
| default | \(-50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1207 \left (1-2 x \right )^{\frac {5}{2}}}{147}-\frac {7346 \left (1-2 x \right )^{\frac {3}{2}}}{189}+\frac {1243 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {83264 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\) | \(75\) |
| pseudoelliptic | \(\frac {83264 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}-51450 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \sqrt {55}+21 \sqrt {1-2 x}\, \left (10863 x^{2}+14848 x +5087\right )}{1029 \left (2+3 x \right )^{3}}\) | \(80\) |
| trager | \(\frac {\left (10863 x^{2}+14848 x +5087\right ) \sqrt {1-2 x}}{49 \left (2+3 x \right )^{3}}+\frac {41632 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1029}+25 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )\) | \(116\) |
-1/49*(21726*x^3+18833*x^2-4674*x-5087)/(2+3*x)^3/(1-2*x)^(1/2)+83264/1029 *arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50*arctanh(1/11*55^(1/2)*(1- 2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {25725 \, \sqrt {55} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 41632 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (10863 \, x^{2} + 14848 \, x + 5087\right )} \sqrt {-2 \, x + 1}}{1029 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/1029*(25725*sqrt(55)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x + sqrt(55)*sq rt(-2*x + 1) - 8)/(5*x + 3)) + 41632*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8) *log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(10863*x^2 + 1484 8*x + 5087)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
Time = 77.24 (sec) , antiderivative size = 578, normalized size of antiderivative = 5.12 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=- \frac {275 \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} + 25 \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 ) + 660 \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 ) - 264 \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 ) + 112 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \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 {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \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 {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) \]
-275*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 21)/3))/7 + 25*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2* x) + sqrt(55)/5)) + 660*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))) - 264*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)*sqr t(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x ) < sqrt(21)/3))) + 112*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqrt(1 - 2*x) /7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 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) - 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 )))
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=25 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {41632}{1029} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (10863 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 51422 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 60907 \, \sqrt {-2 \, x + 1}\right )}}{49 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1 ))) - 41632/1029*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) + 2/49*(10863*(-2*x + 1)^(5/2) - 51422*(-2*x + 1)^(3/2) + 60907*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
Time = 0.33 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=25 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {41632}{1029} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {10863 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 51422 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 60907 \, \sqrt {-2 \, x + 1}}{196 \, {\left (3 \, x + 2\right )}^{3}} \]
25*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr t(-2*x + 1))) - 41632/1029*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/196*(10863*(2*x - 1)^2*sqrt(-2*x + 1) - 51422*(-2*x + 1)^(3/2) + 60907*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 1.39 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {83264\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1029}-50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {2486\,\sqrt {1-2\,x}}{27}-\frac {14692\,{\left (1-2\,x\right )}^{3/2}}{189}+\frac {2414\,{\left (1-2\,x\right )}^{5/2}}{147}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]