Integrand size = 24, antiderivative size = 158 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-335579/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+6650/11*arctanh(1 /11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-48645/98*(1-2*x)^(1/2)/(3+5*x)+1/3*(1 -2*x)^(1/2)/(2+3*x)^3/(3+5*x)+139/42*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)+7261/ 147*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)
Time = 0.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (369116+1692159 x+2583264 x^2+1313415 x^3\right )}{98 (2+3 x)^3 (3+5 x)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-1/98*(Sqrt[1 - 2*x]*(369116 + 1692159*x + 2583264*x^2 + 1313415*x^3))/((2 + 3*x)^3*(3 + 5*x)) - (335579*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]) /49 + 6650*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.25 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {110, 25, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^4 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}-\frac {1}{3} \int -\frac {23-35 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \int \frac {23-35 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \int \frac {2524-3475 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx+\frac {139 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \left (\frac {1}{7} \int \frac {3 (63453-72610 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {14522 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {139 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \int \frac {63453-72610 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {14522 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {139 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-\frac {1}{11} \int \frac {11 (238289-145935 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {48645 \sqrt {1-2 x}}{5 x+3}\right )+\frac {14522 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {139 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-\int \frac {238289-145935 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {48645 \sqrt {1-2 x}}{5 x+3}\right )+\frac {14522 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {139 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (1006737 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-1629250 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {48645 \sqrt {1-2 x}}{5 x+3}\right )+\frac {14522 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {139 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (1629250 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-1006737 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {48645 \sqrt {1-2 x}}{5 x+3}\right )+\frac {14522 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {139 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-671158 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+651700 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-\frac {48645 \sqrt {1-2 x}}{5 x+3}\right )+\frac {14522 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {139 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\) |
Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*x)) + ((139*Sqrt[1 - 2*x])/(14*(2 + 3* x)^2*(3 + 5*x)) + ((14522*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) + (3*((-4 8645*Sqrt[1 - 2*x])/(3 + 5*x) - 671158*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 651700*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/7)/14)/3
3.19.47.3.1 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 = 3.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(\frac {2626830 x^{4}+3853113 x^{3}+801054 x^{2}-953927 x -369116}{98 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (3+5 x \right )}-\frac {335579 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {6650 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}\) | \(81\) |
| derivativedivides | \(\frac {50 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+\frac {6650 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {\frac {196533 \left (1-2 x \right )^{\frac {5}{2}}}{49}-\frac {132276 \left (1-2 x \right )^{\frac {3}{2}}}{7}+22263 \sqrt {1-2 x}}{\left (-4-6 x \right )^{3}}-\frac {335579 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(91\) |
| default | \(\frac {50 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+\frac {6650 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {\frac {196533 \left (1-2 x \right )^{\frac {5}{2}}}{49}-\frac {132276 \left (1-2 x \right )^{\frac {3}{2}}}{7}+22263 \sqrt {1-2 x}}{\left (-4-6 x \right )^{3}}-\frac {335579 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(91\) |
| pseudoelliptic | \(\frac {-7382738 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right ) \sqrt {21}+4561900 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right ) \sqrt {55}-77 \sqrt {1-2 x}\, \left (1313415 x^{3}+2583264 x^{2}+1692159 x +369116\right )}{7546 \left (2+3 x \right )^{3} \left (3+5 x \right )}\) | \(102\) |
| trager | \(-\frac {\left (1313415 x^{3}+2583264 x^{2}+1692159 x +369116\right ) \sqrt {1-2 x}}{98 \left (2+3 x \right )^{3} \left (3+5 x \right )}+\frac {335579 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{686}-\frac {3325 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{11}\) | \(128\) |
1/98*(2626830*x^4+3853113*x^3+801054*x^2-953927*x-369116)/(2+3*x)^3/(1-2*x )^(1/2)/(3+5*x)-335579/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+66 50/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx=\frac {2280950 \, \sqrt {11} \sqrt {5} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 3691369 \, \sqrt {7} \sqrt {3} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (1313415 \, x^{3} + 2583264 \, x^{2} + 1692159 \, x + 369116\right )} \sqrt {-2 \, x + 1}}{7546 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
1/7546*(2280950*sqrt(11)*sqrt(5)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24 )*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 3691369*sq rt(7)*sqrt(3)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((sqrt(7)*sqrt (3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(1313415*x^3 + 2583264*x^2 + 1692159*x + 369116)*sqrt(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)
Time = 88.21 (sec) , antiderivative size = 699, normalized size of antiderivative = 4.42 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx=\frac {3350 \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 {3350 \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} - 6060 \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 ) + 1632 \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 ) - 336 \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 ) - 5500 \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 ) \]
3350*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 21)/3))/7 - 3350*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/11 - 6060*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))) + 1632*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))) - 336*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(2 1)*sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 1/(4 8*(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) < sqr t(21)/3))) - 5500*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)/1 1 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sq rt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {3325}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {335579}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1313415 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 9106773 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 21041937 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 16201507 \, \sqrt {-2 \, x + 1}}{49 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \]
-3325/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 *x + 1))) + 335579/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) + 1/49*(1313415*(-2*x + 1)^(7/2) - 9106773*(-2*x + 1)^(5/2) + 21041937*(-2*x + 1)^(3/2) - 16201507*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 1242*(2*x - 1)^3 + 4284*(2*x - 1)^2 + 13132*x - 2793)
Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {3325}{11} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {335579}{686} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {125 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {3 \, {\left (65511 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 308644 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 363629 \, \sqrt {-2 \, x + 1}\right )}}{392 \, {\left (3 \, x + 2\right )}^{3}} \]
-3325/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 335579/686*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt (-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125*sqrt(-2*x + 1)/(5*x + 3) - 3/392*(65511*(2*x - 1)^2*sqrt(-2*x + 1) - 308644*(-2*x + 1)^(3/2) + 3636 29*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 1.56 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx=\frac {6650\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}-\frac {335579\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {330643\,\sqrt {1-2\,x}}{135}-\frac {111333\,{\left (1-2\,x\right )}^{3/2}}{35}+\frac {3035591\,{\left (1-2\,x\right )}^{5/2}}{2205}-\frac {9729\,{\left (1-2\,x\right )}^{7/2}}{49}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \]
(6650*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/11 - (335579*21^(1/2) *atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - ((330643*(1 - 2*x)^(1/2))/135 - (111333*(1 - 2*x)^(3/2))/35 + (3035591*(1 - 2*x)^(5/2))/2205 - (9729*(1 - 2*x)^(7/2))/49)/((13132*x)/135 + (476*(2*x - 1)^2)/15 + (46*(2*x - 1)^3) /5 + (2*x - 1)^4 - 931/45)