Integrand size = 24, antiderivative size = 153 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {43467}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
43467/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-66325/121*arctanh(1/ 11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-1045/14*(1-2*x)^(1/2)/(3+5*x)^2+1/2*(1 -2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2+139/14*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+346 55/77*(1-2*x)^(1/2)/(3+5*x)
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (788875+3748007 x+5926515 x^2+3118950 x^3\right )}{154 \left (6+19 x+15 x^2\right )^2}+\frac {43467}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(788875 + 3748007*x + 5926515*x^2 + 3118950*x^3))/(154*(6 + 19*x + 15*x^2)^2) + (43467*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (66325*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11
Time = 0.25 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {110, 25, 168, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^3 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}-\frac {1}{2} \int -\frac {23-35 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \int \frac {23-35 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \int \frac {2513-3475 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (-\frac {1}{22} \int \frac {22 (8219-9405 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (-\int \frac {8219-9405 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{11} \int \frac {339517-207930 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {69310 \sqrt {1-2 x}}{11 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{11} \left (2321375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-1434411 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {69310 \sqrt {1-2 x}}{11 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{11} \left (1434411 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-2321375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {69310 \sqrt {1-2 x}}{11 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{11} \left (956274 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-928550 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {69310 \sqrt {1-2 x}}{11 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\) |
Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^2) + ((139*Sqrt[1 - 2*x])/(7*(2 + 3 *x)*(3 + 5*x)^2) + ((-1045*Sqrt[1 - 2*x])/(3 + 5*x)^2 + (69310*Sqrt[1 - 2* x])/(11*(3 + 5*x)) + (956274*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 928550*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11)/7)/2
3.19.55.3.1 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 = 1.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \left (3118950 x^{3}+5926515 x^{2}+3748007 x +788875\right )}{154 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}}+\frac {43467 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}-\frac {66325 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\) | \(79\) |
| derivativedivides | \(\frac {-\frac {24875 \left (1-2 x \right )^{\frac {3}{2}}}{11}+4925 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {66325 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {972 \left (\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{252}-\frac {211 \sqrt {1-2 x}}{108}\right )}{\left (-4-6 x \right )^{2}}+\frac {43467 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(94\) |
| default | \(\frac {-\frac {24875 \left (1-2 x \right )^{\frac {3}{2}}}{11}+4925 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {66325 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {972 \left (\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{252}-\frac {211 \sqrt {1-2 x}}{108}\right )}{\left (-4-6 x \right )^{2}}+\frac {43467 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(94\) |
| pseudoelliptic | \(\frac {10519014 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {21}-6499850 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {55}+77 \sqrt {1-2 x}\, \left (3118950 x^{3}+5926515 x^{2}+3748007 x +788875\right )}{11858 \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(102\) |
| trager | \(\frac {\left (3118950 x^{3}+5926515 x^{2}+3748007 x +788875\right ) \sqrt {1-2 x}}{154 \left (15 x^{2}+19 x +6\right )^{2}}+\frac {175 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7900255\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7900255\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7900255\right )+20845 \sqrt {1-2 x}}{3+5 x}\right )}{242}-\frac {43467 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{98}\) | \(126\) |
-1/154*(-1+2*x)*(3118950*x^3+5926515*x^2+3748007*x+788875)/(15*x^2+19*x+6) ^2/(1-2*x)^(1/2)+43467/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-663 25/121*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 = 1.06 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {3249925 \, \sqrt {11} \sqrt {5} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5259507 \, \sqrt {7} \sqrt {3} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (3118950 \, x^{3} + 5926515 \, x^{2} + 3748007 \, x + 788875\right )} \sqrt {-2 \, x + 1}}{11858 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
1/11858*(3249925*sqrt(11)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 3 6)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 5259507*sq rt(7)*sqrt(3)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(-(sqrt(7)*sqr t(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(3118950*x^3 + 5926515*x^2 + 3748007*x + 788875)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
Time = 76.25 (sec) , antiderivative size = 654, normalized size of antiderivative = 4.27 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=- \frac {3060 \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 {3060 \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} + 3708 \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 ) - 504 \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 ) + 10100 \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 ) + 2200 \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 ) \]
-3060*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3))/7 + 3060*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/11 + 3708*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))) - 504*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))) + 10100*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))) + 2200*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)*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 = 0.95 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {66325}{242} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {43467}{98} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (1559475 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 10604940 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 24027469 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 18137504 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \]
66325/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(- 2*x + 1))) - 43467/98*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) - 2/77*(1559475*(-2*x + 1)^(7/2) - 10604940*(-2*x + 1)^(5/2) + 24027469*(-2*x + 1)^(3/2) - 18137504*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040*(2*x - 1)^3 + 6934*(2*x - 1)^2 + 20944*x - 4543)
Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {66325}{242} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {43467}{98} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (1559475 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 10604940 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 24027469 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 18137504 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]
66325/242*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 43467/98*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt( -2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/77*(1559475*(2*x - 1)^3*sqrt (-2*x + 1) + 10604940*(2*x - 1)^2*sqrt(-2*x + 1) - 24027469*(-2*x + 1)^(3/ 2) + 18137504*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {43467\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {66325\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}+\frac {\frac {471104\,\sqrt {1-2\,x}}{225}-\frac {48054938\,{\left (1-2\,x\right )}^{3/2}}{17325}+\frac {1413992\,{\left (1-2\,x\right )}^{5/2}}{1155}-\frac {13862\,{\left (1-2\,x\right )}^{7/2}}{77}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \]
(43467*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - (66325*55^(1/2)* atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 + ((471104*(1 - 2*x)^(1/2))/225 - (48054938*(1 - 2*x)^(3/2))/17325 + (1413992*(1 - 2*x)^(5/2))/1155 - (138 62*(1 - 2*x)^(7/2))/77)/((20944*x)/225 + (6934*(2*x - 1)^2)/225 + (136*(2* x - 1)^3)/15 + (2*x - 1)^4 - 4543/225)