Integrand size = 24, antiderivative size = 132 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)} \, dx=-\frac {12790}{3773 \sqrt {1-2 x}}+\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {8}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {565}{49 \sqrt {1-2 x} (2+3 x)}+\frac {40140}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
40140/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1250/121*arctanh(1 /11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-12790/3773/(1-2*x)^(1/2)+1/7/(2+3*x)^ 3/(1-2*x)^(1/2)+8/7/(2+3*x)^2/(1-2*x)^(1/2)+565/49/(2+3*x)/(1-2*x)^(1/2)
Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {80863+74556 x-299115 x^2-345330 x^3}{3773 \sqrt {1-2 x} (2+3 x)^3}+\frac {40140}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(80863 + 74556*x - 299115*x^2 - 345330*x^3)/(3773*Sqrt[1 - 2*x]*(2 + 3*x)^ 3) + (40140*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1250*Sqrt[5 /11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11
Time = 0.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {114, 27, 168, 27, 168, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^4 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{21} \int \frac {21 (2-5 x)}{(1-2 x)^{3/2} (3 x+2)^3 (5 x+3)}dx+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {2-5 x}{(1-2 x)^{3/2} (3 x+2)^3 (5 x+3)}dx+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \int \frac {10 (11-40 x)}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}dx+\frac {8}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \int \frac {11-40 x}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}dx+\frac {8}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \int \frac {208-1695 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {113}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {8}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \left (-\frac {2}{77} \int -\frac {31364-19185 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {2558}{77 \sqrt {1-2 x}}\right )+\frac {113}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {8}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \left (\frac {1}{77} \int \frac {31364-19185 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {2558}{77 \sqrt {1-2 x}}\right )+\frac {113}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {8}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \left (\frac {1}{77} \left (214375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-132462 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {2558}{77 \sqrt {1-2 x}}\right )+\frac {113}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {8}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \left (\frac {1}{77} \left (132462 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-214375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {2558}{77 \sqrt {1-2 x}}\right )+\frac {113}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {8}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \left (\frac {1}{77} \left (88308 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-85750 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {2558}{77 \sqrt {1-2 x}}\right )+\frac {113}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {8}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
1/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 8/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*(11 3/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (-2558/(77*Sqrt[1 - 2*x]) + (88308*Sqrt[3/ 7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 85750*Sqrt[5/11]*ArcTanh[Sqrt[5/11]* Sqrt[1 - 2*x]])/77)/7))/7
3.22.14.3.1 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*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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[((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] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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 = 69, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {345330 x^{3}+299115 x^{2}-74556 x -80863}{3773 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {40140 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}-\frac {1250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\) | \(69\) |
| derivativedivides | \(-\frac {1250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {486 \left (\frac {1357 \left (1-2 x \right )^{\frac {5}{2}}}{3}-\frac {57806 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {68453 \sqrt {1-2 x}}{27}\right )}{2401 \left (-4-6 x \right )^{3}}+\frac {40140 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {32}{26411 \sqrt {1-2 x}}\) | \(84\) |
| default | \(-\frac {1250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {486 \left (\frac {1357 \left (1-2 x \right )^{\frac {5}{2}}}{3}-\frac {57806 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {68453 \sqrt {1-2 x}}{27}\right )}{2401 \left (-4-6 x \right )^{3}}+\frac {40140 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {32}{26411 \sqrt {1-2 x}}\) | \(84\) |
| pseudoelliptic | \(\frac {\frac {80863}{3773}+\frac {40140 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}}{2401}-\frac {1250 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \sqrt {55}}{121}-\frac {345330 x^{3}}{3773}-\frac {299115 x^{2}}{3773}+\frac {74556 x}{3773}}{\left (2+3 x \right )^{3} \sqrt {1-2 x}}\) | \(96\) |
| trager | \(\frac {\left (345330 x^{3}+299115 x^{2}-74556 x -80863\right ) \sqrt {1-2 x}}{3773 \left (2+3 x \right )^{3} \left (-1+2 x \right )}+\frac {625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}+\frac {90 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1044309\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1044309\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1044309\right )-4683 \sqrt {1-2 x}}{2+3 x}\right )}{2401}\) | \(129\) |
-1/3773*(345330*x^3+299115*x^2-74556*x-80863)/(2+3*x)^3/(1-2*x)^(1/2)+4014 0/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1250/121*arctanh(1/11* 55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {1500625 \, \sqrt {11} \sqrt {5} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 2428470 \, \sqrt {7} \sqrt {3} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (345330 \, x^{3} + 299115 \, x^{2} - 74556 \, x - 80863\right )} \sqrt {-2 \, x + 1}}{290521 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
1/290521*(1500625*sqrt(11)*sqrt(5)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*l og((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 2428470*sqrt(7 )*sqrt(3)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(-(sqrt(7)*sqrt(3)*sqrt (-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(345330*x^3 + 299115*x^2 - 74556*x - 80863)*sqrt(-2*x + 1))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)
Result contains complex when optimal does not.
Time = 13.24 (sec) , antiderivative size = 6412, normalized size of antiderivative = 48.58 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)} \, dx=\text {Too large to display} \]
29774452055040*sqrt(2)*I*(x - 1/2)**(23/2)/(17566693917696*(x - 1/2)**12 + 204944762373120*(x - 1/2)**11 + 1075960002458880*(x - 1/2)**10 + 33474311 18760960*(x - 1/2)**9 + 6834338534136960*(x - 1/2)**8 + 9568073947791744*( x - 1/2)**7 + 9302294115908640*(x - 1/2)**6 + 6201529410605760*(x - 1/2)** 5 + 2713169117140020*(x - 1/2)**4 + 703414215554820*(x - 1/2)**3 + 8206499 1814729*(x - 1/2)**2) + 313609484643840*sqrt(2)*I*(x - 1/2)**(21/2)/(17566 693917696*(x - 1/2)**12 + 204944762373120*(x - 1/2)**11 + 1075960002458880 *(x - 1/2)**10 + 3347431118760960*(x - 1/2)**9 + 6834338534136960*(x - 1/2 )**8 + 9568073947791744*(x - 1/2)**7 + 9302294115908640*(x - 1/2)**6 + 620 1529410605760*(x - 1/2)**5 + 2713169117140020*(x - 1/2)**4 + 7034142155548 20*(x - 1/2)**3 + 82064991814729*(x - 1/2)**2) + 1468099054743552*sqrt(2)* I*(x - 1/2)**(19/2)/(17566693917696*(x - 1/2)**12 + 204944762373120*(x - 1 /2)**11 + 1075960002458880*(x - 1/2)**10 + 3347431118760960*(x - 1/2)**9 + 6834338534136960*(x - 1/2)**8 + 9568073947791744*(x - 1/2)**7 + 930229411 5908640*(x - 1/2)**6 + 6201529410605760*(x - 1/2)**5 + 2713169117140020*(x - 1/2)**4 + 703414215554820*(x - 1/2)**3 + 82064991814729*(x - 1/2)**2) + 4009033876700160*sqrt(2)*I*(x - 1/2)**(17/2)/(17566693917696*(x - 1/2)**1 2 + 204944762373120*(x - 1/2)**11 + 1075960002458880*(x - 1/2)**10 + 33474 31118760960*(x - 1/2)**9 + 6834338534136960*(x - 1/2)**8 + 956807394779174 4*(x - 1/2)**7 + 9302294115908640*(x - 1/2)**6 + 6201529410605760*(x - ...
Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {625}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20070}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (172665 \, {\left (2 \, x - 1\right )}^{3} + 817110 \, {\left (2 \, x - 1\right )}^{2} + 1934226 \, x - 967897\right )}}{3773 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}} \]
625/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* x + 1))) - 20070/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) + 2/3773*(172665*(2*x - 1)^3 + 817110*(2*x - 1)^2 + 1934226*x - 967897)/(27*(-2*x + 1)^(7/2) - 189*(-2*x + 1)^(5/2) + 441*(-2 *x + 1)^(3/2) - 343*sqrt(-2*x + 1))
Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {625}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {20070}{2401} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {32}{26411 \, \sqrt {-2 \, x + 1}} + \frac {9 \, {\left (12213 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 57806 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 68453 \, \sqrt {-2 \, x + 1}\right )}}{9604 \, {\left (3 \, x + 2\right )}^{3}} \]
625/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 20070/2401*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt( -2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 32/26411/sqrt(-2*x + 1) + 9/96 04*(12213*(2*x - 1)^2*sqrt(-2*x + 1) - 57806*(-2*x + 1)^(3/2) + 68453*sqrt (-2*x + 1))/(3*x + 2)^3
Time = 1.43 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {40140\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {\frac {2924\,x}{77}+\frac {25940\,{\left (2\,x-1\right )}^2}{1617}+\frac {12790\,{\left (2\,x-1\right )}^3}{3773}-\frac {39506}{2079}}{\frac {343\,\sqrt {1-2\,x}}{27}-\frac {49\,{\left (1-2\,x\right )}^{3/2}}{3}+7\,{\left (1-2\,x\right )}^{5/2}-{\left (1-2\,x\right )}^{7/2}}-\frac {1250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121} \]