Integrand size = 24, antiderivative size = 145 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx=-\frac {39520}{11319 (1-2 x)^{3/2}}-\frac {446660}{290521 \sqrt {1-2 x}}+\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac {582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac {127710 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401}-\frac {6250}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-39520/11319/(1-2*x)^(3/2)+1/7/(1-2*x)^(3/2)/(2+3*x)^3+57/49/(1-2*x)^(3/2) /(2+3*x)^2+582/49/(1-2*x)^(3/2)/(2+3*x)+127710/16807*arctanh(1/7*21^(1/2)* (1-2*x)^(1/2))*21^(1/2)-6250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^ (1/2)-446660/290521/(1-2*x)^(1/2)
Time = 0.33 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {8496203-9083055 x-47036214 x^2+26376300 x^3+72358920 x^4}{871563 (1-2 x)^{3/2} (2+3 x)^3}+\frac {127710 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401}-\frac {6250}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(8496203 - 9083055*x - 47036214*x^2 + 26376300*x^3 + 72358920*x^4)/(871563 *(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (127710*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/2401 - (6250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121
Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {114, 27, 168, 27, 168, 27, 169, 27, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^4 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{21} \int \frac {3 (8-45 x)}{(1-2 x)^{5/2} (3 x+2)^3 (5 x+3)}dx+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {8-45 x}{(1-2 x)^{5/2} (3 x+2)^3 (5 x+3)}dx+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{14} \int \frac {14 (4-285 x)}{(1-2 x)^{5/2} (3 x+2)^2 (5 x+3)}dx+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\int \frac {4-285 x}{(1-2 x)^{5/2} (3 x+2)^2 (5 x+3)}dx+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{7} \int -\frac {5 (2910 x+521)}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx+\frac {582}{7 (1-2 x)^{3/2} (3 x+2)}+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {5}{7} \int \frac {2910 x+521}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx+\frac {582}{7 (1-2 x)^{3/2} (3 x+2)}+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{7} \left (-\frac {5}{7} \left (\frac {7904}{231 (1-2 x)^{3/2}}-\frac {2}{231} \int \frac {3 (7307-59280 x)}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )+\frac {582}{7 (1-2 x)^{3/2} (3 x+2)}+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {5}{7} \left (\frac {7904}{231 (1-2 x)^{3/2}}-\frac {1}{77} \int \frac {7307-59280 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )+\frac {582}{7 (1-2 x)^{3/2} (3 x+2)}+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{7} \left (-\frac {5}{7} \left (\frac {1}{77} \left (\frac {2}{77} \int -\frac {1098631-669990 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {89332}{77 \sqrt {1-2 x}}\right )+\frac {7904}{231 (1-2 x)^{3/2}}\right )+\frac {582}{7 (1-2 x)^{3/2} (3 x+2)}+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {5}{7} \left (\frac {1}{77} \left (\frac {89332}{77 \sqrt {1-2 x}}-\frac {1}{77} \int \frac {1098631-669990 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx\right )+\frac {7904}{231 (1-2 x)^{3/2}}\right )+\frac {582}{7 (1-2 x)^{3/2} (3 x+2)}+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{7} \left (-\frac {5}{7} \left (\frac {1}{77} \left (\frac {1}{77} \left (4635873 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-7503125 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {89332}{77 \sqrt {1-2 x}}\right )+\frac {7904}{231 (1-2 x)^{3/2}}\right )+\frac {582}{7 (1-2 x)^{3/2} (3 x+2)}+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{7} \left (-\frac {5}{7} \left (\frac {1}{77} \left (\frac {1}{77} \left (7503125 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-4635873 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {89332}{77 \sqrt {1-2 x}}\right )+\frac {7904}{231 (1-2 x)^{3/2}}\right )+\frac {582}{7 (1-2 x)^{3/2} (3 x+2)}+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{7} \left (-\frac {5}{7} \left (\frac {1}{77} \left (\frac {1}{77} \left (3001250 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-3090582 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {89332}{77 \sqrt {1-2 x}}\right )+\frac {7904}{231 (1-2 x)^{3/2}}\right )+\frac {582}{7 (1-2 x)^{3/2} (3 x+2)}+\frac {57}{7 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}\) |
1/(7*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (57/(7*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + 582/(7*(1 - 2*x)^(3/2)*(2 + 3*x)) - (5*(7904/(231*(1 - 2*x)^(3/2)) + (8933 2/(77*Sqrt[1 - 2*x]) + (-3090582*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x] ] + 3001250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77)/77))/7)/7
3.22.79.3.1 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 = 1.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.64
| method | result | size |
| derivativedivides | \(-\frac {6250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {1458 \left (\frac {1438 \left (1-2 x \right )^{\frac {5}{2}}}{3}-\frac {61250 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {72520 \sqrt {1-2 x}}{27}\right )}{16807 \left (-4-6 x \right )^{3}}+\frac {127710 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{16807}+\frac {32}{79233 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5344}{2033647 \sqrt {1-2 x}}\) | \(93\) |
| default | \(-\frac {6250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {1458 \left (\frac {1438 \left (1-2 x \right )^{\frac {5}{2}}}{3}-\frac {61250 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {72520 \sqrt {1-2 x}}{27}\right )}{16807 \left (-4-6 x \right )^{3}}+\frac {127710 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{16807}+\frac {32}{79233 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5344}{2033647 \sqrt {1-2 x}}\) | \(93\) |
| pseudoelliptic | \(\frac {-\frac {127710 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{3} \sqrt {21}}{16807}+\frac {6250 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{3} \sqrt {55}}{1331}+\frac {24119640 x^{4}}{290521}+\frac {8792100 x^{3}}{290521}-\frac {15678738 x^{2}}{290521}-\frac {3027685 x}{290521}+\frac {8496203}{871563}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{3}}\) | \(111\) |
| trager | \(\frac {\left (72358920 x^{4}+26376300 x^{3}-47036214 x^{2}-9083055 x +8496203\right ) \sqrt {1-2 x}}{871563 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2}}-\frac {63855 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{16807}+\frac {3125 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\) | \(133\) |
-6250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-1458/16807*(1438/ 3*(1-2*x)^(5/2)-61250/27*(1-2*x)^(3/2)+72520/27*(1-2*x)^(1/2))/(-4-6*x)^3+ 127710/16807*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+32/79233/(1-2*x) ^(3/2)+5344/2033647/(1-2*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {157565625 \, \sqrt {11} \sqrt {5} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 254973015 \, \sqrt {7} \sqrt {3} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (72358920 \, x^{4} + 26376300 \, x^{3} - 47036214 \, x^{2} - 9083055 \, x + 8496203\right )} \sqrt {-2 \, x + 1}}{67110351 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
1/67110351*(157565625*sqrt(11)*sqrt(5)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^ 2 + 4*x + 8)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 254973015*sqrt(7)*sqrt(3)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)* log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(72358920* x^4 + 26376300*x^3 - 47036214*x^2 - 9083055*x + 8496203)*sqrt(-2*x + 1))/( 108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)
Result contains complex when optimal does not.
Time = 15.07 (sec) , antiderivative size = 5506, normalized size of antiderivative = 37.97 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx=\text {Too large to display} \]
519900576814080*sqrt(2)*I*(x - 1/2)**(23/2)/(676317715831296*(x - 1/2)**12 + 7101336016228608*(x - 1/2)**11 + 33139568075733504*(x - 1/2)**10 + 9021 3268650607872*(x - 1/2)**9 + 157873220138563776*(x - 1/2)**8 + 18418542349 4991072*(x - 1/2)**7 + 143255329384993056*(x - 1/2)**6 + 71627664692496528 *(x - 1/2)**5 + 20891402201978154*(x - 1/2)**4 + 2708144729886057*(x - 1/2 )**3) + 4868619539857920*sqrt(2)*I*(x - 1/2)**(21/2)/(676317715831296*(x - 1/2)**12 + 7101336016228608*(x - 1/2)**11 + 33139568075733504*(x - 1/2)** 10 + 90213268650607872*(x - 1/2)**9 + 157873220138563776*(x - 1/2)**8 + 18 4185423494991072*(x - 1/2)**7 + 143255329384993056*(x - 1/2)**6 + 71627664 692496528*(x - 1/2)**5 + 20891402201978154*(x - 1/2)**4 + 2708144729886057 *(x - 1/2)**3) + 19946011353295104*sqrt(2)*I*(x - 1/2)**(19/2)/(6763177158 31296*(x - 1/2)**12 + 7101336016228608*(x - 1/2)**11 + 33139568075733504*( x - 1/2)**10 + 90213268650607872*(x - 1/2)**9 + 157873220138563776*(x - 1/ 2)**8 + 184185423494991072*(x - 1/2)**7 + 143255329384993056*(x - 1/2)**6 + 71627664692496528*(x - 1/2)**5 + 20891402201978154*(x - 1/2)**4 + 270814 4729886057*(x - 1/2)**3) + 46692212229919872*sqrt(2)*I*(x - 1/2)**(17/2)/( 676317715831296*(x - 1/2)**12 + 7101336016228608*(x - 1/2)**11 + 331395680 75733504*(x - 1/2)**10 + 90213268650607872*(x - 1/2)**9 + 1578732201385637 76*(x - 1/2)**8 + 184185423494991072*(x - 1/2)**7 + 143255329384993056*(x - 1/2)**6 + 71627664692496528*(x - 1/2)**5 + 20891402201978154*(x - 1/2...
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {3125}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {63855}{16807} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4 \, {\left (9044865 \, {\left (2 \, x - 1\right )}^{4} + 42773535 \, {\left (2 \, x - 1\right )}^{3} + 50533308 \, {\left (2 \, x - 1\right )}^{2} - 315168 \, x + 187768\right )}}{871563 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 343 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
3125/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(- 2*x + 1))) - 63855/16807*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt (21) + 3*sqrt(-2*x + 1))) - 4/871563*(9044865*(2*x - 1)^4 + 42773535*(2*x - 1)^3 + 50533308*(2*x - 1)^2 - 315168*x + 187768)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) + 441*(-2*x + 1)^(5/2) - 343*(-2*x + 1)^(3/2))
Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {3125}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {63855}{16807} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {4 \, {\left (9044865 \, {\left (2 \, x - 1\right )}^{4} + 42773535 \, {\left (2 \, x - 1\right )}^{3} + 50533308 \, {\left (2 \, x - 1\right )}^{2} - 315168 \, x + 187768\right )}}{871563 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}^{3}} \]
3125/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 63855/16807*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sq rt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 4/871563*(9044865*(2*x - 1) ^4 + 42773535*(2*x - 1)^3 + 50533308*(2*x - 1)^2 - 315168*x + 187768)/(3*( -2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))^3
Time = 1.62 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx=\frac {\frac {50928\,{\left (2\,x-1\right )}^2}{5929}-\frac {8576\,x}{160083}+\frac {905260\,{\left (2\,x-1\right )}^3}{124509}+\frac {446660\,{\left (2\,x-1\right )}^4}{290521}+\frac {15328}{480249}}{\frac {343\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {49\,{\left (1-2\,x\right )}^{5/2}}{3}+7\,{\left (1-2\,x\right )}^{7/2}-{\left (1-2\,x\right )}^{9/2}}+\frac {127710\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{16807}-\frac {6250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]
((50928*(2*x - 1)^2)/5929 - (8576*x)/160083 + (905260*(2*x - 1)^3)/124509 + (446660*(2*x - 1)^4)/290521 + 15328/480249)/((343*(1 - 2*x)^(3/2))/27 - (49*(1 - 2*x)^(5/2))/3 + 7*(1 - 2*x)^(7/2) - (1 - 2*x)^(9/2)) + (127710*21 ^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/16807 - (6250*55^(1/2)*atanh(( 55^(1/2)*(1 - 2*x)^(1/2))/11))/1331