Integrand size = 24, antiderivative size = 173 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {91 \sqrt {1-2 x}}{18 (2+3 x)^5}+\frac {2165 \sqrt {1-2 x}}{72 (2+3 x)^4}+\frac {302651 \sqrt {1-2 x}}{1512 (2+3 x)^3}+\frac {31700335 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {736065535 \sqrt {1-2 x}}{49392 (2+3 x)}+\frac {25388847535 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696 \sqrt {21}}-30250 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/18*(1-2*x)^(3/2)/(2+3*x)^6+25388847535/518616*arctanh(1/7*21^(1/2)*(1-2* x)^(1/2))*21^(1/2)-30250*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+91/ 18*(1-2*x)^(1/2)/(2+3*x)^5+2165/72*(1-2*x)^(1/2)/(2+3*x)^4+302651/1512*(1- 2*x)^(1/2)/(2+3*x)^3+31700335/21168*(1-2*x)^(1/2)/(2+3*x)^2+736065535/4939 2*(1-2*x)^(1/2)/(2+3*x)
Time = 0.72 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (24823128464+184131053992 x+546491397114 x^2+811194684822 x^3+602204446665 x^4+178863925005 x^5\right )}{49392 (2+3 x)^6}+\frac {25388847535 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696 \sqrt {21}}-30250 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(24823128464 + 184131053992*x + 546491397114*x^2 + 81119468 4822*x^3 + 602204446665*x^4 + 178863925005*x^5))/(49392*(2 + 3*x)^6) + (25 388847535*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[21]) - 30250*Sqrt[ 55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.16, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {109, 27, 166, 27, 168, 27, 168, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^7 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{18} \int \frac {3 (87-97 x) \sqrt {1-2 x}}{(3 x+2)^6 (5 x+3)}dx+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {(87-97 x) \sqrt {1-2 x}}{(3 x+2)^6 (5 x+3)}dx+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}-\frac {1}{15} \int -\frac {5 (2451-3901 x)}{\sqrt {1-2 x} (3 x+2)^5 (5 x+3)}dx\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \int \frac {2451-3901 x}{\sqrt {1-2 x} (3 x+2)^5 (5 x+3)}dx+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{28} \int \frac {7 (50367-75775 x)}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{4} \int \frac {50367-75775 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{21} \int \frac {5 (1104519-1513255 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {302651 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {5}{21} \int \frac {1104519-1513255 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {302651 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {1}{14} \int \frac {3 (27937479-31700335 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {6340067 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {302651 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \int \frac {27937479-31700335 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {6340067 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {302651 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1201879479-736065535 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {147213107 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {6340067 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {302651 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \left (8217594000 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-5077769507 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {147213107 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {6340067 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {302651 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \left (5077769507 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-8217594000 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {147213107 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {6340067 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {302651 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \left (\frac {10155539014 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-298821600 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {147213107 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {6340067 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {302651 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {2165 \sqrt {1-2 x}}{4 (3 x+2)^4}\right )+\frac {91 \sqrt {1-2 x}}{3 (3 x+2)^5}\right )+\frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
(7*(1 - 2*x)^(3/2))/(18*(2 + 3*x)^6) + ((91*Sqrt[1 - 2*x])/(3*(2 + 3*x)^5) + ((2165*Sqrt[1 - 2*x])/(4*(2 + 3*x)^4) + ((302651*Sqrt[1 - 2*x])/(21*(2 + 3*x)^3) + (5*((6340067*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2) + (3*((147213107* Sqrt[1 - 2*x])/(7*(2 + 3*x)) + ((10155539014*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2* x]])/Sqrt[21] - 298821600*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/7))/ 14))/21)/4)/3)/6
3.20.79.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 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[(((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 = 1.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(-\frac {357727850010 x^{6}+1025544968325 x^{5}+1020184922979 x^{4}+281788109406 x^{3}-178229289130 x^{2}-134484797064 x -24823128464}{49392 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {25388847535 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{518616}-30250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(84\) |
| pseudoelliptic | \(\frac {50777695070 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}-31376268000 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{6} \sqrt {55}+21 \sqrt {1-2 x}\, \left (178863925005 x^{5}+602204446665 x^{4}+811194684822 x^{3}+546491397114 x^{2}+184131053992 x +24823128464\right )}{1037232 \left (2+3 x \right )^{6}}\) | \(95\) |
| derivativedivides | \(-30250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {1458 \left (\frac {736065535 \left (1-2 x \right )^{\frac {11}{2}}}{148176}-\frac {11104383695 \left (1-2 x \right )^{\frac {9}{2}}}{190512}+\frac {1240999441 \left (1-2 x \right )^{\frac {7}{2}}}{4536}-\frac {3744956269 \left (1-2 x \right )^{\frac {5}{2}}}{5832}+\frac {79114433335 \left (1-2 x \right )^{\frac {3}{2}}}{104976}-\frac {37144080785 \sqrt {1-2 x}}{104976}\right )}{\left (-4-6 x \right )^{6}}+\frac {25388847535 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{518616}\) | \(102\) |
| default | \(-30250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {1458 \left (\frac {736065535 \left (1-2 x \right )^{\frac {11}{2}}}{148176}-\frac {11104383695 \left (1-2 x \right )^{\frac {9}{2}}}{190512}+\frac {1240999441 \left (1-2 x \right )^{\frac {7}{2}}}{4536}-\frac {3744956269 \left (1-2 x \right )^{\frac {5}{2}}}{5832}+\frac {79114433335 \left (1-2 x \right )^{\frac {3}{2}}}{104976}-\frac {37144080785 \sqrt {1-2 x}}{104976}\right )}{\left (-4-6 x \right )^{6}}+\frac {25388847535 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{518616}\) | \(102\) |
| trager | \(\frac {\left (178863925005 x^{5}+602204446665 x^{4}+811194684822 x^{3}+546491397114 x^{2}+184131053992 x +24823128464\right ) \sqrt {1-2 x}}{49392 \left (2+3 x \right )^{6}}-15125 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )-\frac {25388847535 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{1037232}\) | \(131\) |
-1/49392*(357727850010*x^6+1025544968325*x^5+1020184922979*x^4+28178810940 6*x^3-178229289130*x^2-134484797064*x-24823128464)/(2+3*x)^6/(1-2*x)^(1/2) +25388847535/518616*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-30250*arc tanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx=\frac {15688134000 \, \sqrt {55} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 25388847535 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (178863925005 \, x^{5} + 602204446665 \, x^{4} + 811194684822 \, x^{3} + 546491397114 \, x^{2} + 184131053992 \, x + 24823128464\right )} \sqrt {-2 \, x + 1}}{1037232 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
1/1037232*(15688134000*sqrt(55)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3) ) + 25388847535*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160* x^2 + 576*x + 64)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21* (178863925005*x^5 + 602204446665*x^4 + 811194684822*x^3 + 546491397114*x^2 + 184131053992*x + 24823128464)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 486 0*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx=\text {Timed out} \]
Time = 0.38 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx=15125 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {25388847535}{1037232} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {178863925005 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2098728518355 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 9851053562658 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 23121360004806 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 27136250633905 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 12740419709255 \, \sqrt {-2 \, x + 1}}{24696 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]
15125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 25388847535/1037232*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/( sqrt(21) + 3*sqrt(-2*x + 1))) - 1/24696*(178863925005*(-2*x + 1)^(11/2) - 2098728518355*(-2*x + 1)^(9/2) + 9851053562658*(-2*x + 1)^(7/2) - 23121360 004806*(-2*x + 1)^(5/2) + 27136250633905*(-2*x + 1)^(3/2) - 12740419709255 *sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)
Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx=15125 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {25388847535}{1037232} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {178863925005 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 2098728518355 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 9851053562658 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 23121360004806 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 27136250633905 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 12740419709255 \, \sqrt {-2 \, x + 1}}{1580544 \, {\left (3 \, x + 2\right )}^{6}} \]
15125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5* sqrt(-2*x + 1))) - 25388847535/1037232*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1580544*(178863925005 *(2*x - 1)^5*sqrt(-2*x + 1) + 2098728518355*(2*x - 1)^4*sqrt(-2*x + 1) + 9 851053562658*(2*x - 1)^3*sqrt(-2*x + 1) + 23121360004806*(2*x - 1)^2*sqrt( -2*x + 1) - 27136250633905*(-2*x + 1)^(3/2) + 12740419709255*sqrt(-2*x + 1 ))/(3*x + 2)^6
Time = 1.46 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx=\frac {25388847535\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{518616}-30250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {37144080785\,\sqrt {1-2\,x}}{52488}-\frac {79114433335\,{\left (1-2\,x\right )}^{3/2}}{52488}+\frac {3744956269\,{\left (1-2\,x\right )}^{5/2}}{2916}-\frac {1240999441\,{\left (1-2\,x\right )}^{7/2}}{2268}+\frac {11104383695\,{\left (1-2\,x\right )}^{9/2}}{95256}-\frac {736065535\,{\left (1-2\,x\right )}^{11/2}}{74088}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]
(25388847535*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/518616 - 30250* 55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((37144080785*(1 - 2*x)^(1 /2))/52488 - (79114433335*(1 - 2*x)^(3/2))/52488 + (3744956269*(1 - 2*x)^( 5/2))/2916 - (1240999441*(1 - 2*x)^(7/2))/2268 + (11104383695*(1 - 2*x)^(9 /2))/95256 - (736065535*(1 - 2*x)^(11/2))/74088)/((67228*x)/81 + (12005*(2 *x - 1)^2)/27 + (6860*(2*x - 1)^3)/27 + (245*(2*x - 1)^4)/3 + 14*(2*x - 1) ^5 + (2*x - 1)^6 - 184877/729)