Integrand size = 24, antiderivative size = 153 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}+\frac {20149879 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/15*(1-2*x)^(3/2)/(2+3*x)^5+20149879/2058*arctanh(1/7*21^(1/2)*(1-2*x)^(1 /2))*21^(1/2)-6050*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+63/10*(1- 2*x)^(1/2)/(2+3*x)^4+1201/30*(1-2*x)^(1/2)/(2+3*x)^3+25159/84*(1-2*x)^(1/2 )/(2+3*x)^2+584179/196*(1-2*x)^(1/2)/(2+3*x)
Time = 0.67 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (147756688+874383298 x+1941349752 x^2+1916515215 x^3+709777485 x^4\right )}{2940 (2+3 x)^5}+\frac {20149879 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(147756688 + 874383298*x + 1941349752*x^2 + 1916515215*x^3 + 709777485*x^4))/(2940*(2 + 3*x)^5) + (20149879*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(98*Sqrt[21]) - 6050*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.26 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {109, 27, 166, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^6 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{15} \int \frac {3 (76-75 x) \sqrt {1-2 x}}{(3 x+2)^5 (5 x+3)}dx+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {(76-75 x) \sqrt {1-2 x}}{(3 x+2)^5 (5 x+3)}dx+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{5} \left (\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}-\frac {1}{12} \int -\frac {6 (1399-2105 x)}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \int \frac {1399-2105 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx+\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{21} \int \frac {35 (4383-6005 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {1201 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {5}{3} \int \frac {4383-6005 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {1201 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {5}{3} \left (\frac {1}{14} \int \frac {3 (110863-125795 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {25159 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {1201 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {5}{3} \left (\frac {3}{14} \int \frac {110863-125795 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {25159 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {1201 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {5}{3} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {4769363-2920895 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {584179 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {25159 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {1201 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {5}{3} \left (\frac {3}{14} \left (\frac {1}{7} \left (32609500 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-20149879 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {584179 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {25159 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {1201 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {5}{3} \left (\frac {3}{14} \left (\frac {1}{7} \left (20149879 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-32609500 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {584179 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {25159 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {1201 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (\frac {5}{3} \left (\frac {3}{14} \left (\frac {1}{7} \left (\frac {40299758 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-1185800 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {584179 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {25159 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {1201 \sqrt {1-2 x}}{3 (3 x+2)^3}\right )+\frac {63 \sqrt {1-2 x}}{2 (3 x+2)^4}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
(7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5) + ((63*Sqrt[1 - 2*x])/(2*(2 + 3*x)^4) + ((1201*Sqrt[1 - 2*x])/(3*(2 + 3*x)^3) + (5*((25159*Sqrt[1 - 2*x])/(14*( 2 + 3*x)^2) + (3*((584179*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + ((40299758*ArcTan h[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 1185800*Sqrt[55]*ArcTanh[Sqrt[5/11] *Sqrt[1 - 2*x]])/7))/14))/3)/2)/5
3.20.78.3.1 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 = 1.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {1419554970 x^{5}+3123252945 x^{4}+1966184289 x^{3}-192583156 x^{2}-578869922 x -147756688}{2940 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {20149879 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2058}-6050 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(79\) |
| pseudoelliptic | \(\frac {201498790 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}-124509000 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{5} \sqrt {55}+7 \sqrt {1-2 x}\, \left (709777485 x^{4}+1916515215 x^{3}+1941349752 x^{2}+874383298 x +147756688\right )}{20580 \left (2+3 x \right )^{5}}\) | \(90\) |
| derivativedivides | \(-6050 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {486 \left (\frac {584179 \left (1-2 x \right )^{\frac {9}{2}}}{588}-\frac {504319 \left (1-2 x \right )^{\frac {7}{2}}}{54}+\frac {13335122 \left (1-2 x \right )^{\frac {5}{2}}}{405}-\frac {75232787 \left (1-2 x \right )^{\frac {3}{2}}}{1458}+\frac {29479429 \sqrt {1-2 x}}{972}\right )}{\left (-4-6 x \right )^{5}}+\frac {20149879 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2058}\) | \(93\) |
| default | \(-6050 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {486 \left (\frac {584179 \left (1-2 x \right )^{\frac {9}{2}}}{588}-\frac {504319 \left (1-2 x \right )^{\frac {7}{2}}}{54}+\frac {13335122 \left (1-2 x \right )^{\frac {5}{2}}}{405}-\frac {75232787 \left (1-2 x \right )^{\frac {3}{2}}}{1458}+\frac {29479429 \sqrt {1-2 x}}{972}\right )}{\left (-4-6 x \right )^{5}}+\frac {20149879 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2058}\) | \(93\) |
| trager | \(\frac {\left (709777485 x^{4}+1916515215 x^{3}+1941349752 x^{2}+874383298 x +147756688\right ) \sqrt {1-2 x}}{2940 \left (2+3 x \right )^{5}}+3025 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )+\frac {20149879 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{4116}\) | \(126\) |
-1/2940*(1419554970*x^5+3123252945*x^4+1966184289*x^3-192583156*x^2-578869 922*x-147756688)/(2+3*x)^5/(1-2*x)^(1/2)+20149879/2058*arctanh(1/7*21^(1/2 )*(1-2*x)^(1/2))*21^(1/2)-6050*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/ 2)
Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.11 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {62254500 \, \sqrt {55} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 100749395 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (709777485 \, x^{4} + 1916515215 \, x^{3} + 1941349752 \, x^{2} + 874383298 \, x + 147756688\right )} \sqrt {-2 \, x + 1}}{20580 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/20580*(62254500*sqrt(55)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 100749395*sqrt (21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x - sqrt (21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 7*(709777485*x^4 + 1916515215*x^3 + 1941349752*x^2 + 874383298*x + 147756688)*sqrt(-2*x + 1))/(243*x^5 + 810*x ^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\text {Timed out} \]
Time = 0.37 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=3025 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20149879}{4116} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {709777485 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 6672140370 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 23523155208 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 36864065630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 21667380315 \, \sqrt {-2 \, x + 1}}{1470 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
3025*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 20149879/4116*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) + 1/1470*(709777485*(-2*x + 1)^(9/2) - 6672140370*( -2*x + 1)^(7/2) + 23523155208*(-2*x + 1)^(5/2) - 36864065630*(-2*x + 1)^(3 /2) + 21667380315*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13 230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)
Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=3025 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {20149879}{4116} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {709777485 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 6672140370 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 23523155208 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 36864065630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 21667380315 \, \sqrt {-2 \, x + 1}}{47040 \, {\left (3 \, x + 2\right )}^{5}} \]
3025*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s qrt(-2*x + 1))) - 20149879/4116*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt( -2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/47040*(709777485*(2*x - 1)^4 *sqrt(-2*x + 1) + 6672140370*(2*x - 1)^3*sqrt(-2*x + 1) + 23523155208*(2*x - 1)^2*sqrt(-2*x + 1) - 36864065630*(-2*x + 1)^(3/2) + 21667380315*sqrt(- 2*x + 1))/(3*x + 2)^5
Time = 0.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {20149879\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2058}-6050\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {29479429\,\sqrt {1-2\,x}}{486}-\frac {75232787\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {26670244\,{\left (1-2\,x\right )}^{5/2}}{405}-\frac {504319\,{\left (1-2\,x\right )}^{7/2}}{27}+\frac {584179\,{\left (1-2\,x\right )}^{9/2}}{294}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
(20149879*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2058 - 6050*55^(1/ 2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((29479429*(1 - 2*x)^(1/2))/486 - (75232787*(1 - 2*x)^(3/2))/729 + (26670244*(1 - 2*x)^(5/2))/405 - (50431 9*(1 - 2*x)^(7/2))/27 + (584179*(1 - 2*x)^(9/2))/294)/((24010*x)/81 + (343 0*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)