Integrand size = 24, antiderivative size = 167 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {47434 \sqrt {1-2 x}}{2917215 (2+3 x)^3}+\frac {23717 \sqrt {1-2 x}}{4084101 (2+3 x)^2}+\frac {23717 \sqrt {1-2 x}}{9529569 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{1323 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^3}{21 (2+3 x)^7}-\frac {2 \sqrt {1-2 x} (54227+88099 x)}{972405 (2+3 x)^5}+\frac {47434 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9529569 \sqrt {21}} \]
47434/200120949*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+47434/2917215 *(1-2*x)^(1/2)/(2+3*x)^3+23717/4084101*(1-2*x)^(1/2)/(2+3*x)^2+23717/95295 69*(1-2*x)^(1/2)/(2+3*x)-53/1323*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^6-1/21*(3 +5*x)^3*(1-2*x)^(1/2)/(2+3*x)^7-2/972405*(54227+88099*x)*(1-2*x)^(1/2)/(2+ 3*x)^5
Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {2 \left (\frac {21 \sqrt {1-2 x} \left (-88036937-361589428 x-306463011 x^2+473987484 x^3+863203932 x^4+413031555 x^5+86448465 x^6\right )}{2 (2+3 x)^7}+118585 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{1000604745} \]
(2*((21*Sqrt[1 - 2*x]*(-88036937 - 361589428*x - 306463011*x^2 + 473987484 *x^3 + 863203932*x^4 + 413031555*x^5 + 86448465*x^6))/(2*(2 + 3*x)^7) + 11 8585*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]))/1000604745
Time = 0.25 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {108, 166, 27, 162, 52, 52, 52, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{21} \int \frac {(12-35 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^7}dx-\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{126} \int \frac {4 (37-910 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^6}dx-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{63} \int \frac {(37-910 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^6}dx-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{63} \left (-\frac {23717}{105} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {\sqrt {1-2 x} (88099 x+54227)}{735 (3 x+2)^5}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{63} \left (-\frac {23717}{105} \left (\frac {5}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (88099 x+54227)}{735 (3 x+2)^5}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{63} \left (-\frac {23717}{105} \left (\frac {5}{21} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (88099 x+54227)}{735 (3 x+2)^5}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{63} \left (-\frac {23717}{105} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (88099 x+54227)}{735 (3 x+2)^5}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{63} \left (-\frac {23717}{105} \left (\frac {5}{21} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (88099 x+54227)}{735 (3 x+2)^5}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{63} \left (-\frac {23717}{105} \left (\frac {5}{21} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (88099 x+54227)}{735 (3 x+2)^5}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{21 (3 x+2)^7}\) |
-1/21*(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^7 + ((-53*Sqrt[1 - 2*x]*(3 + 5 *x)^2)/(63*(2 + 3*x)^6) + (2*(-1/735*(Sqrt[1 - 2*x]*(54227 + 88099*x))/(2 + 3*x)^5 - (23717*(-1/21*Sqrt[1 - 2*x]/(2 + 3*x)^3 + (5*(-1/14*Sqrt[1 - 2* x]/(2 + 3*x)^2 + (3*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTanh[Sqrt[3/7]*S qrt[1 - 2*x]])/(7*Sqrt[21])))/14))/21))/105))/63)/21
3.19.27.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
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_)^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 = 2.97 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.43
| method | result | size |
| risch | \(-\frac {172896930 x^{7}+739614645 x^{6}+1313376309 x^{5}+84771036 x^{4}-1086913506 x^{3}-416715845 x^{2}+185515554 x +88036937}{47647845 \left (2+3 x \right )^{7} \sqrt {1-2 x}}+\frac {47434 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{200120949}\) | \(71\) |
| pseudoelliptic | \(\frac {237170 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{7} \sqrt {21}+21 \sqrt {1-2 x}\, \left (86448465 x^{6}+413031555 x^{5}+863203932 x^{4}+473987484 x^{3}-306463011 x^{2}-361589428 x -88036937\right )}{1000604745 \left (2+3 x \right )^{7}}\) | \(75\) |
| trager | \(\frac {\left (86448465 x^{6}+413031555 x^{5}+863203932 x^{4}+473987484 x^{3}-306463011 x^{2}-361589428 x -88036937\right ) \sqrt {1-2 x}}{47647845 \left (2+3 x \right )^{7}}+\frac {23717 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{200120949}\) | \(92\) |
| derivativedivides | \(\frac {-\frac {426906 \left (1-2 x \right )^{\frac {13}{2}}}{117649}+\frac {948680 \left (1-2 x \right )^{\frac {11}{2}}}{16807}-\frac {13423822 \left (1-2 x \right )^{\frac {9}{2}}}{36015}+\frac {41712416 \left (1-2 x \right )^{\frac {7}{2}}}{36015}-\frac {10203122 \left (1-2 x \right )^{\frac {5}{2}}}{6615}+\frac {201304 \left (1-2 x \right )^{\frac {3}{2}}}{567}+\frac {47434 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{7}}+\frac {47434 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{200120949}\) | \(93\) |
| default | \(\frac {-\frac {426906 \left (1-2 x \right )^{\frac {13}{2}}}{117649}+\frac {948680 \left (1-2 x \right )^{\frac {11}{2}}}{16807}-\frac {13423822 \left (1-2 x \right )^{\frac {9}{2}}}{36015}+\frac {41712416 \left (1-2 x \right )^{\frac {7}{2}}}{36015}-\frac {10203122 \left (1-2 x \right )^{\frac {5}{2}}}{6615}+\frac {201304 \left (1-2 x \right )^{\frac {3}{2}}}{567}+\frac {47434 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{7}}+\frac {47434 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{200120949}\) | \(93\) |
-1/47647845*(172896930*x^7+739614645*x^6+1313376309*x^5+84771036*x^4-10869 13506*x^3-416715845*x^2+185515554*x+88036937)/(2+3*x)^7/(1-2*x)^(1/2)+4743 4/200120949*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {118585 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (86448465 \, x^{6} + 413031555 \, x^{5} + 863203932 \, x^{4} + 473987484 \, x^{3} - 306463011 \, x^{2} - 361589428 \, x - 88036937\right )} \sqrt {-2 \, x + 1}}{1000604745 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
1/1000604745*(118585*sqrt(21)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^ 4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log((3*x - sqrt(21)*sqrt(-2*x + 1 ) - 5)/(3*x + 2)) + 21*(86448465*x^6 + 413031555*x^5 + 863203932*x^4 + 473 987484*x^3 - 306463011*x^2 - 361589428*x - 88036937)*sqrt(-2*x + 1))/(2187 *x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx=-\frac {23717}{200120949} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (86448465 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 1344753900 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 8879858253 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 27592763184 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 36746543883 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 8458290820 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13951406665 \, \sqrt {-2 \, x + 1}\right )}}{47647845 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]
-23717/200120949*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) + 2/47647845*(86448465*(-2*x + 1)^(13/2) - 1344753900*(- 2*x + 1)^(11/2) + 8879858253*(-2*x + 1)^(9/2) - 27592763184*(-2*x + 1)^(7/ 2) + 36746543883*(-2*x + 1)^(5/2) - 8458290820*(-2*x + 1)^(3/2) - 13951406 665*sqrt(-2*x + 1))/(2187*(2*x - 1)^7 + 35721*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972405*(2*x - 1)^4 + 2268945*(2*x - 1)^3 + 3176523*(2*x - 1)^2 + 49 41258*x - 1647086)
Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx=-\frac {23717}{200120949} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {86448465 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + 1344753900 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 8879858253 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 27592763184 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 36746543883 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 8458290820 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13951406665 \, \sqrt {-2 \, x + 1}}{3049462080 \, {\left (3 \, x + 2\right )}^{7}} \]
-23717/200120949*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr t(21) + 3*sqrt(-2*x + 1))) + 1/3049462080*(86448465*(2*x - 1)^6*sqrt(-2*x + 1) + 1344753900*(2*x - 1)^5*sqrt(-2*x + 1) + 8879858253*(2*x - 1)^4*sqrt (-2*x + 1) + 27592763184*(2*x - 1)^3*sqrt(-2*x + 1) + 36746543883*(2*x - 1 )^2*sqrt(-2*x + 1) - 8458290820*(-2*x + 1)^(3/2) - 13951406665*sqrt(-2*x + 1))/(3*x + 2)^7
Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {47434\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{200120949}-\frac {\frac {47434\,\sqrt {1-2\,x}}{177147}+\frac {201304\,{\left (1-2\,x\right )}^{3/2}}{1240029}-\frac {10203122\,{\left (1-2\,x\right )}^{5/2}}{14467005}+\frac {41712416\,{\left (1-2\,x\right )}^{7/2}}{78764805}-\frac {13423822\,{\left (1-2\,x\right )}^{9/2}}{78764805}+\frac {948680\,{\left (1-2\,x\right )}^{11/2}}{36756909}-\frac {47434\,{\left (1-2\,x\right )}^{13/2}}{28588707}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}} \]
(47434*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/200120949 - ((47434*( 1 - 2*x)^(1/2))/177147 + (201304*(1 - 2*x)^(3/2))/1240029 - (10203122*(1 - 2*x)^(5/2))/14467005 + (41712416*(1 - 2*x)^(7/2))/78764805 - (13423822*(1 - 2*x)^(9/2))/78764805 + (948680*(1 - 2*x)^(11/2))/36756909 - (47434*(1 - 2*x)^(13/2))/28588707)/((1647086*x)/729 + (117649*(2*x - 1)^2)/81 + (8403 5*(2*x - 1)^3)/81 + (12005*(2*x - 1)^4)/27 + (343*(2*x - 1)^5)/3 + (49*(2* x - 1)^6)/3 + (2*x - 1)^7 - 1647086/2187)