Integrand size = 24, antiderivative size = 174 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=-\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {4369 \sqrt {1-2 x}}{1210104 (2+3 x)}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}-\frac {4369 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052 \sqrt {21}} \]
-1/21*(1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^7-4369/12706092*arctanh(1/7*21^(1/2) *(1-2*x)^(1/2))*21^(1/2)-4369/518616*(1-2*x)^(1/2)/(2+3*x)^2-4369/1210104* (1-2*x)^(1/2)/(2+3*x)-173/735*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^5+2/7*(3+5*x )^3*(1-2*x)^(1/2)/(2+3*x)^6-1/185220*(146585+237807*x)*(1-2*x)^(1/2)/(2+3* x)^4
Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.46 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (7033976+606784 x-98441652 x^2-182748162 x^3-42669876 x^4+76086135 x^5+15925005 x^6\right )}{2 (2+3 x)^7}-21845 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63530460} \]
((-21*Sqrt[1 - 2*x]*(7033976 + 606784*x - 98441652*x^2 - 182748162*x^3 - 4 2669876*x^4 + 76086135*x^5 + 15925005*x^6))/(2*(2 + 3*x)^7) - 21845*Sqrt[2 1]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/63530460
Time = 0.25 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {108, 27, 166, 27, 166, 162, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^3}{(3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{21} \int \frac {3 (2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^7}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {(2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^7}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{7} \left (\frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^6}-\frac {1}{18} \int -\frac {18 (31-40 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^6}dx\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\int \frac {(31-40 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^6}dx+\frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{105} \int \frac {(2201-2675 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^5}dx+\frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^6}-\frac {173 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{105} \left (\frac {21845}{252} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x} (237807 x+146585)}{252 (3 x+2)^4}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^6}-\frac {173 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{105} \left (\frac {21845}{252} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (237807 x+146585)}{252 (3 x+2)^4}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^6}-\frac {173 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{105} \left (\frac {21845}{252} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (237807 x+146585)}{252 (3 x+2)^4}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^6}-\frac {173 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{105} \left (\frac {21845}{252} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (237807 x+146585)}{252 (3 x+2)^4}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^6}-\frac {173 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{105} \left (\frac {21845}{252} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (237807 x+146585)}{252 (3 x+2)^4}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^6}-\frac {173 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
-1/21*((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^7 + ((-173*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(105*(2 + 3*x)^5) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^6 + (-1/252*(Sqrt[1 - 2*x]*(146585 + 237807*x))/(2 + 3*x)^4 + (21845*(-1/14*Sq rt[1 - 2*x]/(2 + 3*x)^2 + (3*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTanh[Sq rt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14))/252)/105)/7
3.19.94.3.1 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 = 1.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.41
| method | result | size |
| risch | \(\frac {31850010 x^{7}+136247265 x^{6}-161425887 x^{5}-322826448 x^{4}-14135142 x^{3}+99655220 x^{2}+13461168 x -7033976}{6050520 \left (2+3 x \right )^{7} \sqrt {1-2 x}}-\frac {4369 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12706092}\) | \(71\) |
| pseudoelliptic | \(\frac {-43690 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{7} \sqrt {21}-21 \sqrt {1-2 x}\, \left (15925005 x^{6}+76086135 x^{5}-42669876 x^{4}-182748162 x^{3}-98441652 x^{2}+606784 x +7033976\right )}{127060920 \left (2+3 x \right )^{7}}\) | \(75\) |
| trager | \(-\frac {\left (15925005 x^{6}+76086135 x^{5}-42669876 x^{4}-182748162 x^{3}-98441652 x^{2}+606784 x +7033976\right ) \sqrt {1-2 x}}{6050520 \left (2+3 x \right )^{7}}-\frac {4369 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{25412184}\) | \(92\) |
| derivativedivides | \(\frac {\frac {353889 \left (1-2 x \right )^{\frac {13}{2}}}{67228}-\frac {196605 \left (1-2 x \right )^{\frac {11}{2}}}{2401}+\frac {5639843 \left (1-2 x \right )^{\frac {9}{2}}}{20580}+\frac {172608 \left (1-2 x \right )^{\frac {7}{2}}}{1715}-\frac {725323 \left (1-2 x \right )^{\frac {5}{2}}}{420}+\frac {21845 \left (1-2 x \right )^{\frac {3}{2}}}{9}-\frac {30583 \sqrt {1-2 x}}{36}}{\left (-4-6 x \right )^{7}}-\frac {4369 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12706092}\) | \(93\) |
| default | \(\frac {\frac {353889 \left (1-2 x \right )^{\frac {13}{2}}}{67228}-\frac {196605 \left (1-2 x \right )^{\frac {11}{2}}}{2401}+\frac {5639843 \left (1-2 x \right )^{\frac {9}{2}}}{20580}+\frac {172608 \left (1-2 x \right )^{\frac {7}{2}}}{1715}-\frac {725323 \left (1-2 x \right )^{\frac {5}{2}}}{420}+\frac {21845 \left (1-2 x \right )^{\frac {3}{2}}}{9}-\frac {30583 \sqrt {1-2 x}}{36}}{\left (-4-6 x \right )^{7}}-\frac {4369 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12706092}\) | \(93\) |
1/6050520*(31850010*x^7+136247265*x^6-161425887*x^5-322826448*x^4-14135142 *x^3+99655220*x^2+13461168*x-7033976)/(2+3*x)^7/(1-2*x)^(1/2)-4369/1270609 2*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {21845 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (15925005 \, x^{6} + 76086135 \, x^{5} - 42669876 \, x^{4} - 182748162 \, x^{3} - 98441652 \, x^{2} + 606784 \, x + 7033976\right )} \sqrt {-2 \, x + 1}}{127060920 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
1/127060920*(21845*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*(15925005*x^6 + 76086135*x^5 - 42669876*x^4 - 1827481 62*x^3 - 98441652*x^2 + 606784*x + 7033976)*sqrt(-2*x + 1))/(2187*x^7 + 10 206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {4369}{25412184} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {15925005 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 247722300 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 829056921 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 304480512 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 5224501569 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 7342978300 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2570042405 \, \sqrt {-2 \, x + 1}}{3025260 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]
4369/25412184*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq rt(-2*x + 1))) - 1/3025260*(15925005*(-2*x + 1)^(13/2) - 247722300*(-2*x + 1)^(11/2) + 829056921*(-2*x + 1)^(9/2) + 304480512*(-2*x + 1)^(7/2) - 522 4501569*(-2*x + 1)^(5/2) + 7342978300*(-2*x + 1)^(3/2) - 2570042405*sqrt(- 2*x + 1))/(2187*(2*x - 1)^7 + 35721*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972 405*(2*x - 1)^4 + 2268945*(2*x - 1)^3 + 3176523*(2*x - 1)^2 + 4941258*x - 1647086)
Time = 0.30 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {4369}{25412184} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {15925005 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + 247722300 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 829056921 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 304480512 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 5224501569 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 7342978300 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2570042405 \, \sqrt {-2 \, x + 1}}{387233280 \, {\left (3 \, x + 2\right )}^{7}} \]
4369/25412184*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) - 1/387233280*(15925005*(2*x - 1)^6*sqrt(-2*x + 1) + 247722300*(2*x - 1)^5*sqrt(-2*x + 1) + 829056921*(2*x - 1)^4*sqrt(-2*x + 1) - 304480512*(2*x - 1)^3*sqrt(-2*x + 1) - 5224501569*(2*x - 1)^2*sqrt( -2*x + 1) + 7342978300*(-2*x + 1)^(3/2) - 2570042405*sqrt(-2*x + 1))/(3*x + 2)^7
Time = 1.42 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx=-\frac {\frac {21845\,{\left (1-2\,x\right )}^{3/2}}{19683}-\frac {30583\,\sqrt {1-2\,x}}{78732}-\frac {725323\,{\left (1-2\,x\right )}^{5/2}}{918540}+\frac {57536\,{\left (1-2\,x\right )}^{7/2}}{1250235}+\frac {5639843\,{\left (1-2\,x\right )}^{9/2}}{45008460}-\frac {21845\,{\left (1-2\,x\right )}^{11/2}}{583443}+\frac {4369\,{\left (1-2\,x\right )}^{13/2}}{1815156}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}}-\frac {4369\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12706092} \]
- ((21845*(1 - 2*x)^(3/2))/19683 - (30583*(1 - 2*x)^(1/2))/78732 - (725323 *(1 - 2*x)^(5/2))/918540 + (57536*(1 - 2*x)^(7/2))/1250235 + (5639843*(1 - 2*x)^(9/2))/45008460 - (21845*(1 - 2*x)^(11/2))/583443 + (4369*(1 - 2*x)^ (13/2))/1815156)/((1647086*x)/729 + (117649*(2*x - 1)^2)/81 + (84035*(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) - (4369*21^(1/2)*atanh((21^(1/2)*(1 - 2 *x)^(1/2))/7))/12706092