Integrand size = 24, antiderivative size = 181 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {33935 \sqrt {1-2 x}}{2333772 (2+3 x)}-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \sqrt {1-2 x} (187704+301765 x)}{333396 (2+3 x)^3}+\frac {33935 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1166886 \sqrt {21}} \]
-1/21*(1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^7+55/189*(1-2*x)^(3/2)*(3+5*x)^3/(2+ 3*x)^6+33935/24504606*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+33935/2 333772*(1-2*x)^(1/2)/(2+3*x)-3223/2646*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^4+1 1/7*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^5-11/333396*(187704+301765*x)*(1-2*x)^ (1/2)/(2+3*x)^3
Time = 0.46 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.44 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-4005436-12384752 x-39606312 x^2-164222766 x^3-283697388 x^4-141112395 x^5+24738615 x^6\right )}{2 (2+3 x)^7}+33935 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24504606} \]
((21*Sqrt[1 - 2*x]*(-4005436 - 12384752*x - 39606312*x^2 - 164222766*x^3 - 283697388*x^4 - 141112395*x^5 + 24738615*x^6))/(2*(2 + 3*x)^7) + 33935*Sq rt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/24504606
Time = 0.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {108, 27, 166, 27, 166, 27, 166, 27, 162, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^3}{(3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{21} \int -\frac {55 (1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^7}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{21} \int \frac {(1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^7}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {55}{21} \left (-\frac {1}{18} \int \frac {6 (7-3 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^6}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{21} \left (-\frac {1}{3} \int \frac {(7-3 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^6}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {55}{21} \left (\frac {1}{3} \left (\frac {1}{15} \int -\frac {3 (112-125 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^5}dx-\frac {9 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{21} \left (\frac {1}{3} \left (-\frac {1}{5} \int \frac {(112-125 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^5}dx-\frac {9 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {55}{21} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {293 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}-\frac {1}{84} \int \frac {2 (3833-4355 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^4}dx\right )-\frac {9 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{21} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {293 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}-\frac {1}{42} \int \frac {(3833-4355 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^4}dx\right )-\frac {9 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle -\frac {55}{21} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{42} \left (\frac {3085}{126} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {\sqrt {1-2 x} (301765 x+187704)}{126 (3 x+2)^3}\right )+\frac {293 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {9 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {55}{21} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{42} \left (\frac {3085}{126} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (301765 x+187704)}{126 (3 x+2)^3}\right )+\frac {293 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {9 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {55}{21} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{42} \left (\frac {3085}{126} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (301765 x+187704)}{126 (3 x+2)^3}\right )+\frac {293 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {9 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {55}{21} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{42} \left (\frac {3085}{126} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (301765 x+187704)}{126 (3 x+2)^3}\right )+\frac {293 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {9 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^6}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}\) |
-1/21*((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^7 - (55*(-1/9*((1 - 2*x)^(3/ 2)*(3 + 5*x)^3)/(2 + 3*x)^6 + ((-9*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(5*(2 + 3*x) ^5) + ((293*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(42*(2 + 3*x)^4) + ((Sqrt[1 - 2*x]* (187704 + 301765*x))/(126*(2 + 3*x)^3) + (3085*(-1/7*Sqrt[1 - 2*x]/(2 + 3* x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/126)/42)/5)/3))/2 1
3.20.67.3.1 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 = 1.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.39
| method | result | size |
| risch | \(-\frac {49477230 x^{7}-306963405 x^{6}-426282381 x^{5}-44748144 x^{4}+85010142 x^{3}+14836808 x^{2}+4373880 x +4005436}{2333772 \left (2+3 x \right )^{7} \sqrt {1-2 x}}+\frac {33935 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{24504606}\) | \(71\) |
| pseudoelliptic | \(\frac {67870 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{7} \sqrt {21}+21 \sqrt {1-2 x}\, \left (24738615 x^{6}-141112395 x^{5}-283697388 x^{4}-164222766 x^{3}-39606312 x^{2}-12384752 x -4005436\right )}{49009212 \left (2+3 x \right )^{7}}\) | \(75\) |
| trager | \(\frac {\left (24738615 x^{6}-141112395 x^{5}-283697388 x^{4}-164222766 x^{3}-39606312 x^{2}-12384752 x -4005436\right ) \sqrt {1-2 x}}{2333772 \left (2+3 x \right )^{7}}+\frac {33935 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{49009212}\) | \(92\) |
| derivativedivides | \(\frac {-\frac {101805 \left (1-2 x \right )^{\frac {13}{2}}}{4802}-\frac {353950 \left (1-2 x \right )^{\frac {11}{2}}}{3087}+\frac {4931597 \left (1-2 x \right )^{\frac {9}{2}}}{2646}-\frac {3091616 \left (1-2 x \right )^{\frac {7}{2}}}{441}+\frac {1920721 \left (1-2 x \right )^{\frac {5}{2}}}{162}-\frac {2375450 \left (1-2 x \right )^{\frac {3}{2}}}{243}+\frac {1662815 \sqrt {1-2 x}}{486}}{\left (-4-6 x \right )^{7}}+\frac {33935 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{24504606}\) | \(93\) |
| default | \(\frac {-\frac {101805 \left (1-2 x \right )^{\frac {13}{2}}}{4802}-\frac {353950 \left (1-2 x \right )^{\frac {11}{2}}}{3087}+\frac {4931597 \left (1-2 x \right )^{\frac {9}{2}}}{2646}-\frac {3091616 \left (1-2 x \right )^{\frac {7}{2}}}{441}+\frac {1920721 \left (1-2 x \right )^{\frac {5}{2}}}{162}-\frac {2375450 \left (1-2 x \right )^{\frac {3}{2}}}{243}+\frac {1662815 \sqrt {1-2 x}}{486}}{\left (-4-6 x \right )^{7}}+\frac {33935 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{24504606}\) | \(93\) |
-1/2333772*(49477230*x^7-306963405*x^6-426282381*x^5-44748144*x^4+85010142 *x^3+14836808*x^2+4373880*x+4005436)/(2+3*x)^7/(1-2*x)^(1/2)+33935/2450460 6*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {33935 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (24738615 \, x^{6} - 141112395 \, x^{5} - 283697388 \, x^{4} - 164222766 \, x^{3} - 39606312 \, x^{2} - 12384752 \, x - 4005436\right )} \sqrt {-2 \, x + 1}}{49009212 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
1/49009212*(33935*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*(24738615*x^6 - 141112395*x^5 - 283697388*x^4 - 164222 766*x^3 - 39606312*x^2 - 12384752*x - 4005436)*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. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx=-\frac {33935}{49009212} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {24738615 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + 133793100 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2174834277 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 8180415936 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 13834953363 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 11406910900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3992418815 \, \sqrt {-2 \, x + 1}}{1166886 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]
-33935/49009212*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3* sqrt(-2*x + 1))) + 1/1166886*(24738615*(-2*x + 1)^(13/2) + 133793100*(-2*x + 1)^(11/2) - 2174834277*(-2*x + 1)^(9/2) + 8180415936*(-2*x + 1)^(7/2) - 13834953363*(-2*x + 1)^(5/2) + 11406910900*(-2*x + 1)^(3/2) - 3992418815* 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 + 494125 8*x - 1647086)
Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx=-\frac {33935}{49009212} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {24738615 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - 133793100 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - 2174834277 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 8180415936 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 13834953363 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 11406910900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3992418815 \, \sqrt {-2 \, x + 1}}{149361408 \, {\left (3 \, x + 2\right )}^{7}} \]
-33935/49009212*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt (21) + 3*sqrt(-2*x + 1))) + 1/149361408*(24738615*(2*x - 1)^6*sqrt(-2*x + 1) - 133793100*(2*x - 1)^5*sqrt(-2*x + 1) - 2174834277*(2*x - 1)^4*sqrt(-2 *x + 1) - 8180415936*(2*x - 1)^3*sqrt(-2*x + 1) - 13834953363*(2*x - 1)^2* sqrt(-2*x + 1) + 11406910900*(-2*x + 1)^(3/2) - 3992418815*sqrt(-2*x + 1)) /(3*x + 2)^7
Time = 1.56 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {\frac {2375450\,{\left (1-2\,x\right )}^{3/2}}{531441}-\frac {1662815\,\sqrt {1-2\,x}}{1062882}-\frac {1920721\,{\left (1-2\,x\right )}^{5/2}}{354294}+\frac {3091616\,{\left (1-2\,x\right )}^{7/2}}{964467}-\frac {4931597\,{\left (1-2\,x\right )}^{9/2}}{5786802}+\frac {353950\,{\left (1-2\,x\right )}^{11/2}}{6751269}+\frac {33935\,{\left (1-2\,x\right )}^{13/2}}{3500658}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}}+\frac {33935\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{24504606} \]
((2375450*(1 - 2*x)^(3/2))/531441 - (1662815*(1 - 2*x)^(1/2))/1062882 - (1 920721*(1 - 2*x)^(5/2))/354294 + (3091616*(1 - 2*x)^(7/2))/964467 - (49315 97*(1 - 2*x)^(9/2))/5786802 + (353950*(1 - 2*x)^(11/2))/6751269 + (33935*( 1 - 2*x)^(13/2))/3500658)/((1647086*x)/729 + (117649*(2*x - 1)^2)/81 + (84 035*(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) + (33935*21^(1/2)*atanh((21^(1 /2)*(1 - 2*x)^(1/2))/7))/24504606