Integrand size = 26, antiderivative size = 238 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=-\frac {443563 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)^4}+\frac {2199649 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)^3}+\frac {384136145 \sqrt {1-2 x} \sqrt {3+5 x}}{42674688 (2+3 x)^2}+\frac {40175505215 \sqrt {1-2 x} \sqrt {3+5 x}}{597445632 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac {1921 \sqrt {1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}-\frac {1891543995 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2458624 \sqrt {7}} \]
-1/21*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^7+115/756*(1-2*x)^(3/2)*(3+5*x)^ (3/2)/(2+3*x)^6-1891543995/17210368*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5* x)^(1/2))*7^(1/2)+1921/1512*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^5-443563/2 54016*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+2199649/1524096*(1-2*x)^(1/2)* (3+5*x)^(1/2)/(2+3*x)^3+384136145/42674688*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+ 3*x)^2+40175505215/597445632*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.45 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.39 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (11351210112+100906793184 x+373848853744 x^2+738910550592 x^3+821723878536 x^4+487483968610 x^5+120526515645 x^6\right )}{(2+3 x)^7}-1891543995 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{17210368} \]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(11351210112 + 100906793184*x + 3738488537 44*x^2 + 738910550592*x^3 + 821723878536*x^4 + 487483968610*x^5 + 12052651 5645*x^6))/(2 + 3*x)^7 - 1891543995*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]* Sqrt[3 + 5*x])])/17210368
Time = 0.32 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.13, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {108, 27, 166, 27, 166, 27, 166, 27, 168, 27, 168, 27, 168, 27, 104, 217}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{(3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{21} \int -\frac {5 (1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{2 (3 x+2)^7}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{42} \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{(3 x+2)^7}dx-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{42} \left (-\frac {1}{18} \int \frac {3 (419-332 x) \sqrt {1-2 x} \sqrt {5 x+3}}{2 (3 x+2)^6}dx-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{42} \left (-\frac {1}{12} \int \frac {(419-332 x) \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^6}dx-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{15} \int -\frac {(75441-108620 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^5}dx-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (-\frac {1}{30} \int \frac {(75441-108620 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^5}dx-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}-\frac {1}{84} \int \frac {2599301-3799820 x}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}-\frac {1}{168} \int \frac {2599301-3799820 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {1}{168} \left (-\frac {1}{21} \int \frac {35 (13877615-17597192 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {2199649 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {1}{168} \left (-\frac {5}{6} \int \frac {13877615-17597192 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {2199649 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{14} \int \frac {1654003961-1536544580 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {76827229 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {2199649 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \int \frac {1654003961-1536544580 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {76827229 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {2199649 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {91929038157}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {8035101043 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {76827229 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {2199649 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \left (\frac {91929038157}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {8035101043 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {76827229 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {2199649 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \left (\frac {91929038157}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {8035101043 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {76827229 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {2199649 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5}{42} \left (\frac {1}{12} \left (\frac {1}{30} \left (\frac {1}{168} \left (-\frac {5}{6} \left (\frac {1}{28} \left (\frac {8035101043 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {91929038157 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {76827229 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {2199649 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {443563 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {1921 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}\) |
-1/21*((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^7 - (5*((-23*(1 - 2*x)^( 3/2)*(3 + 5*x)^(3/2))/(18*(2 + 3*x)^6) + ((-1921*Sqrt[1 - 2*x]*(3 + 5*x)^( 3/2))/(15*(2 + 3*x)^5) + ((443563*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3* x)^4) + ((-2199649*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) - (5*((768 27229*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((8035101043*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (91929038157*ArcTan[Sqrt[1 - 2*x]/(S qrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6)/168)/30)/12))/42
3.25.10.3.1 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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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_) )^(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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Time = 1.14 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (120526515645 x^{6}+487483968610 x^{5}+821723878536 x^{4}+738910550592 x^{3}+373848853744 x^{2}+100906793184 x +11351210112\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \left (2+3 x \right )^{7} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1891543995 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{34420736 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(144\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (4136806717065 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}+19305098012970 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+38610196025940 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+1687371219030 \sqrt {-10 x^{2}-x +3}\, x^{6}+42900217806600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+6824775560540 x^{5} \sqrt {-10 x^{2}-x +3}+28600145204400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+11504134299504 x^{4} \sqrt {-10 x^{2}-x +3}+11440058081760 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+10344747708288 x^{3} \sqrt {-10 x^{2}-x +3}+2542235129280 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +5233883952416 x^{2} \sqrt {-10 x^{2}-x +3}+242117631360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1412695104576 x \sqrt {-10 x^{2}-x +3}+158916941568 \sqrt {-10 x^{2}-x +3}\right )}{34420736 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{7}}\) | \(394\) |
-1/2458624*(-1+2*x)*(3+5*x)^(1/2)*(120526515645*x^6+487483968610*x^5+82172 3878536*x^4+738910550592*x^3+373848853744*x^2+100906793184*x+11351210112)/ (2+3*x)^7/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+ 1891543995/34420736*7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x) ^2+67+111*x)^(1/2))*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=-\frac {1891543995 \, \sqrt {7} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (120526515645 \, x^{6} + 487483968610 \, x^{5} + 821723878536 \, x^{4} + 738910550592 \, x^{3} + 373848853744 \, x^{2} + 100906793184 \, x + 11351210112\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{34420736 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
-1/34420736*(1891543995*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680* x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*arctan(1/14*sqrt(7)*(37*x + 20) *sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(120526515645*x^6 + 4 87483968610*x^5 + 821723878536*x^4 + 738910550592*x^3 + 373848853744*x^2 + 100906793184*x + 11351210112)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2187*x^7 + 1 0206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.36 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\frac {118356975}{4302592} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{7 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {305 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{588 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {2161 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1176 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {129195 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {4780215 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{307328 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {213042555 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{8605184 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {2892030075}{8605184} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {1891543995}{34420736} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2548112985}{17210368} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {280970415 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{17210368 \, {\left (3 \, x + 2\right )}} \]
118356975/4302592*(-10*x^2 - x + 3)^(3/2) + 1/7*(-10*x^2 - x + 3)^(5/2)/(2 187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344* x + 128) + 305/588*(-10*x^2 - x + 3)^(5/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 2161/1176*(-10*x^2 - x + 3)^(5/2)/(2 43*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 129195/21952*(-10*x^ 2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 4780215/307328 *(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 213042555/8605184* (-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 2892030075/8605184*sqrt(-10*x ^2 - x + 3)*x + 1891543995/34420736*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2548112985/17210368*sqrt(-10*x^2 - x + 3) + 28097041 5/17210368*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (187) = 374\).
Time = 0.84 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.28 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\frac {378308799}{68841472} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {805255 \, \sqrt {10} {\left (2349 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{13} + 4384800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} - 4393081280 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 1503513804800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 272402016768000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 26951436288000000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {1131960324096000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {4527841296384000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1229312 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{7}} \]
378308799/68841472*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5 *x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sq rt(-10*x + 5) - sqrt(22)))) - 805255/1229312*sqrt(10)*(2349*((sqrt(2)*sqrt (-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10* x + 5) - sqrt(22)))^13 + 4384800*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^11 - 43 93081280*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 - 1503513804800*((sqrt(2)*sqr t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10 *x + 5) - sqrt(22)))^7 - 272402016768000*((sqrt(2)*sqrt(-10*x + 5) - sqrt( 22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) ^5 - 26951436288000000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 1131960324096 000000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 45278412963840 00000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt( -10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^7
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^8} \,d x \]