Integrand size = 26, antiderivative size = 207 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=-\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}-\frac {200}{729} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {109715471 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4572288 \sqrt {7}} \]
-1/15*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^5+37/72*(1-2*x)^(3/2)*(3+5*x)^(5 /2)/(2+3*x)^4-109715471/32006016*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^ (1/2))*7^(1/2)-200/729*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-32453/ 36288*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2+2543/1296*(3+5*x)^(5/2)*(1-2*x )^(1/2)/(2+3*x)^3-3248687/1524096*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.36 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (180761312+1044006792 x+2146957188 x^2+1809469170 x^3+490413015 x^4\right )}{(2+3 x)^5}+43904000 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-548577355 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{160030080} \]
((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(180761312 + 1044006792*x + 2146957188*x^ 2 + 1809469170*x^3 + 490413015*x^4))/(2 + 3*x)^5 + 43904000*Sqrt[10]*ArcTa n[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 548577355*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/ (Sqrt[7]*Sqrt[3 + 5*x])])/160030080
Time = 0.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {108, 27, 166, 27, 166, 27, 166, 27, 166, 27, 175, 64, 104, 217, 223}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{15} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)^5}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{6} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{(3 x+2)^5}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{12} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (320 x+1061)}{2 (3 x+2)^4}dx+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (320 x+1061)}{(3 x+2)^4}dx+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}-\frac {1}{9} \int -\frac {(29893-3840 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^3}dx\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \int \frac {(29893-3840 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{42} \int \frac {3 (963429-179200 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}dx-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \int \frac {(963429-179200 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{21} \int \frac {28209157-12544000 x}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \int \frac {28209157-12544000 x}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (\frac {109715471}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {12544000}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (\frac {109715471}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {5017600}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (\frac {219430942}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {5017600}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (-\frac {5017600}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {219430942 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{28} \left (\frac {1}{42} \left (-\frac {2508800}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {219430942 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}\right )+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}\) |
-1/15*((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^5 + ((37*(1 - 2*x)^(3/2) *(3 + 5*x)^(5/2))/(12*(2 + 3*x)^4) + ((2543*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)) /(9*(2 + 3*x)^3) + ((-32453*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(14*(2 + 3*x)^2 ) + ((-3248687*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)) + ((-2508800*Sq rt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 - (219430942*ArcTan[Sqrt[1 - 2* x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/42)/28)/18)/24)/6
3.25.20.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [2/b Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] || PosQ[b])
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[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Time = 1.51 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (490413015 x^{4}+1809469170 x^{3}+2146957188 x^{2}+1044006792 x +180761312\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{7620480 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {100 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{729}-\frac {109715471 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{64012032}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(148\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (133304297265 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}-10668672000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{5}+444347657550 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-35562240000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}+592463543400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-47416320000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+20597346630 x^{4} \sqrt {-10 x^{2}-x +3}+394975695600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-31610880000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+75997705140 x^{3} \sqrt {-10 x^{2}-x +3}+131658565200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -10536960000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +90172201896 x^{2} \sqrt {-10 x^{2}-x +3}+17554475360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1404928000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+43848285264 x \sqrt {-10 x^{2}-x +3}+7591975104 \sqrt {-10 x^{2}-x +3}\right )}{320060160 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(377\) |
-1/7620480*(-1+2*x)*(3+5*x)^(1/2)*(490413015*x^4+1809469170*x^3+2146957188 *x^2+1044006792*x+180761312)/(2+3*x)^5/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)* (3+5*x))^(1/2)/(1-2*x)^(1/2)-(100/729*10^(1/2)*arcsin(20/11*x+1/11)-109715 471/64012032*7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+1 11*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 = 196, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=-\frac {548577355 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 43904000 \, \sqrt {10} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (490413015 \, x^{4} + 1809469170 \, x^{3} + 2146957188 \, x^{2} + 1044006792 \, x + 180761312\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{320060160 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
-1/320060160*(548577355*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/( 10*x^2 + x - 3)) - 43904000*sqrt(10)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x ^2 + 240*x + 32)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(490413015*x^4 + 1809469170*x^3 + 2146957188*x^ 2 + 1044006792*x + 180761312)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810 *x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.29 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\frac {44881}{691488} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {333 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{1960 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {6347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{27440 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {44881 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{768320 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {3156205}{1382976} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {52017151}{24893568} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {9235489 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{13829760 \, {\left (3 \, x + 2\right )}} + \frac {17832215}{1778112} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {100}{729} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {109715471}{64012032} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {49508071}{10668672} \, \sqrt {-10 \, x^{2} - x + 3} \]
44881/691488*(-10*x^2 - x + 3)^(5/2) + 3/35*(-10*x^2 - x + 3)^(7/2)/(243*x ^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 333/1960*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 6347/27440*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 44881/768320*(-10*x^2 - x + 3) ^(7/2)/(9*x^2 + 12*x + 4) - 3156205/1382976*(-10*x^2 - x + 3)^(3/2)*x + 52 017151/24893568*(-10*x^2 - x + 3)^(3/2) - 9235489/13829760*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 17832215/1778112*sqrt(-10*x^2 - x + 3)*x - 100/729*sq rt(10)*arcsin(20/11*x + 1/11) + 109715471/64012032*sqrt(7)*arcsin(37/11*x/ abs(3*x + 2) + 20/11/abs(3*x + 2)) - 49508071/10668672*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (159) = 318\).
Time = 0.74 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.38 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\frac {109715471}{640120320} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {100}{729} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (3248687 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 4238260880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 2165236899840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 364930179712000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {12258004702720000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {49032018810880000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{762048 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
109715471/640120320*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)*s qrt(-10*x + 5) - sqrt(22)))) - 100/729*sqrt(10)*(pi + 2*arctan(-1/4*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)))) - 11/762048*sqrt(10)*(3248687*((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 + 4238260880*((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)))^7 + 216 5236899840*((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 - 364930179712000*((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 - 12258004702720000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 49032018810880000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1 0*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)^5
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^6} \,d x \]