Integrand size = 26, antiderivative size = 172 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {1994287 \sqrt {1-2 x} \sqrt {3+5 x}}{3110400}-\frac {14557 \sqrt {1-2 x} (3+5 x)^{3/2}}{28800}+\frac {4783 \sqrt {1-2 x} (3+5 x)^{5/2}}{32400}+\frac {37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {109715471 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{9331200 \sqrt {10}}+\frac {98}{729} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
37/360*(1-2*x)^(3/2)*(3+5*x)^(5/2)+1/15*(1-2*x)^(5/2)*(3+5*x)^(5/2)+98/729 *arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+109715471/9331200 0*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-14557/28800*(3+5*x)^(3/2)*( 1-2*x)^(1/2)+4783/32400*(3+5*x)^(5/2)*(1-2*x)^(1/2)-1994287/3110400*(1-2*x )^(1/2)*(3+5*x)^(1/2)
Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {30 \sqrt {1-2 x} \left (6495351+36344365 x+6699780 x^2-95229600 x^3+3024000 x^4+103680000 x^5\right )-109715471 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+12544000 \sqrt {7} \sqrt {3+5 x} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{93312000 \sqrt {3+5 x}} \]
(30*Sqrt[1 - 2*x]*(6495351 + 36344365*x + 6699780*x^2 - 95229600*x^3 + 302 4000*x^4 + 103680000*x^5) - 109715471*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5* x]/Sqrt[3 + 5*x]] + 12544000*Sqrt[7]*Sqrt[3 + 5*x]*ArcTan[Sqrt[1 - 2*x]/(S qrt[7]*Sqrt[3 + 5*x])])/(93312000*Sqrt[3 + 5*x])
Time = 0.27 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {112, 27, 171, 27, 171, 27, 171, 27, 171, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 x+2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}-\frac {1}{15} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (37 x+20)}{2 (3 x+2)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (37 x+20)}{3 x+2}dx+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{60} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (4783 x+1882)}{2 (3 x+2)}dx+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (4783 x+1882)}{3 x+2}dx+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{45} \int -\frac {(12374-393039 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)}dx+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {1}{90} \int \frac {(12374-393039 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{90} \left (\frac {1}{12} \int \frac {3 \sqrt {5 x+3} (1994287 x+2061258)}{2 \sqrt {1-2 x} (3 x+2)}dx-\frac {131013}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{90} \left (\frac {1}{8} \int \frac {\sqrt {5 x+3} (1994287 x+2061258)}{\sqrt {1-2 x} (3 x+2)}dx-\frac {131013}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{90} \left (\frac {1}{8} \left (-\frac {1}{6} \int -\frac {109715471 x+70216714}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {1994287}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {131013}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \int \frac {109715471 x+70216714}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {1994287}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {131013}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {109715471}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {8780800}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {1994287}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {131013}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {219430942}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {8780800}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {1994287}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {131013}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {219430942}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {17561600}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {1994287}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {131013}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {219430942}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {2508800}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {1994287}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {131013}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{120} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {109715471}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {2508800}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {1994287}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {131013}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {4783}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {37}{60} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/15 + ((37*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2 ))/60 + ((4783*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/45 + ((-131013*Sqrt[1 - 2*x] *(3 + 5*x)^(3/2))/4 + ((-1994287*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6 + ((109715 471*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + (2508800*Sqrt[7]*ArcTa n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3)/12)/8)/90)/120)/6
3.25.15.3.1 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*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 1.17 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(-\frac {\left (20736000 x^{4}-11836800 x^{3}-11943840 x^{2}+8506260 x +2165117\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3110400 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {109715471 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{186624000}+\frac {49 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{729}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(141\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1244160000 x^{4} \sqrt {-10 x^{2}-x +3}-710208000 x^{3} \sqrt {-10 x^{2}-x +3}-716630400 x^{2} \sqrt {-10 x^{2}-x +3}+109715471 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-12544000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+510375600 x \sqrt {-10 x^{2}-x +3}+129907020 \sqrt {-10 x^{2}-x +3}\right )}{186624000 \sqrt {-10 x^{2}-x +3}}\) | \(149\) |
-1/3110400*(20736000*x^4-11836800*x^3-11943840*x^2+8506260*x+2165117)*(-1+ 2*x)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2* x)^(1/2)-(-109715471/186624000*10^(1/2)*arcsin(20/11*x+1/11)+49/729*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 = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {1}{3110400} \, {\left (20736000 \, x^{4} - 11836800 \, x^{3} - 11943840 \, x^{2} + 8506260 \, x + 2165117\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} + \frac {49}{729} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {109715471}{186624000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/3110400*(20736000*x^4 - 11836800*x^3 - 11943840*x^2 + 8506260*x + 216511 7)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 49/729*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 109715471/18662400 0*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(1 0*x^2 + x - 3))
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{3 x + 2}\, dx \]
Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {1}{15} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {37}{72} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {787}{12960} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {79439}{51840} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {109715471}{186624000} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {49}{729} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {865517}{3110400} \, \sqrt {-10 \, x^{2} - x + 3} \]
1/15*(-10*x^2 - x + 3)^(5/2) + 37/72*(-10*x^2 - x + 3)^(3/2)*x - 787/12960 *(-10*x^2 - x + 3)^(3/2) + 79439/51840*sqrt(-10*x^2 - x + 3)*x + 109715471 /186624000*sqrt(10)*arcsin(20/11*x + 1/11) - 49/729*sqrt(7)*arcsin(37/11*x /abs(3*x + 2) + 20/11/abs(3*x + 2)) + 865517/3110400*sqrt(-10*x^2 - x + 3)
Time = 0.42 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.23 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {49}{7290} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{77760000} \, {\left (12 \, {\left (8 \, {\left (36 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} - 713 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 112817 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 655065 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 9971435 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {109715471}{186624000} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]
-49/7290*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)*sqrt(-10*x + 5) - sqrt(22)))) + 1/77760000*(12*(8*(36*(48*sqrt(5)*(5*x + 3) - 713*sqrt (5))*(5*x + 3) + 112817*sqrt(5))*(5*x + 3) - 655065*sqrt(5))*(5*x + 3) - 9 971435*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 109715471/186624000*sqrt(1 0)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22)) ^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{3\,x+2} \,d x \]