Integrand size = 26, antiderivative size = 171 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {39745 \sqrt {1-2 x} \sqrt {3+5 x}}{4536}+\frac {331 \sqrt {1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {575}{243} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {326717 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{13608 \sqrt {7}} \] Output:
-39745/4536*(1-2*x)^(1/2)*(3+5*x)^(1/2)+331*(1-2*x)^(1/2)*(3+5*x)^(3/2)/(3 36+504*x)-1/9*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^3+181/108*(1-2*x)^(1/2)* (3+5*x)^(5/2)/(2+3*x)^2-575/243*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/ 2)-326717/95256*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (78416+275022 x+286791 x^2+75600 x^3\right )}{(2+3 x)^3}+225400 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-326717 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{95256} \] Input:
Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^4,x]
Output:
((-21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(78416 + 275022*x + 286791*x^2 + 75600*x ^3))/(2 + 3*x)^3 + 225400*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 326717*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/95256
Time = 0.28 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {108, 27, 166, 27, 166, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{(3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{9} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^3}dx-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^3}dx-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{18} \left (\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac {1}{6} \int \frac {(139-4260 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^2}dx\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac {1}{12} \int \frac {(139-4260 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^2}dx\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{12} \left (\frac {2979 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}-\frac {1}{21} \int \frac {3 (2919-158980 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)}dx\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{12} \left (\frac {2979 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}-\frac {1}{14} \int \frac {(2919-158980 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{6} \int -\frac {2 (322000 x+105761)}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {79490}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2979 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (-\frac {1}{3} \int \frac {322000 x+105761}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {79490}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2979 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (\frac {326717}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {322000}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {79490}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2979 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (\frac {326717}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {128800}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {79490}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2979 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (\frac {653434}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {128800}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {79490}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2979 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (-\frac {128800}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {653434 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {79490}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2979 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{12} \left (\frac {1}{14} \left (\frac {1}{3} \left (-\frac {64400}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {653434 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {79490}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2979 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 (3 x+2)}\right )+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
Input:
Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^4,x]
Output:
-1/9*((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^3 + ((181*Sqrt[1 - 2*x]*( 3 + 5*x)^(5/2))/(6*(2 + 3*x)^2) + ((2979*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(7 *(2 + 3*x)) + ((-79490*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3 + ((-64400*Sqrt[10]* ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 - (653434*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7 ]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/3)/14)/12)/18
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[((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 = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (75600 x^{3}+286791 x^{2}+275022 x +78416\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4536 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (-\frac {575 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{486}+\frac {326717 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{190512}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(142\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (8821359 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-6085800 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+17642718 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-12171600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-3175200 x^{3} \sqrt {-10 x^{2}-x +3}+11761812 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -8114400 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -12045222 x^{2} \sqrt {-10 x^{2}-x +3}+2613736 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1803200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-11550924 x \sqrt {-10 x^{2}-x +3}-3293472 \sqrt {-10 x^{2}-x +3}\right )}{190512 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\) | \(270\) |
Input:
int((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4,x,method=_RETURNVERBOSE)
Output:
1/4536*(-1+2*x)*(3+5*x)^(1/2)*(75600*x^3+286791*x^2+275022*x+78416)/(2+3*x )^3/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+(-575/ 486*10^(1/2)*arcsin(20/11*x+1/11)+326717/190512*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 = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {326717 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 225400 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (75600 \, x^{3} + 286791 \, x^{2} + 275022 \, x + 78416\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{190512 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4,x, algorithm="fricas")
Output:
-1/190512*(326717*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7) *(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 225400*sqrt( 10)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*(75600*x^3 + 286791*x^2 + 27502 2*x + 78416)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{4}}\, dx \] Input:
integrate((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**4,x)
Output:
Integral((1 - 2*x)**(3/2)*(5*x + 3)**(5/2)/(3*x + 2)**4, x)
Time = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {865}{2646} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {173 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{588 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {34805}{5292} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {575}{486} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {326717}{190512} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {152917}{31752} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {2507 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3528 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4,x, algorithm="maxima")
Output:
865/2646*(-10*x^2 - x + 3)^(3/2) - 1/21*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 173/588*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 34805/5292*sqrt(-10*x^2 - x + 3)*x - 575/486*sqrt(10)*arcsin(20/11*x + 1/1 1) + 326717/190512*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2 )) - 152917/31752*sqrt(-10*x^2 - x + 3) + 2507/3528*(-10*x^2 - x + 3)^(3/2 )/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (129) = 258\).
Time = 0.37 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.36 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {326717}{1905120} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {575}{486} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {10}{81} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, {\left (2463 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1767360 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 377652800 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{756 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4,x, algorithm="giac")
Output:
326717/1905120*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)))) - 575/486*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)*sqrt(-1 0*x + 5) - sqrt(22)))) - 10/81*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/ 756*(2463*sqrt(10)*((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 + 1767360*sqrt(10)* ((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr t(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 377652800*sqrt(10)*((sqrt(2)*sqrt(-1 0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*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)^3
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^4} \,d x \] Input:
int(((1 - 2*x)^(3/2)*(5*x + 3)^(5/2))/(3*x + 2)^4,x)
Output:
int(((1 - 2*x)^(3/2)*(5*x + 3)^(5/2))/(3*x + 2)^4, x)
Time = 0.26 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.59 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4,x)
Output:
(6085800*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x**3 + 1217160 0*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x**2 + 8114400*sqrt(1 0)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x + 1803200*sqrt(10)*asin((sq rt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 8821359*sqrt(7)*atan((sqrt(33) - sqrt( 35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 1764 2718*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5)) /sqrt(11))/2))/sqrt(2))*x**2 + 11761812*sqrt(7)*atan((sqrt(33) - sqrt(35)* tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 2613736*sqr t(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11 ))/2))/sqrt(2)) - 8821359*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt ( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 - 17642718*sqrt(7)*atan( (sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqr t(2))*x**2 - 11761812*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 2613736*sqrt(7)*atan((sqrt(33 ) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 1587600*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 6022611*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 5775462*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 1646736*sqrt( 5*x + 3)*sqrt( - 2*x + 1))/(95256*(27*x**3 + 54*x**2 + 36*x + 8))