Integrand size = 26, antiderivative size = 209 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt {3+5 x}} \, dx=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}+\frac {21199 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {1729615 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^3}+\frac {302171615 \sqrt {1-2 x} \sqrt {3+5 x}}{338688 (2+3 x)^2}+\frac {31603880465 \sqrt {1-2 x} \sqrt {3+5 x}}{4741632 (2+3 x)}-\frac {13391796605 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \] Output:
7/18*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^6+497/108*(1-2*x)^(1/2)*(3+5*x)^( 1/2)/(2+3*x)^5+21199/864*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+1729615/120 96*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+302171615/338688*(1-2*x)^(1/2)*(3 +5*x)^(1/2)/(2+3*x)^2+31603880465*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(9483264+142 24896*x)-13391796605/1229312*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5 *x)^(1/2))
Time = 0.37 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.44 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt {3+5 x}} \, dx=\frac {1331 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (120549503808+890768460368 x+2634024494432 x^2+3896029345680 x^3+2882422865340 x^4+853304772555 x^5\right )}{1331 (2+3 x)^6}-30184365 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{3687936} \] Input:
Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^7*Sqrt[3 + 5*x]),x]
Output:
(1331*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(120549503808 + 890768460368*x + 263 4024494432*x^2 + 3896029345680*x^3 + 2882422865340*x^4 + 853304772555*x^5) )/(1331*(2 + 3*x)^6) - 30184365*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt [3 + 5*x])]))/3687936
Time = 0.33 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {109, 27, 166, 27, 168, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^7 \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{18} \int \frac {(487-512 x) \sqrt {1-2 x}}{2 (3 x+2)^6 \sqrt {5 x+3}}dx+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \int \frac {(487-512 x) \sqrt {1-2 x}}{(3 x+2)^6 \sqrt {5 x+3}}dx+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{36} \left (\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}-\frac {1}{15} \int -\frac {5 (24323-37712 x)}{2 \sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \int \frac {24323-37712 x}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {1}{28} \int \frac {105 (58777-84796 x)}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {15}{8} \int \frac {58777-84796 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {15}{8} \left (\frac {1}{21} \int \frac {10920161-13836920 x}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {345923 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {15}{8} \left (\frac {1}{42} \int \frac {10920161-13836920 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {345923 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {15}{8} \left (\frac {1}{42} \left (\frac {1}{14} \int \frac {1301134391-1208686460 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {60434323 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {345923 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {15}{8} \left (\frac {1}{42} \left (\frac {1}{28} \int \frac {1301134391-1208686460 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {60434323 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {345923 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {15}{8} \left (\frac {1}{42} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {72315701667}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {6320776093 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {60434323 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {345923 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {15}{8} \left (\frac {1}{42} \left (\frac {1}{28} \left (\frac {72315701667}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {6320776093 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {60434323 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {345923 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {15}{8} \left (\frac {1}{42} \left (\frac {1}{28} \left (\frac {72315701667}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {6320776093 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {60434323 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {345923 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{6} \left (\frac {15}{8} \left (\frac {1}{42} \left (\frac {1}{28} \left (\frac {6320776093 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {72315701667 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {60434323 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {345923 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {21199 \sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\right )+\frac {497 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^5}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}\) |
Input:
Int[(1 - 2*x)^(5/2)/((2 + 3*x)^7*Sqrt[3 + 5*x]),x]
Output:
(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + ((497*Sqrt[1 - 2*x]*S qrt[3 + 5*x])/(3*(2 + 3*x)^5) + ((21199*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4*(2 + 3*x)^4) + (15*((345923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^3) + ((60434323*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((6320776093*Sq rt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (72315701667*ArcTan[Sqrt[1 - 2* x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28)/42))/8)/6)/36
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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (853304772555 x^{5}+2882422865340 x^{4}+3896029345680 x^{3}+2634024494432 x^{2}+890768460368 x +120549503808\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{526848 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {13391796605 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (29287859175135 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+117151436700540 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+195252394500900 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+11946266815770 x^{5} \sqrt {-10 x^{2}-x +3}+173557684000800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+40353920114760 x^{4} \sqrt {-10 x^{2}-x +3}+86778842000400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+54544410839520 x^{3} \sqrt {-10 x^{2}-x +3}+23141024533440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +36876342922048 x^{2} \sqrt {-10 x^{2}-x +3}+2571224948160 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+12470758445152 x \sqrt {-10 x^{2}-x +3}+1687693053312 \sqrt {-10 x^{2}-x +3}\right )}{7375872 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
Input:
int((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/526848*(-1+2*x)*(3+5*x)^(1/2)*(853304772555*x^5+2882422865340*x^4+38960 29345680*x^3+2634024494432*x^2+890768460368*x+120549503808)/(2+3*x)^6/(-(- 1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+13391796605/24 58624*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 = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt {3+5 x}} \, dx=-\frac {40175389815 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (853304772555 \, x^{5} + 2882422865340 \, x^{4} + 3896029345680 \, x^{3} + 2634024494432 \, x^{2} + 890768460368 \, x + 120549503808\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7375872 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x, algorithm="fricas")
Output:
-1/7375872*(40175389815*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqr t(-2*x + 1)/(10*x^2 + x - 3)) - 14*(853304772555*x^5 + 2882422865340*x^4 + 3896029345680*x^3 + 2634024494432*x^2 + 890768460368*x + 120549503808)*sq rt(5*x + 3)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 21 60*x^2 + 576*x + 64)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{7} \sqrt {5 x + 3}}\, dx \] Input:
integrate((1-2*x)**(5/2)/(2+3*x)**7/(3+5*x)**(1/2),x)
Output:
Integral((1 - 2*x)**(5/2)/((3*x + 2)**7*sqrt(5*x + 3)), x)
Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt {3+5 x}} \, dx=\frac {13391796605}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{54 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {469 \, \sqrt {-10 \, x^{2} - x + 3}}{108 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {21199 \, \sqrt {-10 \, x^{2} - x + 3}}{864 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1729615 \, \sqrt {-10 \, x^{2} - x + 3}}{12096 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {302171615 \, \sqrt {-10 \, x^{2} - x + 3}}{338688 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {31603880465 \, \sqrt {-10 \, x^{2} - x + 3}}{4741632 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x, algorithm="maxima")
Output:
13391796605/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 49/54*sqrt(-10*x^2 - x + 3)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^ 3 + 2160*x^2 + 576*x + 64) + 469/108*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810* x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 21199/864*sqrt(-10*x^2 - x + 3)/( 81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1729615/12096*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 302171615/338688*sqrt(-10*x^2 - x + 3)/ (9*x^2 + 12*x + 4) + 31603880465/4741632*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (164) = 328\).
Time = 0.45 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt {3+5 x}} \, dx=\frac {2678359321}{4917248} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {6655 \, \sqrt {10} {\left (20305527 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 17887837240 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 7599643632000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 1749282956467200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 210267345272320000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {10389680589926400000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {41558722359705600000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{263424 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x, algorithm="giac")
Output:
2678359321/4917248*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)))) + 6655/263424*sqrt(10)*(20305527*((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)))^11 + 17887837240*((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 + 7599643632000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 1749282956467200*((s qrt(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 + 210267345272320000*((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 + 10389680589926400000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22) )/sqrt(5*x + 3) - 41558722359705600000*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)^6
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^7\,\sqrt {5\,x+3}} \,d x \] Input:
int((1 - 2*x)^(5/2)/((3*x + 2)^7*(5*x + 3)^(1/2)),x)
Output:
int((1 - 2*x)^(5/2)/((3*x + 2)^7*(5*x + 3)^(1/2)), x)
Time = 0.28 (sec) , antiderivative size = 632, normalized size of antiderivative = 3.02 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt {3+5 x}} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x)
Output:
(29287859175135*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**6 + 117151436700540*sqrt(7)*atan((sq rt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2 ))*x**5 + 195252394500900*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt ( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 + 173557684000800*sqrt(7 )*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/ 2))/sqrt(2))*x**3 + 86778842000400*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(a sin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 + 2314102453344 0*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sq rt(11))/2))/sqrt(2))*x + 2571224948160*sqrt(7)*atan((sqrt(33) - sqrt(35)*t an(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 29287859175135 *sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqr t(11))/2))/sqrt(2))*x**6 - 117151436700540*sqrt(7)*atan((sqrt(33) + sqrt(3 5)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 - 19525 2394500900*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sq rt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 173557684000800*sqrt(7)*atan((sqrt(33 ) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x* *3 - 86778842000400*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2* x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 23141024533440*sqrt(7)*atan( (sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/...