Integrand size = 26, antiderivative size = 138 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac {76 (1-2 x)^{7/2}}{1815 \sqrt {3+5 x}}+\frac {329 \sqrt {1-2 x} \sqrt {3+5 x}}{5000}+\frac {329 (1-2 x)^{3/2} \sqrt {3+5 x}}{16500}+\frac {329 (1-2 x)^{5/2} \sqrt {3+5 x}}{45375}+\frac {3619 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5000 \sqrt {10}} \]
-2/825*(1-2*x)^(7/2)/(3+5*x)^(3/2)+3619/50000*arcsin(1/11*22^(1/2)*(3+5*x) ^(1/2))*10^(1/2)-76/1815*(1-2*x)^(7/2)/(3+5*x)^(1/2)+329/16500*(1-2*x)^(3/ 2)*(3+5*x)^(1/2)+329/45375*(1-2*x)^(5/2)*(3+5*x)^(1/2)+329/5000*(1-2*x)^(1 /2)*(3+5*x)^(1/2)
Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.59 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx=\frac {\frac {5 \sqrt {1-2 x} \left (10633+40930 x+3585 x^2-35100 x^3+36000 x^4\right )}{(3+5 x)^{3/2}}-10857 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{75000} \]
((5*Sqrt[1 - 2*x]*(10633 + 40930*x + 3585*x^2 - 35100*x^3 + 36000*x^4))/(3 + 5*x)^(3/2) - 10857*Sqrt[10]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x])])/75000
Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {100, 27, 87, 60, 60, 60, 64, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^2}{(5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {2}{825} \int \frac {(1-2 x)^{5/2} (1485 x+1081)}{2 (5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{825} \int \frac {(1-2 x)^{5/2} (1485 x+1081)}{(5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{825} \left (\frac {987}{11} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {380 (1-2 x)^{7/2}}{11 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{825} \left (\frac {987}{11} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {380 (1-2 x)^{7/2}}{11 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{825} \left (\frac {987}{11} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {380 (1-2 x)^{7/2}}{11 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{825} \left (\frac {987}{11} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {380 (1-2 x)^{7/2}}{11 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{825} \left (\frac {987}{11} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {380 (1-2 x)^{7/2}}{11 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{825} \left (\frac {987}{11} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {380 (1-2 x)^{7/2}}{11 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}\) |
(-2*(1 - 2*x)^(7/2))/(825*(3 + 5*x)^(3/2)) + ((-380*(1 - 2*x)^(7/2))/(11*S qrt[3 + 5*x]) + (987*(((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/15 + (11*(((1 - 2*x) ^(3/2)*Sqrt[3 + 5*x])/10 + (33*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcS in[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/6))/11)/825
3.25.49.3.1 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_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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 = 147, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(\frac {\left (720000 x^{4} \sqrt {-10 x^{2}-x +3}+271425 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-702000 x^{3} \sqrt {-10 x^{2}-x +3}+325710 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +71700 x^{2} \sqrt {-10 x^{2}-x +3}+97713 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+818600 x \sqrt {-10 x^{2}-x +3}+212660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{300000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(147\) |
1/300000*(720000*x^4*(-10*x^2-x+3)^(1/2)+271425*10^(1/2)*arcsin(20/11*x+1/ 11)*x^2-702000*x^3*(-10*x^2-x+3)^(1/2)+325710*10^(1/2)*arcsin(20/11*x+1/11 )*x+71700*x^2*(-10*x^2-x+3)^(1/2)+97713*10^(1/2)*arcsin(20/11*x+1/11)+8186 00*x*(-10*x^2-x+3)^(1/2)+212660*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^ 2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx=-\frac {10857 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (36000 \, x^{4} - 35100 \, x^{3} + 3585 \, x^{2} + 40930 \, x + 10633\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{300000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
-1/300000*(10857*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 20*(36000*x^4 - 35100 *x^3 + 3585*x^2 + 40930*x + 10633)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (99) = 198\).
Time = 0.30 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.79 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx=\frac {3619}{100000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{125 \, {\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{125 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {1089}{5000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {11 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{750 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{500 \, {\left (5 \, x + 3\right )}} - \frac {121 \, \sqrt {-10 \, x^{2} - x + 3}}{3750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {3113 \, \sqrt {-10 \, x^{2} - x + 3}}{3750 \, {\left (5 \, x + 3\right )}} \]
3619/100000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/125*(-10*x^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*x^2 + 540*x + 81) + 3/125*(-10*x^2 - x + 3)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 3/125*(-10*x^2 - x + 3)^(5/ 2)/(25*x^2 + 30*x + 9) + 1089/5000*sqrt(-10*x^2 - x + 3) - 11/750*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 33/250*(-10*x^2 - x + 3 )^(3/2)/(25*x^2 + 30*x + 9) + 33/500*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 1 21/3750*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 3113/3750*sqrt(-10*x^2 - x + 3)/(5*x + 3)
Time = 0.40 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.34 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx=\frac {1}{125000} \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 135 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9635 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{750000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {3619}{50000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {1353 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{62500 \, \sqrt {5 \, x + 3}} + \frac {11 \, \sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {369 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{46875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
1/125000*(12*(8*sqrt(5)*(5*x + 3) - 135*sqrt(5))*(5*x + 3) + 9635*sqrt(5)) *sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/750000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 3619/50000*sqrt(10)*arcsin(1/11*sqrt(22 )*sqrt(5*x + 3)) - 1353/62500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22) )/sqrt(5*x + 3) + 11/46875*sqrt(10)*(5*x + 3)^(3/2)*(369*(sqrt(2)*sqrt(-10 *x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^ 3
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2}{{\left (5\,x+3\right )}^{5/2}} \,d x \]