Integrand size = 19, antiderivative size = 160 \[ \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx=\frac {161051 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}+\frac {14641 (1-2 x)^{3/2} \sqrt {3+5 x}}{30720}+\frac {1331 (1-2 x)^{5/2} \sqrt {3+5 x}}{7680}-\frac {121}{256} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {1771561 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400 \sqrt {10}} \]
-11/48*(1-2*x)^(7/2)*(3+5*x)^(3/2)-1/12*(1-2*x)^(7/2)*(3+5*x)^(5/2)+177156 1/1024000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+14641/30720*(1-2*x) ^(3/2)*(3+5*x)^(1/2)+1331/7680*(1-2*x)^(5/2)*(3+5*x)^(1/2)-121/256*(1-2*x) ^(7/2)*(3+5*x)^(1/2)+161051/102400*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.49 \[ \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \sqrt {3+5 x} \left (96003+1895020 x-748640 x^2-4905600 x^3+1280000 x^4+5120000 x^5\right )+5314683 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right )}{3072000} \]
(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(96003 + 1895020*x - 748640*x^2 - 4905600* x^3 + 1280000*x^4 + 5120000*x^5) + 5314683*Sqrt[10]*ArcTan[Sqrt[6/5 + 2*x] /Sqrt[1 - 2*x]])/3072000
Time = 0.24 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {60, 60, 60, 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.
| \(\displaystyle \int (1-2 x)^{5/2} (5 x+3)^{5/2} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {55}{24} \int (1-2 x)^{5/2} (5 x+3)^{3/2}dx-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {55}{24} \left (\frac {33}{20} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
-1/12*((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2)) + (55*(-1/10*((1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)) + (33*(-1/8*((1 - 2*x)^(7/2)*Sqrt[3 + 5*x]) + (11*(((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*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5 *Sqrt[10])))/20))/6))/16))/20))/24
3.25.14.3.1 Defintions of rubi rules used
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[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 = 1.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {\left (5120000 x^{5}+1280000 x^{4}-4905600 x^{3}-748640 x^{2}+1895020 x +96003\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307200 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1771561 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2048000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(113\) |
| default | \(\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {7}{2}}}{30}+\frac {11 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {7}{2}}}{300}+\frac {121 \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}{4000}-\frac {1331 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{48000}-\frac {14641 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{76800}-\frac {161051 \sqrt {1-2 x}\, \sqrt {3+5 x}}{102400}+\frac {1771561 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2048000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(136\) |
-1/307200*(5120000*x^5+1280000*x^4-4905600*x^3-748640*x^2+1895020*x+96003) *(-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)+1771561/2048000*10^(1/2)*arcsin(20/11*x+1/11)*((1-2*x)*(3+5* x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx=\frac {1}{307200} \, {\left (5120000 \, x^{5} + 1280000 \, x^{4} - 4905600 \, x^{3} - 748640 \, x^{2} + 1895020 \, x + 96003\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1771561}{2048000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/307200*(5120000*x^5 + 1280000*x^4 - 4905600*x^3 - 748640*x^2 + 1895020*x + 96003)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1771561/2048000*sqrt(10)*arctan(1 /20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
Result contains complex when optimal does not.
Time = 168.95 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.22 \[ \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx=\begin {cases} \frac {500 i \left (x + \frac {3}{5}\right )^{\frac {13}{2}}}{3 \sqrt {10 x - 5}} - \frac {1925 i \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{3 \sqrt {10 x - 5}} + \frac {40535 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{48 \sqrt {10 x - 5}} - \frac {73205 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{192 \sqrt {10 x - 5}} - \frac {14641 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{7680 \sqrt {10 x - 5}} - \frac {161051 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{30720 \sqrt {10 x - 5}} + \frac {1771561 i \sqrt {x + \frac {3}{5}}}{102400 \sqrt {10 x - 5}} - \frac {1771561 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{1024000} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {1771561 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{1024000} - \frac {500 \left (x + \frac {3}{5}\right )^{\frac {13}{2}}}{3 \sqrt {5 - 10 x}} + \frac {1925 \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{3 \sqrt {5 - 10 x}} - \frac {40535 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{48 \sqrt {5 - 10 x}} + \frac {73205 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{192 \sqrt {5 - 10 x}} + \frac {14641 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{7680 \sqrt {5 - 10 x}} + \frac {161051 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{30720 \sqrt {5 - 10 x}} - \frac {1771561 \sqrt {x + \frac {3}{5}}}{102400 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]
Piecewise((500*I*(x + 3/5)**(13/2)/(3*sqrt(10*x - 5)) - 1925*I*(x + 3/5)** (11/2)/(3*sqrt(10*x - 5)) + 40535*I*(x + 3/5)**(9/2)/(48*sqrt(10*x - 5)) - 73205*I*(x + 3/5)**(7/2)/(192*sqrt(10*x - 5)) - 14641*I*(x + 3/5)**(5/2)/ (7680*sqrt(10*x - 5)) - 161051*I*(x + 3/5)**(3/2)/(30720*sqrt(10*x - 5)) + 1771561*I*sqrt(x + 3/5)/(102400*sqrt(10*x - 5)) - 1771561*sqrt(10)*I*acos h(sqrt(110)*sqrt(x + 3/5)/11)/1024000, Abs(x + 3/5) > 11/10), (1771561*sqr t(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/1024000 - 500*(x + 3/5)**(13/2)/(3* sqrt(5 - 10*x)) + 1925*(x + 3/5)**(11/2)/(3*sqrt(5 - 10*x)) - 40535*(x + 3 /5)**(9/2)/(48*sqrt(5 - 10*x)) + 73205*(x + 3/5)**(7/2)/(192*sqrt(5 - 10*x )) + 14641*(x + 3/5)**(5/2)/(7680*sqrt(5 - 10*x)) + 161051*(x + 3/5)**(3/2 )/(30720*sqrt(5 - 10*x)) - 1771561*sqrt(x + 3/5)/(102400*sqrt(5 - 10*x)), True))
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.62 \[ \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx=\frac {1}{6} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {1}{120} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {121}{192} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {121}{3840} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {14641}{5120} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1771561}{2048000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {14641}{102400} \, \sqrt {-10 \, x^{2} - x + 3} \]
1/6*(-10*x^2 - x + 3)^(5/2)*x + 1/120*(-10*x^2 - x + 3)^(5/2) + 121/192*(- 10*x^2 - x + 3)^(3/2)*x + 121/3840*(-10*x^2 - x + 3)^(3/2) + 14641/5120*sq rt(-10*x^2 - x + 3)*x - 1771561/2048000*sqrt(10)*arcsin(-20/11*x - 1/11) + 14641/102400*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (115) = 230\).
Time = 0.35 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.22 \[ \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx=\frac {1}{76800000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{2400000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {47}{1920000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {69}{40000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
1/76800000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 15385695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sq rt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1 /2400000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt (2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 47/1920000*sqrt(5)*(2*(4*(8*(60 *x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 69/40000*sqrt(5 )*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785 *sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 27/2000*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt (5*x + 3))) + 27/50*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3) ) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
Timed out. \[ \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]