Integrand size = 19, antiderivative size = 138 \[ \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}+\frac {1331 (1-2 x)^{3/2} \sqrt {3+5 x}}{6400}+\frac {121 (1-2 x)^{5/2} \sqrt {3+5 x}}{1600}-\frac {33}{160} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {483153 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000 \sqrt {10}} \]
-1/10*(1-2*x)^(7/2)*(3+5*x)^(3/2)+483153/640000*arcsin(1/11*22^(1/2)*(3+5* x)^(1/2))*10^(1/2)+1331/6400*(1-2*x)^(3/2)*(3+5*x)^(1/2)+121/1600*(1-2*x)^ (5/2)*(3+5*x)^(1/2)-33/160*(1-2*x)^(7/2)*(3+5*x)^(1/2)+43923/64000*(1-2*x) ^(1/2)*(3+5*x)^(1/2)
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.54 \[ \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \sqrt {3+5 x} \left (29673+116420 x-177440 x^2-124800 x^3+256000 x^4\right )+483153 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right )}{640000} \]
(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(29673 + 116420*x - 177440*x^2 - 124800*x^ 3 + 256000*x^4) + 483153*Sqrt[10]*ArcTan[Sqrt[6/5 + 2*x]/Sqrt[1 - 2*x]])/6 40000
Time = 0.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {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.
| \(\displaystyle \int (1-2 x)^{5/2} (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {33}{20} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {33}{20} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\) |
-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[Sq rt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/6))/16))/20
3.25.2.3.1 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 = 1.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(-\frac {\left (256000 x^{4}-124800 x^{3}-177440 x^{2}+116420 x +29673\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{64000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {483153 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1280000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
| default | \(\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{25}+\frac {11 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{200}+\frac {121 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{2000}-\frac {1331 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{16000}-\frac {43923 \sqrt {1-2 x}\, \sqrt {3+5 x}}{64000}+\frac {483153 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1280000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(120\) |
-1/64000*(256000*x^4-124800*x^3-177440*x^2+116420*x+29673)*(-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)+48 3153/1280000*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 = 77, normalized size of antiderivative = 0.56 \[ \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {1}{64000} \, {\left (256000 \, x^{4} - 124800 \, x^{3} - 177440 \, x^{2} + 116420 \, x + 29673\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {483153}{1280000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/64000*(256000*x^4 - 124800*x^3 - 177440*x^2 + 116420*x + 29673)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 483153/1280000*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 = 31.11 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.24 \[ \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx=\begin {cases} \frac {40 i \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {10 x - 5}} - \frac {319 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {10 x - 5}} + \frac {8833 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{40 \sqrt {10 x - 5}} - \frac {171699 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{1600 \sqrt {10 x - 5}} - \frac {14641 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{6400 \sqrt {10 x - 5}} + \frac {483153 i \sqrt {x + \frac {3}{5}}}{64000 \sqrt {10 x - 5}} - \frac {483153 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{640000} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {483153 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{640000} - \frac {40 \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {5 - 10 x}} + \frac {319 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {5 - 10 x}} - \frac {8833 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{40 \sqrt {5 - 10 x}} + \frac {171699 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{1600 \sqrt {5 - 10 x}} + \frac {14641 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{6400 \sqrt {5 - 10 x}} - \frac {483153 \sqrt {x + \frac {3}{5}}}{64000 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]
Piecewise((40*I*(x + 3/5)**(11/2)/sqrt(10*x - 5) - 319*I*(x + 3/5)**(9/2)/ (2*sqrt(10*x - 5)) + 8833*I*(x + 3/5)**(7/2)/(40*sqrt(10*x - 5)) - 171699* I*(x + 3/5)**(5/2)/(1600*sqrt(10*x - 5)) - 14641*I*(x + 3/5)**(3/2)/(6400* sqrt(10*x - 5)) + 483153*I*sqrt(x + 3/5)/(64000*sqrt(10*x - 5)) - 483153*s qrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/640000, Abs(x + 3/5) > 11/10), (483153*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/640000 - 40*(x + 3/5)** (11/2)/sqrt(5 - 10*x) + 319*(x + 3/5)**(9/2)/(2*sqrt(5 - 10*x)) - 8833*(x + 3/5)**(7/2)/(40*sqrt(5 - 10*x)) + 171699*(x + 3/5)**(5/2)/(1600*sqrt(5 - 10*x)) + 14641*(x + 3/5)**(3/2)/(6400*sqrt(5 - 10*x)) - 483153*sqrt(x + 3 /5)/(64000*sqrt(5 - 10*x)), True))
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.61 \[ \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {1}{25} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {11}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {11}{800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3993}{3200} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {483153}{1280000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3993}{64000} \, \sqrt {-10 \, x^{2} - x + 3} \]
1/25*(-10*x^2 - x + 3)^(5/2) + 11/40*(-10*x^2 - x + 3)^(3/2)*x + 11/800*(- 10*x^2 - x + 3)^(3/2) + 3993/3200*sqrt(-10*x^2 - x + 3)*x - 483153/1280000 *sqrt(10)*arcsin(-20/11*x - 1/11) + 3993/64000*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (99) = 198\).
Time = 0.38 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {1}{9600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{480000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {59}{120000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {3}{1000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
1/9600000*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*sqr t(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/480000*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))) - 59/120000*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))) - 3/1000*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))) + 9/50*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. \[ \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \]