Integrand size = 19, antiderivative size = 94 \[ \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx=\frac {121}{400} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {11}{120} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {1331 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}} \]
1331/4000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+11/120*(1-2*x)^(3/2 )*(3+5*x)^(1/2)-1/6*(1-2*x)^(5/2)*(3+5*x)^(1/2)+121/400*(1-2*x)^(1/2)*(3+5 *x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.68 \[ \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \sqrt {3+5 x} \left (273+580 x-800 x^2\right )+3993 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right )}{12000} \]
(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(273 + 580*x - 800*x^2) + 3993*Sqrt[10]*Ar cTan[Sqrt[6/5 + 2*x]/Sqrt[1 - 2*x]])/12000
Time = 0.17 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {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.
\(\displaystyle \int (1-2 x)^{3/2} \sqrt {5 x+3} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {11}{12} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {11}{12} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {11}{12} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\) |
-1/6*((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]) + (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))/12
3.24.30.3.1 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 = 1.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{15}+\frac {11 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{100}-\frac {121 \sqrt {1-2 x}\, \sqrt {3+5 x}}{400}+\frac {1331 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{8000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(88\) |
risch | \(\frac {\left (800 x^{2}-580 x -273\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1200 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1331 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{8000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(98\) |
1/15*(1-2*x)^(3/2)*(3+5*x)^(3/2)+11/100*(3+5*x)^(3/2)*(1-2*x)^(1/2)-121/40 0*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1331/8000*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 = 67, normalized size of antiderivative = 0.71 \[ \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx=-\frac {1}{1200} \, {\left (800 \, x^{2} - 580 \, x - 273\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1331}{8000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-1/1200*(800*x^2 - 580*x - 273)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1331/8000*s qrt(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 = 3.34 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.43 \[ \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx=\begin {cases} - \frac {20 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{3 \sqrt {10 x - 5}} + \frac {121 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{6 \sqrt {10 x - 5}} - \frac {2057 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{120 \sqrt {10 x - 5}} + \frac {1331 i \sqrt {x + \frac {3}{5}}}{400 \sqrt {10 x - 5}} - \frac {1331 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{4000} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {1331 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{4000} + \frac {20 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{3 \sqrt {5 - 10 x}} - \frac {121 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{6 \sqrt {5 - 10 x}} + \frac {2057 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{120 \sqrt {5 - 10 x}} - \frac {1331 \sqrt {x + \frac {3}{5}}}{400 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]
Piecewise((-20*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) + 121*I*(x + 3/5)**(5 /2)/(6*sqrt(10*x - 5)) - 2057*I*(x + 3/5)**(3/2)/(120*sqrt(10*x - 5)) + 13 31*I*sqrt(x + 3/5)/(400*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(110)* sqrt(x + 3/5)/11)/4000, Abs(x + 3/5) > 11/10), (1331*sqrt(10)*asin(sqrt(11 0)*sqrt(x + 3/5)/11)/4000 + 20*(x + 3/5)**(7/2)/(3*sqrt(5 - 10*x)) - 121*( x + 3/5)**(5/2)/(6*sqrt(5 - 10*x)) + 2057*(x + 3/5)**(3/2)/(120*sqrt(5 - 1 0*x)) - 1331*sqrt(x + 3/5)/(400*sqrt(5 - 10*x)), True))
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.59 \[ \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx=\frac {1}{15} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {11}{20} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1331}{8000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {11}{400} \, \sqrt {-10 \, x^{2} - x + 3} \]
1/15*(-10*x^2 - x + 3)^(3/2) + 11/20*sqrt(-10*x^2 - x + 3)*x - 1331/8000*s qrt(10)*arcsin(-20/11*x - 1/11) + 11/400*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (67) = 134\).
Time = 0.34 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.49 \[ \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx=-\frac {1}{12000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
-1/12000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-1 0*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/2000*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))) + 3/50*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. \[ \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx=\int {\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3} \,d x \]