Integrand size = 19, antiderivative size = 116 \[ \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx=\frac {1331}{512} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {14641 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{512 \sqrt {10}} \]
-55/96*(1-2*x)^(3/2)*(3+5*x)^(3/2)-1/8*(1-2*x)^(3/2)*(3+5*x)^(5/2)+14641/5 120*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-605/256*(1-2*x)^(3/2)*(3+ 5*x)^(1/2)+1331/512*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.59 \[ \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \sqrt {3+5 x} \left (-4005+5836 x+15520 x^2+9600 x^3\right )+43923 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right )}{15360} \]
(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-4005 + 5836*x + 15520*x^2 + 9600*x^3) + 43923*Sqrt[10]*ArcTan[Sqrt[6/5 + 2*x]/Sqrt[1 - 2*x]])/15360
Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {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.
\(\displaystyle \int \sqrt {1-2 x} (5 x+3)^{5/2} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {55}{16} \int \sqrt {1-2 x} (5 x+3)^{3/2}dx-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {55}{16} \left (\frac {11}{4} \int \sqrt {1-2 x} \sqrt {5 x+3}dx-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
-1/8*((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)) + (55*(-1/6*((1 - 2*x)^(3/2)*(3 + 5 *x)^(3/2)) + (11*(-1/4*((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (11*((Sqrt[1 - 2* x]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10]))) /8))/4))/16
3.23.90.3.1 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 = 1.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {\left (9600 x^{3}+15520 x^{2}+5836 x -4005\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1536 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {14641 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{10240 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
default | \(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{8}-\frac {55 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{96}-\frac {605 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{256}+\frac {1331 \sqrt {1-2 x}\, \sqrt {3+5 x}}{512}+\frac {14641 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{10240 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(104\) |
-1/1536*(9600*x^3+15520*x^2+5836*x-4005)*(-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)+14641/10240*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 = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.62 \[ \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx=\frac {1}{1536} \, {\left (9600 \, x^{3} + 15520 \, x^{2} + 5836 \, x - 4005\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {14641}{10240} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/1536*(9600*x^3 + 15520*x^2 + 5836*x - 4005)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 14641/10240*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 = 19.05 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.33 \[ \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx=\begin {cases} \frac {125 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {10 x - 5}} - \frac {1925 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{24 \sqrt {10 x - 5}} - \frac {605 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{192 \sqrt {10 x - 5}} - \frac {6655 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{768 \sqrt {10 x - 5}} + \frac {14641 i \sqrt {x + \frac {3}{5}}}{512 \sqrt {10 x - 5}} - \frac {14641 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{5120} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {14641 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{5120} - \frac {125 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {5 - 10 x}} + \frac {1925 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{24 \sqrt {5 - 10 x}} + \frac {605 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{192 \sqrt {5 - 10 x}} + \frac {6655 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{768 \sqrt {5 - 10 x}} - \frac {14641 \sqrt {x + \frac {3}{5}}}{512 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]
Piecewise((125*I*(x + 3/5)**(9/2)/(2*sqrt(10*x - 5)) - 1925*I*(x + 3/5)**( 7/2)/(24*sqrt(10*x - 5)) - 605*I*(x + 3/5)**(5/2)/(192*sqrt(10*x - 5)) - 6 655*I*(x + 3/5)**(3/2)/(768*sqrt(10*x - 5)) + 14641*I*sqrt(x + 3/5)/(512*s qrt(10*x - 5)) - 14641*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/5120, Abs(x + 3/5) > 11/10), (14641*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/51 20 - 125*(x + 3/5)**(9/2)/(2*sqrt(5 - 10*x)) + 1925*(x + 3/5)**(7/2)/(24*s qrt(5 - 10*x)) + 605*(x + 3/5)**(5/2)/(192*sqrt(5 - 10*x)) + 6655*(x + 3/5 )**(3/2)/(768*sqrt(5 - 10*x)) - 14641*sqrt(x + 3/5)/(512*sqrt(5 - 10*x)), True))
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.60 \[ \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx=-\frac {5}{8} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {91}{96} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {605}{128} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {14641}{10240} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {121}{512} \, \sqrt {-10 \, x^{2} - x + 3} \]
-5/8*(-10*x^2 - x + 3)^(3/2)*x - 91/96*(-10*x^2 - x + 3)^(3/2) + 605/128*s qrt(-10*x^2 - x + 3)*x - 14641/10240*sqrt(10)*arcsin(-20/11*x - 1/11) + 12 1/512*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (83) = 166\).
Time = 0.35 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.75 \[ \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx=\frac {1}{76800} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{1600} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{400} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
1/76800*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))) + 3/1600*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))) + 27/400*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. \[ \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx=\int \sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2} \,d x \]