Integrand size = 19, antiderivative size = 96 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {605}{64} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {55}{48} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1331}{64} \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
1331/128*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-55/48*(3+5*x)^(3/2)* (1-2*x)^(1/2)-1/6*(3+5*x)^(5/2)*(1-2*x)^(1/2)-605/64*(1-2*x)^(1/2)*(3+5*x) ^(1/2)
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\frac {1}{384} \left (-2 \sqrt {1-2 x} \sqrt {3+5 x} \left (2763+2060 x+800 x^2\right )+3993 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right )\right ) \]
(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2763 + 2060*x + 800*x^2) + 3993*Sqrt[10]* ArcTan[Sqrt[6/5 + 2*x]/Sqrt[1 - 2*x]])/384
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08, 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.
\(\displaystyle \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {55}{12} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}}dx-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {55}{12} \left (\frac {33}{8} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {55}{12} \left (\frac {33}{8} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {55}{12} \left (\frac {33}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {55}{12} \left (\frac {33}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\) |
-1/6*(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)) + (55*(-1/4*(Sqrt[1 - 2*x]*(3 + 5*x)^ (3/2)) + (33*(-1/2*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (11*ArcSin[Sqrt[2/11]*S qrt[3 + 5*x]])/(2*Sqrt[10])))/8))/12
3.25.81.3.1 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 = 1.17 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {\left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{6}-\frac {55 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{48}-\frac {605 \sqrt {1-2 x}\, \sqrt {3+5 x}}{64}+\frac {1331 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{256 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(88\) |
risch | \(\frac {\left (800 x^{2}+2060 x +2763\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{192 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1331 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{256 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(98\) |
-1/6*(3+5*x)^(5/2)*(1-2*x)^(1/2)-55/48*(3+5*x)^(3/2)*(1-2*x)^(1/2)-605/64* (1-2*x)^(1/2)*(3+5*x)^(1/2)+1331/256*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 = 73, normalized size of antiderivative = 0.76 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{192} \, {\left (800 \, x^{2} + 2060 \, x + 2763\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1331}{256} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-1/192*(800*x^2 + 2060*x + 2763)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1331/256*s qrt(5)*sqrt(2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(- 2*x + 1)/(10*x^2 + x - 3))
Result contains complex when optimal does not.
Time = 8.17 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.38 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\begin {cases} - \frac {125 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{3 \sqrt {10 x - 5}} - \frac {275 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{24 \sqrt {10 x - 5}} - \frac {3025 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{96 \sqrt {10 x - 5}} + \frac {6655 i \sqrt {x + \frac {3}{5}}}{64 \sqrt {10 x - 5}} - \frac {1331 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{128} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {1331 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{128} + \frac {125 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{3 \sqrt {5 - 10 x}} + \frac {275 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{24 \sqrt {5 - 10 x}} + \frac {3025 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{96 \sqrt {5 - 10 x}} - \frac {6655 \sqrt {x + \frac {3}{5}}}{64 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]
Piecewise((-125*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) - 275*I*(x + 3/5)**( 5/2)/(24*sqrt(10*x - 5)) - 3025*I*(x + 3/5)**(3/2)/(96*sqrt(10*x - 5)) + 6 655*I*sqrt(x + 3/5)/(64*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(110)* sqrt(x + 3/5)/11)/128, Abs(x + 3/5) > 11/10), (1331*sqrt(10)*asin(sqrt(110 )*sqrt(x + 3/5)/11)/128 + 125*(x + 3/5)**(7/2)/(3*sqrt(5 - 10*x)) + 275*(x + 3/5)**(5/2)/(24*sqrt(5 - 10*x)) + 3025*(x + 3/5)**(3/2)/(96*sqrt(5 - 10 *x)) - 6655*sqrt(x + 3/5)/(64*sqrt(5 - 10*x)), True))
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.60 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {25}{6} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {515}{48} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1331}{256} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {921}{64} \, \sqrt {-10 \, x^{2} - x + 3} \]
-25/6*sqrt(-10*x^2 - x + 3)*x^2 - 515/48*sqrt(-10*x^2 - x + 3)*x - 1331/25 6*sqrt(10)*arcsin(-20/11*x - 1/11) - 921/64*sqrt(-10*x^2 - x + 3)
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.56 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{1920} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x + 79\right )} {\left (5 \, x + 3\right )} + 1815\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 19965 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
-1/1920*sqrt(5)*(2*(4*(40*x + 79)*(5*x + 3) + 1815)*sqrt(5*x + 3)*sqrt(-10 *x + 5) - 19965*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]