Integrand size = 19, antiderivative size = 72 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {33}{16} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{4} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {363 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{16 \sqrt {10}} \] Output:
-33/16*(1-2*x)^(1/2)*(3+5*x)^(1/2)-1/4*(1-2*x)^(1/2)*(3+5*x)^(3/2)+363/160 *arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {5}{16} \sqrt {1-2 x} (9+4 x) \sqrt {3+5 x}+\frac {363 \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right )}{16 \sqrt {10}} \] Input:
Integrate[(3 + 5*x)^(3/2)/Sqrt[1 - 2*x],x]
Output:
(-5*Sqrt[1 - 2*x]*(9 + 4*x)*Sqrt[3 + 5*x])/16 + (363*ArcTan[Sqrt[6/5 + 2*x ]/Sqrt[1 - 2*x]])/(16*Sqrt[10])
Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {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.
\(\displaystyle \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {33}{8} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {33}{8} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {33}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {33}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
Input:
Int[(3 + 5*x)^(3/2)/Sqrt[1 - 2*x],x]
Output:
-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]*Sqrt[3 + 5*x]])/(2*Sqrt[10])))/8
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 = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \left (3+5 x \right )^{\frac {3}{2}}}{4}-\frac {33 \sqrt {1-2 x}\, \sqrt {3+5 x}}{16}+\frac {363 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{320 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(72\) |
risch | \(\frac {5 \left (9+4 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{16 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {363 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{320 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(93\) |
Input:
int((3+5*x)^(3/2)/(1-2*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(1-2*x)^(1/2)*(3+5*x)^(3/2)-33/16*(1-2*x)^(1/2)*(3+5*x)^(1/2)+363/320 *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 = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {5}{16} \, \sqrt {5 \, x + 3} {\left (4 \, x + 9\right )} \sqrt {-2 \, x + 1} - \frac {363}{320} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="fricas")
Output:
-5/16*sqrt(5*x + 3)*(4*x + 9)*sqrt(-2*x + 1) - 363/320*sqrt(10)*arctan(1/2 0*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 = 2.17 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.57 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=\begin {cases} - \frac {25 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{2 \sqrt {10 x - 5}} - \frac {55 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{8 \sqrt {10 x - 5}} + \frac {363 i \sqrt {x + \frac {3}{5}}}{16 \sqrt {10 x - 5}} - \frac {363 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{160} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {363 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{160} + \frac {25 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{2 \sqrt {5 - 10 x}} + \frac {55 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{8 \sqrt {5 - 10 x}} - \frac {363 \sqrt {x + \frac {3}{5}}}{16 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \] Input:
integrate((3+5*x)**(3/2)/(1-2*x)**(1/2),x)
Output:
Piecewise((-25*I*(x + 3/5)**(5/2)/(2*sqrt(10*x - 5)) - 55*I*(x + 3/5)**(3/ 2)/(8*sqrt(10*x - 5)) + 363*I*sqrt(x + 3/5)/(16*sqrt(10*x - 5)) - 363*sqrt (10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/160, Abs(x + 3/5) > 11/10), (363* sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/160 + 25*(x + 3/5)**(5/2)/(2*sqr t(5 - 10*x)) + 55*(x + 3/5)**(3/2)/(8*sqrt(5 - 10*x)) - 363*sqrt(x + 3/5)/ (16*sqrt(5 - 10*x)), True))
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.57 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {5}{4} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {363}{320} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {45}{16} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="maxima")
Output:
-5/4*sqrt(-10*x^2 - x + 3)*x - 363/320*sqrt(10)*arcsin(-20/11*x - 1/11) - 45/16*sqrt(-10*x^2 - x + 3)
Time = 0.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.62 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{160} \, \sqrt {5} {\left (10 \, \sqrt {5 \, x + 3} {\left (4 \, x + 9\right )} \sqrt {-10 \, x + 5} - 363 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="giac")
Output:
-1/160*sqrt(5)*(10*sqrt(5*x + 3)*(4*x + 9)*sqrt(-10*x + 5) - 363*sqrt(2)*a rcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}} \,d x \] Input:
int((5*x + 3)^(3/2)/(1 - 2*x)^(1/2),x)
Output:
int((5*x + 3)^(3/2)/(1 - 2*x)^(1/2), x)
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.67 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {363 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{160}-\frac {5 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{4}-\frac {45 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{16} \] Input:
int((3+5*x)^(3/2)/(1-2*x)^(1/2),x)
Output:
( - 363*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) - 200*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 450*sqrt(5*x + 3)*sqrt( - 2*x + 1))/160