Integrand size = 17, antiderivative size = 53 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815}{8} \sqrt {1-2 x}-\frac {275}{8} (1-2 x)^{3/2}+\frac {25}{8} (1-2 x)^{5/2} \] Output:
1331/8/(1-2*x)^(1/2)+1815/8*(1-2*x)^(1/2)-275/8*(1-2*x)^(3/2)+25/8*(1-2*x) ^(5/2)
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.47 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {362-335 x-100 x^2-25 x^3}{\sqrt {1-2 x}} \] Input:
Integrate[(3 + 5*x)^3/(1 - 2*x)^(3/2),x]
Output:
(362 - 335*x - 100*x^2 - 25*x^3)/Sqrt[1 - 2*x]
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {53, 2009}
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.
\(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (-\frac {125}{8} (1-2 x)^{3/2}+\frac {825}{8} \sqrt {1-2 x}-\frac {1815}{8 \sqrt {1-2 x}}+\frac {1331}{8 (1-2 x)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {25}{8} (1-2 x)^{5/2}-\frac {275}{8} (1-2 x)^{3/2}+\frac {1815}{8} \sqrt {1-2 x}+\frac {1331}{8 \sqrt {1-2 x}}\) |
Input:
Int[(3 + 5*x)^3/(1 - 2*x)^(3/2),x]
Output:
1331/(8*Sqrt[1 - 2*x]) + (1815*Sqrt[1 - 2*x])/8 - (275*(1 - 2*x)^(3/2))/8 + (25*(1 - 2*x)^(5/2))/8
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45
method | result | size |
pseudoelliptic | \(\frac {-25 x^{3}-100 x^{2}-335 x +362}{\sqrt {1-2 x}}\) | \(24\) |
gosper | \(-\frac {25 x^{3}+100 x^{2}+335 x -362}{\sqrt {1-2 x}}\) | \(25\) |
risch | \(-\frac {25 x^{3}+100 x^{2}+335 x -362}{\sqrt {1-2 x}}\) | \(25\) |
orering | \(\frac {\left (-1+2 x \right ) \left (25 x^{3}+100 x^{2}+335 x -362\right )}{\left (1-2 x \right )^{\frac {3}{2}}}\) | \(29\) |
trager | \(\frac {\left (25 x^{3}+100 x^{2}+335 x -362\right ) \sqrt {1-2 x}}{-1+2 x}\) | \(31\) |
derivativedivides | \(\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815 \sqrt {1-2 x}}{8}-\frac {275 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{8}\) | \(38\) |
default | \(\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815 \sqrt {1-2 x}}{8}-\frac {275 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{8}\) | \(38\) |
meijerg | \(-\frac {27 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-135 \sqrt {\pi }+\frac {135 \sqrt {\pi }\, \left (-8 x +8\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {225 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}+\frac {-50 \sqrt {\pi }+\frac {25 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{64 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(122\) |
Input:
int((3+5*x)^3/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
(-25*x^3-100*x^2-335*x+362)/(1-2*x)^(1/2)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.57 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {{\left (25 \, x^{3} + 100 \, x^{2} + 335 \, x - 362\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \] Input:
integrate((3+5*x)^3/(1-2*x)^(3/2),x, algorithm="fricas")
Output:
(25*x^3 + 100*x^2 + 335*x - 362)*sqrt(-2*x + 1)/(2*x - 1)
Result contains complex when optimal does not.
Time = 1.07 (sec) , antiderivative size = 434, normalized size of antiderivative = 8.19 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\begin {cases} \frac {125 \sqrt {55} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {275 \sqrt {55} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {1210 \sqrt {55} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {26620 \sqrt {5} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {2662 \sqrt {55} i \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {29282 \sqrt {5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {125 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {275 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {1210 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {2662 \sqrt {55} \sqrt {5 - 10 x}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {26620 \sqrt {5} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {29282 \sqrt {5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} & \text {otherwise} \end {cases} \] Input:
integrate((3+5*x)**3/(1-2*x)**(3/2),x)
Output:
Piecewise((125*sqrt(55)*I*(x + 3/5)**3*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/ 5) - 55*sqrt(11)) + 275*sqrt(55)*I*(x + 3/5)**2*sqrt(10*x - 5)/(50*sqrt(11 )*(x + 3/5) - 55*sqrt(11)) + 1210*sqrt(55)*I*(x + 3/5)*sqrt(10*x - 5)/(50* sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 26620*sqrt(5)*(x + 3/5)/(50*sqrt(11)*( x + 3/5) - 55*sqrt(11)) - 2662*sqrt(55)*I*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 29282*sqrt(5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) , Abs(x + 3/5) > 11/10), (125*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)**3/(50*sqr t(11)*(x + 3/5) - 55*sqrt(11)) + 275*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)**2/ (50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 1210*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 2662*sqrt(55)*sqrt(5 - 10*x)/ (50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 26620*sqrt(5)*(x + 3/5)/(50*sqrt(1 1)*(x + 3/5) - 55*sqrt(11)) + 29282*sqrt(5)/(50*sqrt(11)*(x + 3/5) - 55*sq rt(11)), True))
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {25}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {275}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1815}{8} \, \sqrt {-2 \, x + 1} + \frac {1331}{8 \, \sqrt {-2 \, x + 1}} \] Input:
integrate((3+5*x)^3/(1-2*x)^(3/2),x, algorithm="maxima")
Output:
25/8*(-2*x + 1)^(5/2) - 275/8*(-2*x + 1)^(3/2) + 1815/8*sqrt(-2*x + 1) + 1 331/8/sqrt(-2*x + 1)
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {25}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {275}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1815}{8} \, \sqrt {-2 \, x + 1} + \frac {1331}{8 \, \sqrt {-2 \, x + 1}} \] Input:
integrate((3+5*x)^3/(1-2*x)^(3/2),x, algorithm="giac")
Output:
25/8*(2*x - 1)^2*sqrt(-2*x + 1) - 275/8*(-2*x + 1)^(3/2) + 1815/8*sqrt(-2* x + 1) + 1331/8/sqrt(-2*x + 1)
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {1331}{8\,\sqrt {1-2\,x}}+\frac {1815\,\sqrt {1-2\,x}}{8}-\frac {275\,{\left (1-2\,x\right )}^{3/2}}{8}+\frac {25\,{\left (1-2\,x\right )}^{5/2}}{8} \] Input:
int((5*x + 3)^3/(1 - 2*x)^(3/2),x)
Output:
1331/(8*(1 - 2*x)^(1/2)) + (1815*(1 - 2*x)^(1/2))/8 - (275*(1 - 2*x)^(3/2) )/8 + (25*(1 - 2*x)^(5/2))/8
Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {-25 x^{3}-100 x^{2}-335 x +362}{\sqrt {-2 x +1}} \] Input:
int((3+5*x)^3/(1-2*x)^(3/2),x)
Output:
( - 25*x**3 - 100*x**2 - 335*x + 362)/sqrt( - 2*x + 1)