Integrand size = 26, antiderivative size = 138 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{80000 \sqrt {10}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {91, 81, 52, 56, 222} \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\frac {236313 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80000 \sqrt {10}}-\frac {9}{200} \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {5 x+3}}+\frac {651 \sqrt {5 x+3} (1-2 x)^{5/2}}{22000}+\frac {651 \sqrt {5 x+3} (1-2 x)^{3/2}}{8000}+\frac {21483 \sqrt {5 x+3} \sqrt {1-2 x}}{80000} \]
[In]
[Out]
Rule 52
Rule 56
Rule 81
Rule 91
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {2}{275} \int \frac {(1-2 x)^{5/2} \left (\frac {351}{2}+\frac {495 x}{2}\right )}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {1953 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{4400} \\ & = -\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {651}{800} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {21483 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{16000} \\ & = -\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{160000} \\ & = -\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{80000 \sqrt {5}} \\ & = -\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{80000 \sqrt {10}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\frac {10 \sqrt {1-2 x} \left (79699+134625 x-112620 x^2-77600 x^3+144000 x^4\right )-236313 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{800000 \sqrt {3+5 x}} \]
[In]
[Out]
Time = 1.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\left (2880000 x^{4} \sqrt {-10 x^{2}-x +3}-1552000 x^{3} \sqrt {-10 x^{2}-x +3}+1181565 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -2252400 x^{2} \sqrt {-10 x^{2}-x +3}+708939 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2692500 x \sqrt {-10 x^{2}-x +3}+1593980 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{1600000 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(133\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=-\frac {236313 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (144000 \, x^{4} - 77600 \, x^{3} - 112620 \, x^{2} + 134625 \, x + 79699\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1600000 \, {\left (5 \, x + 3\right )}} \]
[In]
[Out]
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2}}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=-\frac {18 \, x^{5}}{5 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {187 \, x^{4}}{50 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3691 \, x^{3}}{2000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {38187 \, x^{2}}{8000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {236313}{1600000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {24773 \, x}{80000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {79699}{80000 \, \sqrt {-10 \, x^{2} - x + 3}} \]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\frac {1}{2000000} \, {\left (4 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 529 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 16905 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 61545 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {236313}{800000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {121 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{31250 \, \sqrt {5 \, x + 3}} + \frac {242 \, \sqrt {10} \sqrt {5 \, x + 3}}{15625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
[In]
[Out]