Integrand size = 19, antiderivative size = 74 \[ \int \frac {(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}+\frac {4}{25} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {49, 56, 222} \[ \int \frac {(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\frac {4}{25} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {5 x+3}} \]
[In]
[Out]
Rule 49
Rule 56
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {2}{5} \int \frac {\sqrt {1-2 x}}{(3+5 x)^{3/2}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}+\frac {4}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}+\frac {8 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25 \sqrt {5}} \\ & = -\frac {2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}+\frac {4}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\frac {2}{375} \left (\frac {5 \sqrt {1-2 x} (13+40 x)}{(3+5 x)^{3/2}}+6 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right )\right ) \]
[In]
[Out]
\[\int \frac {\left (1-2 x \right )^{\frac {3}{2}}}{\left (3+5 x \right )^{\frac {5}{2}}}d x\]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \frac {(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {5} \sqrt {2} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 5 \, {\left (40 \, x + 13\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}\right )}}{375 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.86 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.76 \[ \int \frac {(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\begin {cases} \frac {16 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{375} - \frac {22 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{1875 \left (x + \frac {3}{5}\right )} + \frac {2 \sqrt {10} i \log {\left (\frac {1}{x + \frac {3}{5}} \right )}}{125} + \frac {2 \sqrt {10} i \log {\left (x + \frac {3}{5} \right )}}{125} + \frac {4 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{125} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\\frac {16 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{375} - \frac {22 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{1875 \left (x + \frac {3}{5}\right )} + \frac {2 \sqrt {10} i \log {\left (\frac {1}{x + \frac {3}{5}} \right )}}{125} - \frac {4 \sqrt {10} i \log {\left (\sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} + 1 \right )}}{125} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.26 \[ \int \frac {(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\frac {2}{125} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{15 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} - \frac {11 \, \sqrt {-10 \, x^{2} - x + 3}}{75 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {14 \, \sqrt {-10 \, x^{2} - x + 3}}{75 \, {\left (5 \, x + 3\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (51) = 102\).
Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.89 \[ \int \frac {(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{6000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {4}{125} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{100 \, \sqrt {5 \, x + 3}} - \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {15 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
[In]
[Out]