Integrand size = 19, antiderivative size = 72 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {33}{16} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{4} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {363 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{16 \sqrt {10}} \]
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Time = 0.01 (sec) , antiderivative size = 72, 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 = {52, 56, 222} \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=\frac {363 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{16 \sqrt {10}}-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {33}{16} \sqrt {1-2 x} \sqrt {5 x+3} \]
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Rule 52
Rule 56
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {33}{8} \int \frac {\sqrt {3+5 x}}{\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}{32} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 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 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{16 \sqrt {5}} \\ & = -\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 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{16 \sqrt {10}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {5}{16} \sqrt {1-2 x} (9+4 x) \sqrt {3+5 x}+\frac {363 \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right )}{16 \sqrt {10}} \]
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Time = 1.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{4}-\frac {33 \sqrt {1-2 x}\, \sqrt {3+5 x}}{16}+\frac {363 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{320 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(72\) |
risch | \(\frac {5 \left (9+4 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{16 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {363 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{320 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(93\) |
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Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {5}{16} \, \sqrt {5 \, x + 3} {\left (4 \, x + 9\right )} \sqrt {-2 \, x + 1} - \frac {363}{320} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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Result contains complex when optimal does not.
Time = 2.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.57 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=\begin {cases} - \frac {25 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{2 \sqrt {10 x - 5}} - \frac {55 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{8 \sqrt {10 x - 5}} + \frac {363 i \sqrt {x + \frac {3}{5}}}{16 \sqrt {10 x - 5}} - \frac {363 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{160} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {363 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{160} + \frac {25 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{2 \sqrt {5 - 10 x}} + \frac {55 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{8 \sqrt {5 - 10 x}} - \frac {363 \sqrt {x + \frac {3}{5}}}{16 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.57 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {5}{4} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {363}{320} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {45}{16} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.62 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{160} \, \sqrt {5} {\left (10 \, \sqrt {5 \, x + 3} {\left (4 \, x + 9\right )} \sqrt {-10 \, x + 5} - 363 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
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Timed out. \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}} \,d x \]
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