Integrand size = 24, antiderivative size = 45 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {2}{55 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {74 \sqrt {3+5 x}}{605 \sqrt {1-2 x}} \]
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Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {79, 37} \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {74 \sqrt {5 x+3}}{605 \sqrt {1-2 x}}-\frac {2}{55 \sqrt {1-2 x} \sqrt {5 x+3}} \]
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Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{55 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {37}{55} \int \frac {1}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx \\ & = -\frac {2}{55 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {74 \sqrt {3+5 x}}{605 \sqrt {1-2 x}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.60 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {2 (20+37 x)}{121 \sqrt {1-2 x} \sqrt {3+5 x}} \]
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Time = 1.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.49
method | result | size |
gosper | \(\frac {\frac {74 x}{121}+\frac {40}{121}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(22\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (37 x +20\right )}{121 \sqrt {3+5 x}\, \left (-1+2 x \right )}\) | \(29\) |
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Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.69 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{121 \, {\left (10 \, x^{2} + x - 3\right )}} \]
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\[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {3 x + 2}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.67 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {74 \, x}{121 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {40}{121 \, \sqrt {-10 \, x^{2} - x + 3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (33) = 66\).
Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.93 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1210 \, \sqrt {5 \, x + 3}} - \frac {14 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{605 \, {\left (2 \, x - 1\right )}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{605 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
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Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {\sqrt {5\,x+3}\,\left (\frac {74\,x}{605}+\frac {8}{121}\right )}{x\,\sqrt {1-2\,x}+\frac {3\,\sqrt {1-2\,x}}{5}} \]
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