Integrand size = 25, antiderivative size = 53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {65}{8 \sqrt {3+2 x}}-\frac {109}{8} \sqrt {3+2 x}+\frac {47}{24} (3+2 x)^{3/2}-\frac {3}{40} (3+2 x)^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {3}{40} (2 x+3)^{5/2}+\frac {47}{24} (2 x+3)^{3/2}-\frac {109}{8} \sqrt {2 x+3}-\frac {65}{8 \sqrt {2 x+3}} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {65}{8 (3+2 x)^{3/2}}-\frac {109}{8 \sqrt {3+2 x}}+\frac {47}{8} \sqrt {3+2 x}-\frac {3}{8} (3+2 x)^{3/2}\right ) \, dx \\ & = -\frac {65}{8 \sqrt {3+2 x}}-\frac {109}{8} \sqrt {3+2 x}+\frac {47}{24} (3+2 x)^{3/2}-\frac {3}{40} (3+2 x)^{5/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {501+117 x-77 x^2+9 x^3}{15 \sqrt {3+2 x}} \]
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Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.47
method | result | size |
gosper | \(-\frac {9 x^{3}-77 x^{2}+117 x +501}{15 \sqrt {3+2 x}}\) | \(25\) |
trager | \(-\frac {9 x^{3}-77 x^{2}+117 x +501}{15 \sqrt {3+2 x}}\) | \(25\) |
risch | \(-\frac {9 x^{3}-77 x^{2}+117 x +501}{15 \sqrt {3+2 x}}\) | \(25\) |
pseudoelliptic | \(\frac {-9 x^{3}+77 x^{2}-117 x -501}{15 \sqrt {3+2 x}}\) | \(25\) |
derivativedivides | \(\frac {47 \left (3+2 x \right )^{\frac {3}{2}}}{24}-\frac {3 \left (3+2 x \right )^{\frac {5}{2}}}{40}-\frac {65}{8 \sqrt {3+2 x}}-\frac {109 \sqrt {3+2 x}}{8}\) | \(38\) |
default | \(\frac {47 \left (3+2 x \right )^{\frac {3}{2}}}{24}-\frac {3 \left (3+2 x \right )^{\frac {5}{2}}}{40}-\frac {65}{8 \sqrt {3+2 x}}-\frac {109 \sqrt {3+2 x}}{8}\) | \(38\) |
meijerg | \(\frac {10 \sqrt {3}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1+\frac {2 x}{3}}}\right )}{3 \sqrt {\pi }}+\frac {15 \sqrt {3}\, \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {8}{9} x^{2}+\frac {16}{3} x +16\right )}{6 \sqrt {1+\frac {2 x}{3}}}\right )}{2 \sqrt {\pi }}+\frac {23 \sqrt {3}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (\frac {8 x}{3}+8\right )}{4 \sqrt {1+\frac {2 x}{3}}}\right )}{2 \sqrt {\pi }}-\frac {27 \sqrt {3}\, \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (\frac {64}{27} x^{3}-\frac {64}{9} x^{2}+\frac {128}{3} x +128\right )}{40 \sqrt {1+\frac {2 x}{3}}}\right )}{8 \sqrt {\pi }}\) | \(134\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {9 \, x^{3} - 77 \, x^{2} + 117 \, x + 501}{15 \, \sqrt {2 \, x + 3}} \]
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Time = 0.91 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=- \frac {3 \left (2 x + 3\right )^{\frac {5}{2}}}{40} + \frac {47 \left (2 x + 3\right )^{\frac {3}{2}}}{24} - \frac {109 \sqrt {2 x + 3}}{8} - \frac {65}{8 \sqrt {2 x + 3}} \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {3}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {47}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {109}{8} \, \sqrt {2 \, x + 3} - \frac {65}{8 \, \sqrt {2 \, x + 3}} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {3}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {47}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {109}{8} \, \sqrt {2 \, x + 3} - \frac {65}{8 \, \sqrt {2 \, x + 3}} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=\frac {47\,{\left (2\,x+3\right )}^{3/2}}{24}-\frac {109\,\sqrt {2\,x+3}}{8}-\frac {65}{8\,\sqrt {2\,x+3}}-\frac {3\,{\left (2\,x+3\right )}^{5/2}}{40} \]
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