Integrand size = 25, antiderivative size = 53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=\frac {65}{8} \sqrt {3+2 x}-\frac {109}{24} (3+2 x)^{3/2}+\frac {47}{40} (3+2 x)^{5/2}-\frac {3}{56} (3+2 x)^{7/2} \]
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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} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {3}{56} (2 x+3)^{7/2}+\frac {47}{40} (2 x+3)^{5/2}-\frac {109}{24} (2 x+3)^{3/2}+\frac {65}{8} \sqrt {2 x+3} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {65}{8 \sqrt {3+2 x}}-\frac {109}{8} \sqrt {3+2 x}+\frac {47}{8} (3+2 x)^{3/2}-\frac {3}{8} (3+2 x)^{5/2}\right ) \, dx \\ & = \frac {65}{8} \sqrt {3+2 x}-\frac {109}{24} (3+2 x)^{3/2}+\frac {47}{40} (3+2 x)^{5/2}-\frac {3}{56} (3+2 x)^{7/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {1}{105} \sqrt {3+2 x} \left (-381-223 x-291 x^2+45 x^3\right ) \]
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Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45
method | result | size |
trager | \(\left (-\frac {3}{7} x^{3}+\frac {97}{35} x^{2}+\frac {223}{105} x +\frac {127}{35}\right ) \sqrt {3+2 x}\) | \(24\) |
gosper | \(-\frac {\left (45 x^{3}-291 x^{2}-223 x -381\right ) \sqrt {3+2 x}}{105}\) | \(25\) |
risch | \(-\frac {\left (45 x^{3}-291 x^{2}-223 x -381\right ) \sqrt {3+2 x}}{105}\) | \(25\) |
pseudoelliptic | \(-\frac {\left (45 x^{3}-291 x^{2}-223 x -381\right ) \sqrt {3+2 x}}{105}\) | \(25\) |
derivativedivides | \(-\frac {109 \left (3+2 x \right )^{\frac {3}{2}}}{24}+\frac {47 \left (3+2 x \right )^{\frac {5}{2}}}{40}-\frac {3 \left (3+2 x \right )^{\frac {7}{2}}}{56}+\frac {65 \sqrt {3+2 x}}{8}\) | \(38\) |
default | \(-\frac {109 \left (3+2 x \right )^{\frac {3}{2}}}{24}+\frac {47 \left (3+2 x \right )^{\frac {5}{2}}}{40}-\frac {3 \left (3+2 x \right )^{\frac {7}{2}}}{56}+\frac {65 \sqrt {3+2 x}}{8}\) | \(38\) |
meijerg | \(\frac {5 \sqrt {3}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1+\frac {2 x}{3}}\right )}{\sqrt {\pi }}+\frac {45 \sqrt {3}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{15}\right )}{4 \sqrt {\pi }}+\frac {69 \sqrt {3}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {8 x}{3}+8\right ) \sqrt {1+\frac {2 x}{3}}}{6}\right )}{4 \sqrt {\pi }}-\frac {81 \sqrt {3}\, \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{140}\right )}{16 \sqrt {\pi }}\) | \(136\) |
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {1}{105} \, {\left (45 \, x^{3} - 291 \, x^{2} - 223 \, x - 381\right )} \sqrt {2 \, x + 3} \]
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Time = 0.75 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=- \frac {3 \left (2 x + 3\right )^{\frac {7}{2}}}{56} + \frac {47 \left (2 x + 3\right )^{\frac {5}{2}}}{40} - \frac {109 \left (2 x + 3\right )^{\frac {3}{2}}}{24} + \frac {65 \sqrt {2 x + 3}}{8} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {3}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {47}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {109}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {65}{8} \, \sqrt {2 \, x + 3} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {3}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {47}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {109}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {65}{8} \, \sqrt {2 \, x + 3} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=\frac {65\,\sqrt {2\,x+3}}{8}-\frac {109\,{\left (2\,x+3\right )}^{3/2}}{24}+\frac {47\,{\left (2\,x+3\right )}^{5/2}}{40}-\frac {3\,{\left (2\,x+3\right )}^{7/2}}{56} \]
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