Integrand size = 22, antiderivative size = 53 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {539}{8} \sqrt {1-2 x}+\frac {707}{24} (1-2 x)^{3/2}-\frac {309}{40} (1-2 x)^{5/2}+\frac {45}{56} (1-2 x)^{7/2} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=\frac {45}{56} (1-2 x)^{7/2}-\frac {309}{40} (1-2 x)^{5/2}+\frac {707}{24} (1-2 x)^{3/2}-\frac {539}{8} \sqrt {1-2 x} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {539}{8 \sqrt {1-2 x}}-\frac {707}{8} \sqrt {1-2 x}+\frac {309}{8} (1-2 x)^{3/2}-\frac {45}{8} (1-2 x)^{5/2}\right ) \, dx \\ & = -\frac {539}{8} \sqrt {1-2 x}+\frac {707}{24} (1-2 x)^{3/2}-\frac {309}{40} (1-2 x)^{5/2}+\frac {45}{56} (1-2 x)^{7/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.53 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{105} \sqrt {1-2 x} \left (4708+3448 x+2232 x^2+675 x^3\right ) \]
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Time = 1.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45
method | result | size |
trager | \(\left (-\frac {45}{7} x^{3}-\frac {744}{35} x^{2}-\frac {3448}{105} x -\frac {4708}{105}\right ) \sqrt {1-2 x}\) | \(24\) |
gosper | \(-\frac {\sqrt {1-2 x}\, \left (675 x^{3}+2232 x^{2}+3448 x +4708\right )}{105}\) | \(25\) |
pseudoelliptic | \(-\frac {\sqrt {1-2 x}\, \left (675 x^{3}+2232 x^{2}+3448 x +4708\right )}{105}\) | \(25\) |
risch | \(\frac {\left (-1+2 x \right ) \left (675 x^{3}+2232 x^{2}+3448 x +4708\right )}{105 \sqrt {1-2 x}}\) | \(30\) |
derivativedivides | \(\frac {707 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {309 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {45 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {539 \sqrt {1-2 x}}{8}\) | \(38\) |
default | \(\frac {707 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {309 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {45 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {539 \sqrt {1-2 x}}{8}\) | \(38\) |
meijerg | \(-\frac {6 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {\frac {56 \sqrt {\pi }}{3}-\frac {7 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{3}}{\sqrt {\pi }}-\frac {87 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {18 \sqrt {\pi }}{7}-\frac {9 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{448}}{\sqrt {\pi }}\) | \(124\) |
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Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{105} \, {\left (675 \, x^{3} + 2232 \, x^{2} + 3448 \, x + 4708\right )} \sqrt {-2 \, x + 1} \]
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Time = 0.62 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=\frac {45 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {309 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} + \frac {707 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} - \frac {539 \sqrt {1 - 2 x}}{8} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=\frac {45}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {309}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {707}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {539}{8} \, \sqrt {-2 \, x + 1} \]
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {45}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {309}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {707}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {539}{8} \, \sqrt {-2 \, x + 1} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^2 (3+5 x)}{\sqrt {1-2 x}} \, dx=\frac {707\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {539\,\sqrt {1-2\,x}}{8}-\frac {309\,{\left (1-2\,x\right )}^{5/2}}{40}+\frac {45\,{\left (1-2\,x\right )}^{7/2}}{56} \]
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