Integrand size = 22, antiderivative size = 66 \[ \int \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {3773}{16} \sqrt {1-2 x}+\frac {3283}{24} (1-2 x)^{3/2}-\frac {1071}{20} (1-2 x)^{5/2}+\frac {621}{56} (1-2 x)^{7/2}-\frac {15}{16} (1-2 x)^{9/2} \]
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Time = 0.01 (sec) , antiderivative size = 66, 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} \[ \int \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {15}{16} (1-2 x)^{9/2}+\frac {621}{56} (1-2 x)^{7/2}-\frac {1071}{20} (1-2 x)^{5/2}+\frac {3283}{24} (1-2 x)^{3/2}-\frac {3773}{16} \sqrt {1-2 x} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3773}{16 \sqrt {1-2 x}}-\frac {3283}{8} \sqrt {1-2 x}+\frac {1071}{4} (1-2 x)^{3/2}-\frac {621}{8} (1-2 x)^{5/2}+\frac {135}{16} (1-2 x)^{7/2}\right ) \, dx \\ & = -\frac {3773}{16} \sqrt {1-2 x}+\frac {3283}{24} (1-2 x)^{3/2}-\frac {1071}{20} (1-2 x)^{5/2}+\frac {621}{56} (1-2 x)^{7/2}-\frac {15}{16} (1-2 x)^{9/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.50 \[ \int \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{105} \sqrt {1-2 x} \left (14954+12434 x+10881 x^2+6165 x^3+1575 x^4\right ) \]
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Time = 1.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.44
method | result | size |
trager | \(\left (-15 x^{4}-\frac {411}{7} x^{3}-\frac {3627}{35} x^{2}-\frac {12434}{105} x -\frac {14954}{105}\right ) \sqrt {1-2 x}\) | \(29\) |
gosper | \(-\frac {\sqrt {1-2 x}\, \left (1575 x^{4}+6165 x^{3}+10881 x^{2}+12434 x +14954\right )}{105}\) | \(30\) |
pseudoelliptic | \(-\frac {\sqrt {1-2 x}\, \left (1575 x^{4}+6165 x^{3}+10881 x^{2}+12434 x +14954\right )}{105}\) | \(30\) |
risch | \(\frac {\left (-1+2 x \right ) \left (1575 x^{4}+6165 x^{3}+10881 x^{2}+12434 x +14954\right )}{105 \sqrt {1-2 x}}\) | \(35\) |
derivativedivides | \(\frac {3283 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {1071 \left (1-2 x \right )^{\frac {5}{2}}}{20}+\frac {621 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {15 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {3773 \sqrt {1-2 x}}{16}\) | \(47\) |
default | \(\frac {3283 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {1071 \left (1-2 x \right )^{\frac {5}{2}}}{20}+\frac {621 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {15 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {3773 \sqrt {1-2 x}}{16}\) | \(47\) |
meijerg | \(-\frac {12 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {\frac {148 \sqrt {\pi }}{3}-\frac {37 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{6}}{\sqrt {\pi }}-\frac {171 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{4 \sqrt {\pi }}+\frac {\frac {702 \sqrt {\pi }}{35}-\frac {351 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{2240}}{\sqrt {\pi }}-\frac {135 \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{315}\right )}{32 \sqrt {\pi }}\) | \(167\) |
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Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.44 \[ \int \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{105} \, {\left (1575 \, x^{4} + 6165 \, x^{3} + 10881 \, x^{2} + 12434 \, x + 14954\right )} \sqrt {-2 \, x + 1} \]
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Time = 0.65 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx=- \frac {15 \left (1 - 2 x\right )^{\frac {9}{2}}}{16} + \frac {621 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {1071 \left (1 - 2 x\right )^{\frac {5}{2}}}{20} + \frac {3283 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} - \frac {3773 \sqrt {1 - 2 x}}{16} \]
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Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {15}{16} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {621}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1071}{20} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {3283}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {3773}{16} \, \sqrt {-2 \, x + 1} \]
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Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {15}{16} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {621}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1071}{20} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {3283}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {3773}{16} \, \sqrt {-2 \, x + 1} \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^3 (3+5 x)}{\sqrt {1-2 x}} \, dx=\frac {3283\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {3773\,\sqrt {1-2\,x}}{16}-\frac {1071\,{\left (1-2\,x\right )}^{5/2}}{20}+\frac {621\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {15\,{\left (1-2\,x\right )}^{9/2}}{16} \]
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