Integrand size = 20, antiderivative size = 40 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {77}{4} \sqrt {1-2 x}+\frac {17}{3} (1-2 x)^{3/2}-\frac {3}{4} (1-2 x)^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {3}{4} (1-2 x)^{5/2}+\frac {17}{3} (1-2 x)^{3/2}-\frac {77}{4} \sqrt {1-2 x} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {77}{4 \sqrt {1-2 x}}-17 \sqrt {1-2 x}+\frac {15}{4} (1-2 x)^{3/2}\right ) \, dx \\ & = -\frac {77}{4} \sqrt {1-2 x}+\frac {17}{3} (1-2 x)^{3/2}-\frac {3}{4} (1-2 x)^{5/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{3} \sqrt {1-2 x} \left (43+25 x+9 x^2\right ) \]
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Time = 1.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48
method | result | size |
trager | \(\left (-3 x^{2}-\frac {25}{3} x -\frac {43}{3}\right ) \sqrt {1-2 x}\) | \(19\) |
gosper | \(-\frac {\sqrt {1-2 x}\, \left (9 x^{2}+25 x +43\right )}{3}\) | \(20\) |
pseudoelliptic | \(-\frac {\sqrt {1-2 x}\, \left (9 x^{2}+25 x +43\right )}{3}\) | \(20\) |
risch | \(\frac {\left (-1+2 x \right ) \left (9 x^{2}+25 x +43\right )}{3 \sqrt {1-2 x}}\) | \(25\) |
derivativedivides | \(\frac {17 \left (1-2 x \right )^{\frac {3}{2}}}{3}-\frac {3 \left (1-2 x \right )^{\frac {5}{2}}}{4}-\frac {77 \sqrt {1-2 x}}{4}\) | \(29\) |
default | \(\frac {17 \left (1-2 x \right )^{\frac {3}{2}}}{3}-\frac {3 \left (1-2 x \right )^{\frac {5}{2}}}{4}-\frac {77 \sqrt {1-2 x}}{4}\) | \(29\) |
meijerg | \(-\frac {3 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {\frac {19 \sqrt {\pi }}{3}-\frac {19 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{24}}{\sqrt {\pi }}-\frac {15 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}\) | \(86\) |
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Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {1}{3} \, {\left (9 \, x^{2} + 25 \, x + 43\right )} \sqrt {-2 \, x + 1} \]
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Time = 0.57 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=- \frac {3 \left (1 - 2 x\right )^{\frac {5}{2}}}{4} + \frac {17 \left (1 - 2 x\right )^{\frac {3}{2}}}{3} - \frac {77 \sqrt {1 - 2 x}}{4} \]
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Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {3}{4} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {17}{3} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {77}{4} \, \sqrt {-2 \, x + 1} \]
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Time = 0.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {3}{4} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {17}{3} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {77}{4} \, \sqrt {-2 \, x + 1} \]
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Time = 1.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int \frac {(2+3 x) (3+5 x)}{\sqrt {1-2 x}} \, dx=-\frac {\sqrt {1-2\,x}\,\left (136\,x+9\,{\left (2\,x-1\right )}^2+163\right )}{12} \]
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