Integrand size = 24, antiderivative size = 67 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \]
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Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {89, 65, 212} \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}}-\frac {27}{20} \sqrt {1-2 x}-\frac {784}{121 \sqrt {1-2 x}}+\frac {343}{132 (1-2 x)^{3/2}} \]
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Rule 65
Rule 89
Rule 212
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {343}{44 (1-2 x)^{5/2}}-\frac {784}{121 (1-2 x)^{3/2}}+\frac {27}{20 \sqrt {1-2 x}}+\frac {1}{605 \sqrt {1-2 x} (3+5 x)}\right ) \, dx \\ & = \frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}+\frac {1}{605} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}-\frac {1}{605} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {-\frac {55 \left (9494-33321 x+9801 x^2\right )}{(1-2 x)^{3/2}}-6 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{99825} \]
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Time = 1.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {343}{132 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{33275}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27 \sqrt {1-2 x}}{20}\) | \(47\) |
default | \(\frac {343}{132 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{33275}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27 \sqrt {1-2 x}}{20}\) | \(47\) |
pseudoelliptic | \(\frac {-\frac {9494}{1815}+\frac {2 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \sqrt {55}}{33275}-\frac {27 x^{2}}{5}+\frac {11107 x}{605}}{\left (1-2 x \right )^{\frac {3}{2}}}\) | \(49\) |
trager | \(-\frac {\left (9801 x^{2}-33321 x +9494\right ) \sqrt {1-2 x}}{1815 \left (-1+2 x \right )^{2}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{33275}\) | \(72\) |
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Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {3 \, \sqrt {55} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (9801 \, x^{2} - 33321 \, x + 9494\right )} \sqrt {-2 \, x + 1}}{99825 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
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Time = 2.52 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=- \frac {27 \sqrt {1 - 2 x}}{20} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{33275} - \frac {784}{121 \sqrt {1 - 2 x}} + \frac {343}{132 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {1}{33275} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {27}{20} \, \sqrt {-2 \, x + 1} + \frac {49 \, {\left (384 \, x - 115\right )}}{1452 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {1}{33275} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {27}{20} \, \sqrt {-2 \, x + 1} - \frac {49 \, {\left (384 \, x - 115\right )}}{1452 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]
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Time = 1.35 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.61 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {\frac {1568\,x}{121}-\frac {5635}{1452}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{33275}-\frac {27\,\sqrt {1-2\,x}}{20} \]
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