Integrand size = 24, antiderivative size = 88 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {49}{22 \sqrt {1-2 x} (3+5 x)^2}-\frac {613 \sqrt {1-2 x}}{605 (3+5 x)^2}-\frac {2589 \sqrt {1-2 x}}{13310 (3+5 x)}-\frac {2589 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6655 \sqrt {55}} \]
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Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 44, 65, 212} \[ \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=-\frac {2589 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6655 \sqrt {55}}-\frac {2589 \sqrt {1-2 x}}{13310 (5 x+3)}-\frac {613 \sqrt {1-2 x}}{605 (5 x+3)^2}+\frac {49}{22 \sqrt {1-2 x} (5 x+3)^2} \]
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Rule 44
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {49}{22 \sqrt {1-2 x} (3+5 x)^2}-\frac {1}{22} \int \frac {-431+99 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx \\ & = \frac {49}{22 \sqrt {1-2 x} (3+5 x)^2}-\frac {613 \sqrt {1-2 x}}{605 (3+5 x)^2}+\frac {2589 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx}{1210} \\ & = \frac {49}{22 \sqrt {1-2 x} (3+5 x)^2}-\frac {613 \sqrt {1-2 x}}{605 (3+5 x)^2}-\frac {2589 \sqrt {1-2 x}}{13310 (3+5 x)}+\frac {2589 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{13310} \\ & = \frac {49}{22 \sqrt {1-2 x} (3+5 x)^2}-\frac {613 \sqrt {1-2 x}}{605 (3+5 x)^2}-\frac {2589 \sqrt {1-2 x}}{13310 (3+5 x)}-\frac {2589 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{13310} \\ & = \frac {49}{22 \sqrt {1-2 x} (3+5 x)^2}-\frac {613 \sqrt {1-2 x}}{605 (3+5 x)^2}-\frac {2589 \sqrt {1-2 x}}{13310 (3+5 x)}-\frac {2589 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6655 \sqrt {55}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {\frac {55 \left (8392+29561 x+25890 x^2\right )}{\sqrt {1-2 x} (3+5 x)^2}-5178 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{732050} \]
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Time = 1.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {25890 x^{2}+29561 x +8392}{13310 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {2589 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{366025}\) | \(46\) |
pseudoelliptic | \(-\frac {2589 \left (\sqrt {55}\, \left (x +\frac {3}{5}\right )^{2} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )-11 x^{2}-\frac {325171 x}{25890}-\frac {46156}{12945}\right )}{14641 \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}\) | \(56\) |
derivativedivides | \(\frac {\frac {139 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {141 \sqrt {1-2 x}}{605}}{\left (-6-10 x \right )^{2}}-\frac {2589 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{366025}+\frac {98}{1331 \sqrt {1-2 x}}\) | \(57\) |
default | \(\frac {\frac {139 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {141 \sqrt {1-2 x}}{605}}{\left (-6-10 x \right )^{2}}-\frac {2589 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{366025}+\frac {98}{1331 \sqrt {1-2 x}}\) | \(57\) |
trager | \(-\frac {\left (25890 x^{2}+29561 x +8392\right ) \sqrt {1-2 x}}{13310 \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {2589 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{732050}\) | \(79\) |
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Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {2589 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (25890 \, x^{2} + 29561 \, x + 8392\right )} \sqrt {-2 \, x + 1}}{732050 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
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Time = 107.70 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.89 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {49 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{14641} - \frac {272 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{605} + \frac {8 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{55} + \frac {98}{1331 \sqrt {1 - 2 x}} \]
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Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {2589}{732050} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {12945 \, {\left (2 \, x - 1\right )}^{2} + 110902 \, x + 3839}{6655 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {2589}{732050} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {98}{1331 \, \sqrt {-2 \, x + 1}} + \frac {695 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1551 \, \sqrt {-2 \, x + 1}}{26620 \, {\left (5 \, x + 3\right )}^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {\frac {10082\,x}{15125}+\frac {2589\,{\left (2\,x-1\right )}^2}{33275}+\frac {349}{15125}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}-\frac {2589\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{366025} \]
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