Integrand size = 24, antiderivative size = 101 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {2873}{73205 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}-\frac {2873}{39930 \sqrt {1-2 x} (3+5 x)}-\frac {2873 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641 \sqrt {55}} \]
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Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {91, 79, 44, 53, 65, 212} \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=-\frac {2873 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641 \sqrt {55}}+\frac {2873}{73205 \sqrt {1-2 x}}-\frac {2873}{39930 \sqrt {1-2 x} (5 x+3)}-\frac {614}{1815 \sqrt {1-2 x} (5 x+3)^2}+\frac {49}{66 (1-2 x)^{3/2} (5 x+3)^2} \]
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {1}{66} \int \frac {-313+297 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx \\ & = \frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}+\frac {2873 \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx}{3630} \\ & = \frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}-\frac {2873}{39930 \sqrt {1-2 x} (3+5 x)}+\frac {2873 \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx}{13310} \\ & = \frac {2873}{73205 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}-\frac {2873}{39930 \sqrt {1-2 x} (3+5 x)}+\frac {2873 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{29282} \\ & = \frac {2873}{73205 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}-\frac {2873}{39930 \sqrt {1-2 x} (3+5 x)}-\frac {2873 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{29282} \\ & = \frac {2873}{73205 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^2}-\frac {614}{1815 \sqrt {1-2 x} (3+5 x)^2}-\frac {2873}{39930 \sqrt {1-2 x} (3+5 x)}-\frac {2873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641 \sqrt {55}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {-\frac {55 \left (-47568-107127 x+57460 x^2+172380 x^3\right )}{2 (1-2 x)^{3/2} (3+5 x)^2}-8619 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{2415765} \]
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Time = 3.54 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.57
method | result | size |
risch | \(\frac {172380 x^{3}+57460 x^{2}-107127 x -47568}{87846 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {2873 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{805255}\) | \(58\) |
derivativedivides | \(\frac {\frac {65 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {145 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {2873 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{805255}+\frac {98}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {546}{14641 \sqrt {1-2 x}}\) | \(66\) |
default | \(\frac {\frac {65 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {145 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {2873 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{805255}+\frac {98}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {546}{14641 \sqrt {1-2 x}}\) | \(66\) |
pseudoelliptic | \(\frac {\frac {2873 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \left (3+5 x \right )^{2} \sqrt {55}}{805255}-\frac {28730 x^{3}}{14641}-\frac {28730 x^{2}}{43923}+\frac {35709 x}{29282}+\frac {7928}{14641}}{\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{2}}\) | \(69\) |
trager | \(-\frac {\left (172380 x^{3}+57460 x^{2}-107127 x -47568\right ) \sqrt {1-2 x}}{87846 \left (10 x^{2}+x -3\right )^{2}}+\frac {2873 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1610510}\) | \(80\) |
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Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.98 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {8619 \, \sqrt {55} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (172380 \, x^{3} + 57460 \, x^{2} - 107127 \, x - 47568\right )} \sqrt {-2 \, x + 1}}{4831530 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
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Time = 109.03 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.50 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {273 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{161051} - \frac {280 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{1331} + \frac {8 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{121} + \frac {546}{14641 \sqrt {1 - 2 x}} + \frac {98}{3993 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {2873}{1610510} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {43095 \, {\left (2 \, x - 1\right )}^{3} + 158015 \, {\left (2 \, x - 1\right )}^{2} + 159236 \, x - 210056}{43923 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {2873}{1610510} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {28 \, {\left (117 \, x - 97\right )}}{43923 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {5 \, {\left (13 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 29 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \]
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Time = 1.58 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=-\frac {2873\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{805255}-\frac {\frac {1316\,x}{9075}+\frac {2873\,{\left (2\,x-1\right )}^2}{19965}+\frac {2873\,{\left (2\,x-1\right )}^3}{73205}-\frac {1736}{9075}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}} \]
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