Integrand size = 24, antiderivative size = 128 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac {258 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]
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Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {99, 158, 152, 52, 65, 212} \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {258 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625}-\frac {(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac {11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac {2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac {(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac {258 \sqrt {1-2 x}}{15625} \]
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Rule 52
Rule 65
Rule 99
Rule 152
Rule 158
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}+\frac {1}{5} \int \frac {(6-33 x) \sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx \\ & = \frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {1}{225} \int \frac {(-243-18 x) \sqrt {1-2 x} (2+3 x)^2}{3+5 x} \, dx \\ & = -\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}+\frac {\int \frac {\sqrt {1-2 x} (2+3 x) (17010+25641 x)}{3+5 x} \, dx}{7875} \\ & = -\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}+\frac {129 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{3125} \\ & = \frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}+\frac {1419 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15625} \\ & = \frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac {1419 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15625} \\ & = \frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac {258 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {5 \sqrt {1-2 x} \left (161312-143235 x-924335 x^2+157275 x^3+1395000 x^4+787500 x^5\right )+1806 \sqrt {55} (3+5 x) \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{546875 (3+5 x)} \]
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Time = 1.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {1575000 x^{6}+2002500 x^{5}-1080450 x^{4}-2005945 x^{3}+637865 x^{2}+465859 x -161312}{109375 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {258 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}\) | \(66\) |
pseudoelliptic | \(\frac {-1806 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-5 \sqrt {1-2 x}\, \left (787500 x^{5}+1395000 x^{4}+157275 x^{3}-924335 x^{2}-143235 x +161312\right )}{1640625+2734375 x}\) | \(67\) |
derivativedivides | \(-\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{100}+\frac {999 \left (1-2 x \right )^{\frac {7}{2}}}{1750}-\frac {12393 \left (1-2 x \right )^{\frac {5}{2}}}{12500}+\frac {8 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {52 \sqrt {1-2 x}}{3125}+\frac {22 \sqrt {1-2 x}}{78125 \left (-\frac {6}{5}-2 x \right )}-\frac {258 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}\) | \(81\) |
default | \(-\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{100}+\frac {999 \left (1-2 x \right )^{\frac {7}{2}}}{1750}-\frac {12393 \left (1-2 x \right )^{\frac {5}{2}}}{12500}+\frac {8 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {52 \sqrt {1-2 x}}{3125}+\frac {22 \sqrt {1-2 x}}{78125 \left (-\frac {6}{5}-2 x \right )}-\frac {258 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}\) | \(81\) |
trager | \(-\frac {\left (787500 x^{5}+1395000 x^{4}+157275 x^{3}-924335 x^{2}-143235 x +161312\right ) \sqrt {1-2 x}}{109375 \left (3+5 x \right )}-\frac {129 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{78125}\) | \(87\) |
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Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {903 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (787500 \, x^{5} + 1395000 \, x^{4} + 157275 \, x^{3} - 924335 \, x^{2} - 143235 \, x + 161312\right )} \sqrt {-2 \, x + 1}}{546875 \, {\left (5 \, x + 3\right )}} \]
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Time = 42.42 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.73 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=- \frac {9 \left (1 - 2 x\right )^{\frac {9}{2}}}{100} + \frac {999 \left (1 - 2 x\right )^{\frac {7}{2}}}{1750} - \frac {12393 \left (1 - 2 x\right )^{\frac {5}{2}}}{12500} + \frac {8 \left (1 - 2 x\right )^{\frac {3}{2}}}{3125} + \frac {52 \sqrt {1 - 2 x}}{3125} + \frac {128 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{78125} - \frac {484 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15625} \]
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Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {9}{100} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {999}{1750} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {12393}{12500} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {8}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {129}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {52}{3125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {9}{100} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {999}{1750} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {12393}{12500} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {8}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {129}{78125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {52}{3125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {52\,\sqrt {1-2\,x}}{3125}-\frac {22\,\sqrt {1-2\,x}}{78125\,\left (2\,x+\frac {6}{5}\right )}+\frac {8\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {12393\,{\left (1-2\,x\right )}^{5/2}}{12500}+\frac {999\,{\left (1-2\,x\right )}^{7/2}}{1750}-\frac {9\,{\left (1-2\,x\right )}^{9/2}}{100}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,258{}\mathrm {i}}{78125} \]
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