Integrand size = 24, antiderivative size = 80 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {136419 \sqrt {1-2 x}}{5000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {81}{280} (1-2 x)^{7/2}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \]
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Time = 0.02 (sec) , antiderivative size = 80, 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 = {90, 65, 212} \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}}+\frac {81}{280} (1-2 x)^{7/2}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {136419 \sqrt {1-2 x}}{5000} \]
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Rule 65
Rule 90
Rule 212
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {136419}{5000 \sqrt {1-2 x}}-\frac {34371 \sqrt {1-2 x}}{1000}+\frac {2889}{200} (1-2 x)^{3/2}-\frac {81}{40} (1-2 x)^{5/2}+\frac {1}{625 \sqrt {1-2 x} (3+5 x)}\right ) \, dx \\ & = -\frac {136419 \sqrt {1-2 x}}{5000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {81}{280} (1-2 x)^{7/2}+\frac {1}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {136419 \sqrt {1-2 x}}{5000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {81}{280} (1-2 x)^{7/2}-\frac {1}{625} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {136419 \sqrt {1-2 x}}{5000}+\frac {11457 (1-2 x)^{3/2}}{1000}-\frac {2889 (1-2 x)^{5/2}}{1000}+\frac {81}{280} (1-2 x)^{7/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {3 \sqrt {1-2 x} \left (26872+19095 x+11790 x^2+3375 x^3\right )}{4375}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \]
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Time = 1.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.55
method | result | size |
pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{34375}-\frac {3 \sqrt {1-2 x}\, \left (3375 x^{3}+11790 x^{2}+19095 x +26872\right )}{4375}\) | \(44\) |
risch | \(\frac {3 \left (3375 x^{3}+11790 x^{2}+19095 x +26872\right ) \left (-1+2 x \right )}{4375 \sqrt {1-2 x}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{34375}\) | \(49\) |
derivativedivides | \(\frac {11457 \left (1-2 x \right )^{\frac {3}{2}}}{1000}-\frac {2889 \left (1-2 x \right )^{\frac {5}{2}}}{1000}+\frac {81 \left (1-2 x \right )^{\frac {7}{2}}}{280}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{34375}-\frac {136419 \sqrt {1-2 x}}{5000}\) | \(56\) |
default | \(\frac {11457 \left (1-2 x \right )^{\frac {3}{2}}}{1000}-\frac {2889 \left (1-2 x \right )^{\frac {5}{2}}}{1000}+\frac {81 \left (1-2 x \right )^{\frac {7}{2}}}{280}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{34375}-\frac {136419 \sqrt {1-2 x}}{5000}\) | \(56\) |
trager | \(\left (-\frac {81}{35} x^{3}-\frac {7074}{875} x^{2}-\frac {11457}{875} x -\frac {80616}{4375}\right ) \sqrt {1-2 x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{34375}\) | \(69\) |
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Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {3}{4375} \, {\left (3375 \, x^{3} + 11790 \, x^{2} + 19095 \, x + 26872\right )} \sqrt {-2 \, x + 1} + \frac {1}{34375} \, \sqrt {55} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \]
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Time = 1.97 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.09 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx=\frac {81 \left (1 - 2 x\right )^{\frac {7}{2}}}{280} - \frac {2889 \left (1 - 2 x\right )^{\frac {5}{2}}}{1000} + \frac {11457 \left (1 - 2 x\right )^{\frac {3}{2}}}{1000} - \frac {136419 \sqrt {1 - 2 x}}{5000} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{34375} \]
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Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx=\frac {81}{280} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {2889}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {11457}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{34375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {136419}{5000} \, \sqrt {-2 \, x + 1} \]
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Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.12 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {81}{280} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {2889}{1000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {11457}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{34375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {136419}{5000} \, \sqrt {-2 \, x + 1} \]
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Time = 1.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx=\frac {11457\,{\left (1-2\,x\right )}^{3/2}}{1000}-\frac {136419\,\sqrt {1-2\,x}}{5000}-\frac {2889\,{\left (1-2\,x\right )}^{5/2}}{1000}+\frac {81\,{\left (1-2\,x\right )}^{7/2}}{280}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{34375} \]
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