Integrand size = 24, antiderivative size = 77 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=-\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}-6 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {105, 162, 65, 212} \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=-6 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-\frac {5 \sqrt {1-2 x}}{11 (5 x+3)} \]
[In]
[Out]
Rule 65
Rule 105
Rule 162
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}-\frac {1}{11} \int \frac {23-15 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx \\ & = -\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}+9 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {160}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}-9 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {160}{11} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}-6 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=-\frac {5 \sqrt {1-2 x}}{33+55 x}-6 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
[In]
[Out]
Time = 1.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {64 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(54\) |
default | \(\frac {2 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {64 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(54\) |
risch | \(\frac {-\frac {5}{11}+\frac {10 x}{11}}{\left (3+5 x \right ) \sqrt {1-2 x}}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}+\frac {64 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\) | \(59\) |
pseudoelliptic | \(\frac {-726 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right ) \sqrt {21}+448 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-385 \sqrt {1-2 x}}{2541+4235 x}\) | \(65\) |
trager | \(-\frac {5 \sqrt {1-2 x}}{11 \left (3+5 x \right )}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{7}-\frac {32 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}\) | \(106\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=\frac {224 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 363 \, \sqrt {7} \sqrt {3} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 385 \, \sqrt {-2 \, x + 1}}{847 \, {\left (5 \, x + 3\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 5.67 (sec) , antiderivative size = 515, normalized size of antiderivative = 6.69 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=- \frac {140 \sqrt {55} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {4340 \sqrt {55} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {7260 \sqrt {21} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {2170 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {3630 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {154 \sqrt {55} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {4774 \sqrt {55} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {7986 \sqrt {21} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {2387 \sqrt {55} i \pi \sqrt {x - \frac {1}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {3993 \sqrt {21} i \pi \sqrt {x - \frac {1}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {770 \sqrt {2} i \left (x - \frac {1}{2}\right )}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=-\frac {32}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {3}{7} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {5 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=-\frac {32}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {3}{7} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {5 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=\frac {64\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {6\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {2\,\sqrt {1-2\,x}}{11\,\left (2\,x+\frac {6}{5}\right )} \]
[In]
[Out]