Integrand size = 39, antiderivative size = 232 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=-x-\frac {8 \text {arctanh}\left (\frac {-\frac {39}{\sqrt {217}}-\frac {52 x}{\sqrt {217}}+\frac {8 \sqrt {-3+5 x+12 x^2}}{\sqrt {217}}}{3+4 x}\right )}{3 \sqrt {217}}-\frac {4}{3} \text {RootSum}\left [144-156 \text {$\#$1}+145 \text {$\#$1}^2+13 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {12 \log (3+4 x)-12 \log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right )+13 \log (3+4 x) \text {$\#$1}-13 \log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right ) \text {$\#$1}-\log (3+4 x) \text {$\#$1}^2+\log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right ) \text {$\#$1}^2}{-156+290 \text {$\#$1}+39 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=\int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-1+\frac {2}{1+((-1+3 x) (3+4 x))^{3/2}}\right ) \, dx \\ & = -x+2 \int \frac {1}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx \\ & = -x+2 \int \left (-\frac {1}{3 \left (-4+5 x+12 x^2\right )}+\frac {\sqrt {-3+5 x+12 x^2}}{3 \left (-4+5 x+12 x^2\right )}+\frac {4 \sqrt {-3+5 x+12 x^2}}{3 \left (7-25 x-35 x^2+120 x^3+144 x^4\right )}-\frac {5 x \sqrt {-3+5 x+12 x^2}}{3 \left (7-25 x-35 x^2+120 x^3+144 x^4\right )}-\frac {4 x^2 \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4}+\frac {-1+5 x+12 x^2}{3 \left (7-25 x-35 x^2+120 x^3+144 x^4\right )}\right ) \, dx \\ & = -x-\frac {2}{3} \int \frac {1}{-4+5 x+12 x^2} \, dx+\frac {2}{3} \int \frac {\sqrt {-3+5 x+12 x^2}}{-4+5 x+12 x^2} \, dx+\frac {2}{3} \int \frac {-1+5 x+12 x^2}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx+\frac {8}{3} \int \frac {\sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-\frac {10}{3} \int \frac {x \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-8 \int \frac {x^2 \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx \\ & = -x-\frac {1}{18} \int -\frac {12}{\left (-4+5 x+12 x^2\right ) \sqrt {-3+5 x+12 x^2}} \, dx+\frac {2}{3} \int \frac {1}{\sqrt {-3+5 x+12 x^2}} \, dx+\frac {2}{3} \text {Subst}\left (\int \frac {48 \left (-73+576 x^2\right )}{22753-167040 x^2+331776 x^4} \, dx,x,\frac {5}{24}+x\right )+\frac {4}{3} \text {Subst}\left (\int \frac {1}{217-x^2} \, dx,x,5+24 x\right )+\frac {8}{3} \int \frac {\sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-\frac {10}{3} \int \frac {x \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-8 \int \frac {x^2 \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx \\ & = -x+\frac {4 \text {arctanh}\left (\frac {5+24 x}{\sqrt {217}}\right )}{3 \sqrt {217}}+\frac {2}{3} \int \frac {1}{\left (-4+5 x+12 x^2\right ) \sqrt {-3+5 x+12 x^2}} \, dx+\frac {4}{3} \text {Subst}\left (\int \frac {1}{48-x^2} \, dx,x,\frac {5+24 x}{\sqrt {-3+5 x+12 x^2}}\right )+\frac {8}{3} \int \frac {\sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-\frac {10}{3} \int \frac {x \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-8 \int \frac {x^2 \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx+32 \text {Subst}\left (\int \frac {-73+576 x^2}{22753-167040 x^2+331776 x^4} \, dx,x,\frac {5}{24}+x\right ) \\ & = -x+\frac {4 \text {arctanh}\left (\frac {5+24 x}{\sqrt {217}}\right )}{3 \sqrt {217}}+\frac {\text {arctanh}\left (\frac {5+24 x}{4 \sqrt {3} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {3}}+\frac {8}{3} \int \frac {\sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-\frac {10}{3} \int \frac {x \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-\frac {20}{3} \text {Subst}\left (\int \frac {1}{1085-5 x^2} \, dx,x,\frac {5+24 x}{\sqrt {-3+5 x+12 x^2}}\right )-8 \int \frac {x^2 \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx+\frac {1}{3} \sqrt {\frac {2}{22753 \left (145+\sqrt {22753}\right )}} \text {Subst}\left (\int \frac {-\frac {73}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )}-\left (-73-\sqrt {22753}\right ) x}{\frac {\sqrt {22753}}{576}-\frac {1}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )} x+x^2} \, dx,x,\frac {5}{24}+x\right )+\frac {1}{3} \sqrt {\frac {2}{22753 \left (145+\sqrt {22753}\right )}} \text {Subst}\left (\int \frac {-\frac {73}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )}+\left (-73-\sqrt {22753}\right ) x}{\frac {\sqrt {22753}}{576}+\frac {1}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )} x+x^2} \, dx,x,\frac {5}{24}+x\right ) \\ & = -x+\frac {4 \text {arctanh}\left (\frac {5+24 x}{\sqrt {217}}\right )}{3 \sqrt {217}}+\frac {\text {arctanh}\left (\frac {5+24 x}{4 \sqrt {3} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {3}}-\frac {4 \text {arctanh}\left (\frac {5+24 x}{\sqrt {217} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {217}}+\frac {8}{3} \int \frac {\sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-\frac {10}{3} \int \frac {x \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-8 \int \frac {x^2 \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx+\frac {1}{36} \sqrt {\frac {14041-73 \sqrt {22753}}{45506}} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {22753}}{576}-\frac {1}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )} x+x^2} \, dx,x,\frac {5}{24}+x\right )+\frac {1}{36} \sqrt {\frac {14041-73 \sqrt {22753}}{45506}} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {22753}}{576}+\frac {1}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )} x+x^2} \, dx,x,\frac {5}{24}+x\right )+\frac {\left (-73-\sqrt {22753}\right ) \text {Subst}\left (\int \frac {\frac {1}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )}+2 x}{\frac {\sqrt {22753}}{576}+\frac {1}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )} x+x^2} \, dx,x,\frac {5}{24}+x\right )}{3 \sqrt {45506 \left (145+\sqrt {22753}\right )}}+\frac {\left (73+\sqrt {22753}\right ) \text {Subst}\left (\int \frac {-\frac {1}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )}+2 x}{\frac {\sqrt {22753}}{576}-\frac {1}{12} \sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )} x+x^2} \, dx,x,\frac {5}{24}+x\right )}{3 \sqrt {45506 \left (145+\sqrt {22753}\right )}} \\ & = -x+\frac {4 \text {arctanh}\left (\frac {5+24 x}{\sqrt {217}}\right )}{3 \sqrt {217}}+\frac {\text {arctanh}\left (\frac {5+24 x}{4 \sqrt {3} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {3}}-\frac {4 \text {arctanh}\left (\frac {5+24 x}{\sqrt {217} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {217}}+\frac {1}{3} \sqrt {-\frac {217}{22753}+\frac {2}{\sqrt {22753}}} \log \left (\sqrt {22753}-\sqrt {2 \left (145+\sqrt {22753}\right )} (5+24 x)+(5+24 x)^2\right )-\frac {1}{3} \sqrt {-\frac {217}{22753}+\frac {2}{\sqrt {22753}}} \log \left (\sqrt {22753}+\sqrt {2 \left (145+\sqrt {22753}\right )} (5+24 x)+(5+24 x)^2\right )+\frac {8}{3} \int \frac {\sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-\frac {10}{3} \int \frac {x \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-8 \int \frac {x^2 \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-\frac {1}{18} \sqrt {\frac {14041-73 \sqrt {22753}}{45506}} \text {Subst}\left (\int \frac {1}{\frac {1}{288} \left (145-\sqrt {22753}\right )-x^2} \, dx,x,\frac {1}{12} \left (5-\sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )}+24 x\right )\right )-\frac {1}{18} \sqrt {\frac {14041-73 \sqrt {22753}}{45506}} \text {Subst}\left (\int \frac {1}{\frac {1}{288} \left (145-\sqrt {22753}\right )-x^2} \, dx,x,\frac {1}{12} \left (5+\sqrt {\frac {1}{2} \left (145+\sqrt {22753}\right )}+24 x\right )\right ) \\ & = -x+\frac {2}{3} \sqrt {\frac {217+2 \sqrt {22753}}{22753}} \arctan \left (\frac {10-\sqrt {2 \left (145+\sqrt {22753}\right )}+48 x}{\sqrt {2 \left (-145+\sqrt {22753}\right )}}\right )+\frac {2}{3} \sqrt {\frac {217+2 \sqrt {22753}}{22753}} \arctan \left (\frac {10+\sqrt {2 \left (145+\sqrt {22753}\right )}+48 x}{\sqrt {2 \left (-145+\sqrt {22753}\right )}}\right )+\frac {4 \text {arctanh}\left (\frac {5+24 x}{\sqrt {217}}\right )}{3 \sqrt {217}}+\frac {\text {arctanh}\left (\frac {5+24 x}{4 \sqrt {3} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {3}}-\frac {4 \text {arctanh}\left (\frac {5+24 x}{\sqrt {217} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {217}}+\frac {1}{3} \sqrt {-\frac {217}{22753}+\frac {2}{\sqrt {22753}}} \log \left (\sqrt {22753}-\sqrt {2 \left (145+\sqrt {22753}\right )} (5+24 x)+(5+24 x)^2\right )-\frac {1}{3} \sqrt {-\frac {217}{22753}+\frac {2}{\sqrt {22753}}} \log \left (\sqrt {22753}+\sqrt {2 \left (145+\sqrt {22753}\right )} (5+24 x)+(5+24 x)^2\right )+\frac {8}{3} \int \frac {\sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-\frac {10}{3} \int \frac {x \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx-8 \int \frac {x^2 \sqrt {-3+5 x+12 x^2}}{7-25 x-35 x^2+120 x^3+144 x^4} \, dx \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.95 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=-x-\frac {8 \text {arctanh}\left (\frac {-39-52 x+8 \sqrt {-3+5 x+12 x^2}}{\sqrt {217} (3+4 x)}\right )}{3 \sqrt {217}}-\frac {4}{3} \text {RootSum}\left [144-156 \text {$\#$1}+145 \text {$\#$1}^2+13 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {12 \log (3+4 x)-12 \log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right )+13 \log (3+4 x) \text {$\#$1}-13 \log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right ) \text {$\#$1}-\log (3+4 x) \text {$\#$1}^2+\log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right ) \text {$\#$1}^2}{-156+290 \text {$\#$1}+39 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
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Time = 3.77 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.34
method | result | size |
trager | \(-x -\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-217\right ) \ln \left (\frac {24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-217\right ) x +217 \sqrt {12 x^{2}+5 x -3}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-217\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-217\right ) x -5 x +8}\right )}{651}+4 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {2019920328 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{3} x +420816735 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{3}+1706475 \sqrt {12 x^{2}+5 x -3}\, \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+1594416 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right ) x +332170 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )+16977 \sqrt {12 x^{2}+5 x -3}}{237131766 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{3} x +1023885 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2} x +46509 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right ) x +2055 x -1694}\right )+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4659291081 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) \ln \left (-\frac {88776 x \operatorname {RootOf}\left (\textit {\_Z}^{2}+4659291081 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+18495 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4659291081 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+5119425 \sqrt {12 x^{2}+5 x -3}\, \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+24 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4659291081 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4659291081 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right )-45506 \sqrt {12 x^{2}+5 x -3}}{3474 x \operatorname {RootOf}\left (\textit {\_Z}^{2}+4659291081 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+1023885 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4659291081 \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) x -970 x +1694}\right )}{68259}\) | \(544\) |
default | \(\text {Expression too large to display}\) | \(2632\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.42 (sec) , antiderivative size = 623, normalized size of antiderivative = 2.69 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=\frac {1}{136518} \, \sqrt {22753} \sqrt {968 i \, \sqrt {3} - 1736} \log \left (\sqrt {22753} \sqrt {968 i \, \sqrt {3} - 1736} {\left (169 i \, \sqrt {3} - 73\right )} + 4368576 \, x + 910120\right ) - \frac {1}{136518} \, \sqrt {22753} \sqrt {968 i \, \sqrt {3} - 1736} \log \left (\sqrt {22753} \sqrt {968 i \, \sqrt {3} - 1736} {\left (-169 i \, \sqrt {3} + 73\right )} + 4368576 \, x + 910120\right ) - \frac {1}{136518} \, \sqrt {22753} \sqrt {-968 i \, \sqrt {3} - 1736} \log \left (\sqrt {22753} {\left (169 i \, \sqrt {3} + 73\right )} \sqrt {-968 i \, \sqrt {3} - 1736} + 4368576 \, x + 910120\right ) + \frac {1}{136518} \, \sqrt {22753} \sqrt {-968 i \, \sqrt {3} - 1736} \log \left (\sqrt {22753} {\left (-169 i \, \sqrt {3} - 73\right )} \sqrt {-968 i \, \sqrt {3} - 1736} + 4368576 \, x + 910120\right ) - \frac {1}{273036} \, \sqrt {22753} \sqrt {968 i \, \sqrt {3} - 1736} \log \left (-\frac {\sqrt {22753} \sqrt {12 \, x^{2} + 5 \, x - 3} {\left (411 \, \sqrt {3} {\left (24 i \, x + 5 i\right )} + 8664 \, x + 1805\right )} \sqrt {968 i \, \sqrt {3} - 1736} - 409007928 \, x^{2} + 45506 \, \sqrt {3} {\left (1428 i \, x^{2} + 595 i \, x + 150 i\right )} - 170419970 \, x + 109942496}{x^{2}}\right ) + \frac {1}{273036} \, \sqrt {22753} \sqrt {968 i \, \sqrt {3} - 1736} \log \left (-\frac {\sqrt {22753} \sqrt {12 \, x^{2} + 5 \, x - 3} {\left (411 \, \sqrt {3} {\left (-24 i \, x - 5 i\right )} - 8664 \, x - 1805\right )} \sqrt {968 i \, \sqrt {3} - 1736} - 409007928 \, x^{2} + 45506 \, \sqrt {3} {\left (1428 i \, x^{2} + 595 i \, x + 150 i\right )} - 170419970 \, x + 109942496}{x^{2}}\right ) + \frac {1}{273036} \, \sqrt {22753} \sqrt {-968 i \, \sqrt {3} - 1736} \log \left (-\frac {\sqrt {22753} \sqrt {12 \, x^{2} + 5 \, x - 3} {\left (411 \, \sqrt {3} {\left (24 i \, x + 5 i\right )} - 8664 \, x - 1805\right )} \sqrt {-968 i \, \sqrt {3} - 1736} - 409007928 \, x^{2} + 45506 \, \sqrt {3} {\left (-1428 i \, x^{2} - 595 i \, x - 150 i\right )} - 170419970 \, x + 109942496}{x^{2}}\right ) - \frac {1}{273036} \, \sqrt {22753} \sqrt {-968 i \, \sqrt {3} - 1736} \log \left (-\frac {\sqrt {22753} \sqrt {12 \, x^{2} + 5 \, x - 3} {\left (411 \, \sqrt {3} {\left (-24 i \, x - 5 i\right )} + 8664 \, x + 1805\right )} \sqrt {-968 i \, \sqrt {3} - 1736} - 409007928 \, x^{2} + 45506 \, \sqrt {3} {\left (-1428 i \, x^{2} - 595 i \, x - 150 i\right )} - 170419970 \, x + 109942496}{x^{2}}\right ) + \frac {1}{651} \, \sqrt {217} \log \left (\frac {16112016 \, x^{4} + 13426680 \, x^{3} - 4 \, \sqrt {217} {\left (76320 \, x^{3} + 47700 \, x^{2} - 8399 \, x - 3130\right )} \sqrt {12 \, x^{2} + 5 \, x - 3} - 2423639 \, x^{2} - 2175360 \, x + 326776}{144 \, x^{4} + 120 \, x^{3} - 71 \, x^{2} - 40 \, x + 16}\right ) + \frac {2}{651} \, \sqrt {217} \log \left (\frac {288 \, x^{2} + \sqrt {217} {\left (24 \, x + 5\right )} + 120 \, x + 121}{12 \, x^{2} + 5 \, x - 4}\right ) - x \]
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Not integrable
Time = 21.79 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.18 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=- \int \left (- \frac {3 \sqrt {12 x^{2} + 5 x - 3}}{12 x^{2} \sqrt {12 x^{2} + 5 x - 3} + 5 x \sqrt {12 x^{2} + 5 x - 3} - 3 \sqrt {12 x^{2} + 5 x - 3} + 1}\right )\, dx - \int \frac {5 x \sqrt {12 x^{2} + 5 x - 3}}{12 x^{2} \sqrt {12 x^{2} + 5 x - 3} + 5 x \sqrt {12 x^{2} + 5 x - 3} - 3 \sqrt {12 x^{2} + 5 x - 3} + 1}\, dx - \int \frac {12 x^{2} \sqrt {12 x^{2} + 5 x - 3}}{12 x^{2} \sqrt {12 x^{2} + 5 x - 3} + 5 x \sqrt {12 x^{2} + 5 x - 3} - 3 \sqrt {12 x^{2} + 5 x - 3} + 1}\, dx - \int \left (- \frac {1}{12 x^{2} \sqrt {12 x^{2} + 5 x - 3} + 5 x \sqrt {12 x^{2} + 5 x - 3} - 3 \sqrt {12 x^{2} + 5 x - 3} + 1}\right )\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.16 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=\int { -\frac {\left ({\left (4 \, x + 3\right )} {\left (3 \, x - 1\right )}\right )^{\frac {3}{2}} - 1}{\left ({\left (4 \, x + 3\right )} {\left (3 \, x - 1\right )}\right )^{\frac {3}{2}} + 1} \,d x } \]
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Not integrable
Time = 46.09 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.32 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=\int { -\frac {\left ({\left (4 \, x + 3\right )} {\left (3 \, x - 1\right )}\right )^{\frac {3}{2}} - 1}{\left ({\left (4 \, x + 3\right )} {\left (3 \, x - 1\right )}\right )^{\frac {3}{2}} + 1} \,d x } \]
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Not integrable
Time = 7.75 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.16 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=\int -\frac {{\left (\left (3\,x-1\right )\,\left (4\,x+3\right )\right )}^{3/2}-1}{{\left (\left (3\,x-1\right )\,\left (4\,x+3\right )\right )}^{3/2}+1} \,d x \]
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