\(\int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx\) [2633]

Optimal result

Integrand size = 39, antiderivative size = 232 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=-x-\frac {8 \text {arctanh}\left (\frac {-\frac {39}{\sqrt {217}}-\frac {52 x}{\sqrt {217}}+\frac {8 \sqrt {-3+5 x+12 x^2}}{\sqrt {217}}}{3+4 x}\right )}{3 \sqrt {217}}-\frac {4}{3} \text {RootSum}\left [144-156 \text {$\#$1}+145 \text {$\#$1}^2+13 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {12 \log (3+4 x)-12 \log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right )+13 \log (3+4 x) \text {$\#$1}-13 \log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right ) \text {$\#$1}-\log (3+4 x) \text {$\#$1}^2+\log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right ) \text {$\#$1}^2}{-156+290 \text {$\#$1}+39 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=\int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx \]

[In]

Int[(1 - ((-1 + 3*x)*(3 + 4*x))^(3/2))/(1 + ((-1 + 3*x)*(3 + 4*x))^(3/2)),x]

[Out]

-x + (2*Sqrt[(217 + 2*Sqrt[22753])/22753]*ArcTan[(10 - Sqrt[2*(145 + Sqrt[22753])] + 48*x)/Sqrt[2*(-145 + Sqrt
[22753])]])/3 + (2*Sqrt[(217 + 2*Sqrt[22753])/22753]*ArcTan[(10 + Sqrt[2*(145 + Sqrt[22753])] + 48*x)/Sqrt[2*(
-145 + Sqrt[22753])]])/3 + (4*ArcTanh[(5 + 24*x)/Sqrt[217]])/(3*Sqrt[217]) + ArcTanh[(5 + 24*x)/(4*Sqrt[3]*Sqr
t[-3 + 5*x + 12*x^2])]/(3*Sqrt[3]) - (4*ArcTanh[(5 + 24*x)/(Sqrt[217]*Sqrt[-3 + 5*x + 12*x^2])])/(3*Sqrt[217])
+ (Sqrt[(-217 + 2*Sqrt[22753])/22753]*Log[Sqrt[22753] - Sqrt[2*(145 + Sqrt[22753])]*(5 + 24*x) + (5 + 24*x)^2
])/3 - (Sqrt[(-217 + 2*Sqrt[22753])/22753]*Log[Sqrt[22753] + Sqrt[2*(145 + Sqrt[22753])]*(5 + 24*x) + (5 + 24*
x)^2])/3 + (8*Defer[Int][Sqrt[-3 + 5*x + 12*x^2]/(7 - 25*x - 35*x^2 + 120*x^3 + 144*x^4), x])/3 - (10*Defer[In
t][(x*Sqrt[-3 + 5*x + 12*x^2])/(7 - 25*x - 35*x^2 + 120*x^3 + 144*x^4), x])/3 - 8*Defer[Int][(x^2*Sqrt[-3 + 5*
x + 12*x^2])/(7 - 25*x - 35*x^2 + 120*x^3 + 144*x^4), x]

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*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.95 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=-x-\frac {8 \text {arctanh}\left (\frac {-39-52 x+8 \sqrt {-3+5 x+12 x^2}}{\sqrt {217} (3+4 x)}\right )}{3 \sqrt {217}}-\frac {4}{3} \text {RootSum}\left [144-156 \text {$\#$1}+145 \text {$\#$1}^2+13 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {12 \log (3+4 x)-12 \log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right )+13 \log (3+4 x) \text {$\#$1}-13 \log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right ) \text {$\#$1}-\log (3+4 x) \text {$\#$1}^2+\log \left (4 \sqrt {-3+5 x+12 x^2}-3 \text {$\#$1}-4 x \text {$\#$1}\right ) \text {$\#$1}^2}{-156+290 \text {$\#$1}+39 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[In]

Integrate[(1 - ((-1 + 3*x)*(3 + 4*x))^(3/2))/(1 + ((-1 + 3*x)*(3 + 4*x))^(3/2)),x]

[Out]

-x - (8*ArcTanh[(-39 - 52*x + 8*Sqrt[-3 + 5*x + 12*x^2])/(Sqrt[217]*(3 + 4*x))])/(3*Sqrt[217]) - (4*RootSum[14
4 - 156*#1 + 145*#1^2 + 13*#1^3 + #1^4 & , (12*Log[3 + 4*x] - 12*Log[4*Sqrt[-3 + 5*x + 12*x^2] - 3*#1 - 4*x*#1
] + 13*Log[3 + 4*x]*#1 - 13*Log[4*Sqrt[-3 + 5*x + 12*x^2] - 3*#1 - 4*x*#1]*#1 - Log[3 + 4*x]*#1^2 + Log[4*Sqrt
[-3 + 5*x + 12*x^2] - 3*#1 - 4*x*#1]*#1^2)/(-156 + 290*#1 + 39*#1^2 + 4*#1^3) & ])/3

Maple [N/A]

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 [phone] \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}[phone] \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}[phone] \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}[phone] \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}[phone] \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \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}[phone] \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}[phone] \operatorname {RootOf}\left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) x -970 x +1694}\right )}{68259}\)	\(544\)
default	\(\text {Expression too large to display}\)	\(2632\)

[In]

int((1-((-1+3*x)*(3+4*x))^(3/2))/(1+((-1+3*x)*(3+4*x))^(3/2)),x,method=_RETURNVERBOSE)

[Out]

-x-4/651*RootOf(_Z^2-217)*ln((24*RootOf(_Z^2-217)*x+217*(12*x^2+5*x-3)^(1/2)+5*RootOf(_Z^2-217))/(RootOf(_Z^2-
217)*x-5*x+8))+4*RootOf(1842993*_Z^4+1953*_Z^2+1)*ln((2019920328*RootOf(1842993*_Z^4+1953*_Z^2+1)^3*x+42081673
5*RootOf(1842993*_Z^4+1953*_Z^2+1)^3+1706475*(12*x^2+5*x-3)^(1/2)*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+1594416*R
ootOf(1842993*_Z^4+1953*_Z^2+1)*x+332170*RootOf(1842993*_Z^4+1953*_Z^2+1)+16977*(12*x^2+5*x-3)^(1/2))/(2371317
66*RootOf(1842993*_Z^4+1953*_Z^2+1)^3*x+1023885*RootOf(1842993*_Z^4+1953*_Z^2+1)^2*x+46509*RootOf(1842993*_Z^4
+1953*_Z^2+1)*x+2055*x-1694))+4/68259*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+4937401)*ln(-(
88776*x*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+4937401)*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+
18495*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+4937401)*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+51
19425*(12*x^2+5*x-3)^(1/2)*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+24*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+19
53*_Z^2+1)^2+4937401)*x+5*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+4937401)-45506*(12*x^2+5*x
-3)^(1/2))/(3474*x*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+4937401)*RootOf(1842993*_Z^4+1953
*_Z^2+1)^2+1023885*RootOf(1842993*_Z^4+1953*_Z^2+1)^2*x+3*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+1953*_Z^2
+1)^2+4937401)*x-970*x+1694))

Fricas [C] (verification not implemented)

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 \]

[In]

integrate((1-((-1+3*x)*(3+4*x))^(3/2))/(1+((-1+3*x)*(3+4*x))^(3/2)),x, algorithm="fricas")

[Out]

1/136518*sqrt(22753)*sqrt(968*I*sqrt(3) - 1736)*log(sqrt(22753)*sqrt(968*I*sqrt(3) - 1736)*(169*I*sqrt(3) - 73
) + 4368576*x + 910120) - 1/136518*sqrt(22753)*sqrt(968*I*sqrt(3) - 1736)*log(sqrt(22753)*sqrt(968*I*sqrt(3) -
1736)*(-169*I*sqrt(3) + 73) + 4368576*x + 910120) - 1/136518*sqrt(22753)*sqrt(-968*I*sqrt(3) - 1736)*log(sqrt
(22753)*(169*I*sqrt(3) + 73)*sqrt(-968*I*sqrt(3) - 1736) + 4368576*x + 910120) + 1/136518*sqrt(22753)*sqrt(-96
8*I*sqrt(3) - 1736)*log(sqrt(22753)*(-169*I*sqrt(3) - 73)*sqrt(-968*I*sqrt(3) - 1736) + 4368576*x + 910120) -
1/273036*sqrt(22753)*sqrt(968*I*sqrt(3) - 1736)*log(-(sqrt(22753)*sqrt(12*x^2 + 5*x - 3)*(411*sqrt(3)*(24*I*x
+ 5*I) + 8664*x + 1805)*sqrt(968*I*sqrt(3) - 1736) - 409007928*x^2 + 45506*sqrt(3)*(1428*I*x^2 + 595*I*x + 150
*I) - 170419970*x + 109942496)/x^2) + 1/273036*sqrt(22753)*sqrt(968*I*sqrt(3) - 1736)*log(-(sqrt(22753)*sqrt(1
2*x^2 + 5*x - 3)*(411*sqrt(3)*(-24*I*x - 5*I) - 8664*x - 1805)*sqrt(968*I*sqrt(3) - 1736) - 409007928*x^2 + 45
506*sqrt(3)*(1428*I*x^2 + 595*I*x + 150*I) - 170419970*x + 109942496)/x^2) + 1/273036*sqrt(22753)*sqrt(-968*I*
sqrt(3) - 1736)*log(-(sqrt(22753)*sqrt(12*x^2 + 5*x - 3)*(411*sqrt(3)*(24*I*x + 5*I) - 8664*x - 1805)*sqrt(-96
8*I*sqrt(3) - 1736) - 409007928*x^2 + 45506*sqrt(3)*(-1428*I*x^2 - 595*I*x - 150*I) - 170419970*x + 109942496)
/x^2) - 1/273036*sqrt(22753)*sqrt(-968*I*sqrt(3) - 1736)*log(-(sqrt(22753)*sqrt(12*x^2 + 5*x - 3)*(411*sqrt(3)
*(-24*I*x - 5*I) + 8664*x + 1805)*sqrt(-968*I*sqrt(3) - 1736) - 409007928*x^2 + 45506*sqrt(3)*(-1428*I*x^2 - 5
95*I*x - 150*I) - 170419970*x + 109942496)/x^2) + 1/651*sqrt(217)*log((16112016*x^4 + 13426680*x^3 - 4*sqrt(21
7)*(76320*x^3 + 47700*x^2 - 8399*x - 3130)*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)) + 2/651*sqrt(217)*log((288*x^2 + sqrt(217)*(24*x + 5) + 120*x + 121)/(12*x^2
+ 5*x - 4)) - x

Sympy [N/A]

Not integrable

Time = 21.79 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.18 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=- \int \left (- \frac {3 \sqrt {12 x^{2} + 5 x - 3}}{12 x^{2} \sqrt {12 x^{2} + 5 x - 3} + 5 x \sqrt {12 x^{2} + 5 x - 3} - 3 \sqrt {12 x^{2} + 5 x - 3} + 1}\right )\, dx - \int \frac {5 x \sqrt {12 x^{2} + 5 x - 3}}{12 x^{2} \sqrt {12 x^{2} + 5 x - 3} + 5 x \sqrt {12 x^{2} + 5 x - 3} - 3 \sqrt {12 x^{2} + 5 x - 3} + 1}\, dx - \int \frac {12 x^{2} \sqrt {12 x^{2} + 5 x - 3}}{12 x^{2} \sqrt {12 x^{2} + 5 x - 3} + 5 x \sqrt {12 x^{2} + 5 x - 3} - 3 \sqrt {12 x^{2} + 5 x - 3} + 1}\, dx - \int \left (- \frac {1}{12 x^{2} \sqrt {12 x^{2} + 5 x - 3} + 5 x \sqrt {12 x^{2} + 5 x - 3} - 3 \sqrt {12 x^{2} + 5 x - 3} + 1}\right )\, dx \]

[In]

integrate((1-((-1+3*x)*(3+4*x))**(3/2))/(1+((-1+3*x)*(3+4*x))**(3/2)),x)

[Out]

-Integral(-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(1
2*x**2 + 5*x - 3) + 1), x) - Integral(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), x) - Integral(12*x**2*sqrt(12*x**2 + 5*x - 3)/(12*x**2*sq
rt(12*x**2 + 5*x - 3) + 5*x*sqrt(12*x**2 + 5*x - 3) - 3*sqrt(12*x**2 + 5*x - 3) + 1), x) - Integral(-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), x)

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.16 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=\int { -\frac {\left ({\left (4 \, x + 3\right )} {\left (3 \, x - 1\right )}\right )^{\frac {3}{2}} - 1}{\left ({\left (4 \, x + 3\right )} {\left (3 \, x - 1\right )}\right )^{\frac {3}{2}} + 1} \,d x } \]

[In]

integrate((1-((-1+3*x)*(3+4*x))^(3/2))/(1+((-1+3*x)*(3+4*x))^(3/2)),x, algorithm="maxima")

[Out]

-integrate((((4*x + 3)*(3*x - 1))^(3/2) - 1)/(((4*x + 3)*(3*x - 1))^(3/2) + 1), x)

Giac [N/A]

Not integrable

Time = 46.09 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.32 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=\int { -\frac {\left ({\left (4 \, x + 3\right )} {\left (3 \, x - 1\right )}\right )^{\frac {3}{2}} - 1}{\left ({\left (4 \, x + 3\right )} {\left (3 \, x - 1\right )}\right )^{\frac {3}{2}} + 1} \,d x } \]

[In]

integrate((1-((-1+3*x)*(3+4*x))^(3/2))/(1+((-1+3*x)*(3+4*x))^(3/2)),x, algorithm="giac")

[Out]

-1/68259*sqrt(45506*sqrt(22753) - 4937401)*log(387923590497058811515901983830460365251005010667651224037333579
03935757466552483657422977213267641344*(48*sqrt(22753)*x + 6960*x + sqrt(22753)*sqrt(2*sqrt(22753) + 290) + 10
*sqrt(22753) + 145*sqrt(2*sqrt(22753) + 290) + 1450)^2 + 13406639287578352525989572561180710223074733168674026
3027302484916001977804405383520053809249052968484864*sqrt(22753) + 1943962696698861116268488021371202982345836
3094577338138958860312820286781638780610407802341112680430305280) + 1/68259*sqrt(45506*sqrt(22753) - 4937401)*
log(38792359049705881151590198383046036525100501066765122403733357903935757466552483657422977213267641344*(48*
sqrt(22753)*x + 6960*x - sqrt(22753)*sqrt(2*sqrt(22753) + 290) + 10*sqrt(22753) - 145*sqrt(2*sqrt(22753) + 290
) + 1450)^2 + 134066392875783525259895725611807102230747331686740263027302484916001977804405383520053809249052
968484864*sqrt(22753) + 19439626966988611162684880213712029823458363094577338138958860312820286781638780610407
802341112680430305280) - 2/651*sqrt(217)*log(abs(24*x - sqrt(217) + 5)/abs(24*x + sqrt(217) + 5)) + undef + 24
2/68259*sqrt(3)*sqrt(45506*sqrt(22753) - 4937401)*arctan(1/24*(48*x*(sqrt(22753) + 145) + sqrt(22753)*sqrt(2*s
qrt(22753) + 290) + 10*sqrt(22753) + 145*sqrt(2*sqrt(22753) + 290) + 1450)/sqrt(6*sqrt(22753) + 870))/(2*sqrt(
22753) - 217) + 242/68259*sqrt(3)*sqrt(45506*sqrt(22753) - 4937401)*arctan(1/24*(48*x*(sqrt(22753) + 145) - sq
rt(22753)*sqrt(2*sqrt(22753) + 290) + 10*sqrt(22753) - 145*sqrt(2*sqrt(22753) + 290) + 1450)/sqrt(6*sqrt(22753
) + 870))/(2*sqrt(22753) - 217)

Mupad [N/A]

Not integrable

Time = 7.75 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.16 \[ \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx=\int -\frac {{\left (\left (3\,x-1\right )\,\left (4\,x+3\right )\right )}^{3/2}-1}{{\left (\left (3\,x-1\right )\,\left (4\,x+3\right )\right )}^{3/2}+1} \,d x \]

[In]

int(-(((3*x - 1)*(4*x + 3))^(3/2) - 1)/(((3*x - 1)*(4*x + 3))^(3/2) + 1),x)

[Out]

int(-(((3*x - 1)*(4*x + 3))^(3/2) - 1)/(((3*x - 1)*(4*x + 3))^(3/2) + 1), x)