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

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

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Rubi [F]  time = 1.89, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \int \frac {1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx &=\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} \operatorname {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} \operatorname {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 \tanh ^{-1}\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} \operatorname {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 \operatorname {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 \tanh ^{-1}\left (\frac {5+24 x}{\sqrt {217}}\right )}{3 \sqrt {217}}+\frac {\tanh ^{-1}\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} \operatorname {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 )}} \operatorname {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 )}} \operatorname {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 \tanh ^{-1}\left (\frac {5+24 x}{\sqrt {217}}\right )}{3 \sqrt {217}}+\frac {\tanh ^{-1}\left (\frac {5+24 x}{4 \sqrt {3} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {3}}-\frac {4 \tanh ^{-1}\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}} \operatorname {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}} \operatorname {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 ) \operatorname {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 ) \operatorname {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 \tanh ^{-1}\left (\frac {5+24 x}{\sqrt {217}}\right )}{3 \sqrt {217}}+\frac {\tanh ^{-1}\left (\frac {5+24 x}{4 \sqrt {3} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {3}}-\frac {4 \tanh ^{-1}\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}} \operatorname {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}} \operatorname {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}} \tan ^{-1}\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}} \tan ^{-1}\left (\frac {10+\sqrt {2 \left (145+\sqrt {22753}\right )}+48 x}{\sqrt {2 \left (-145+\sqrt {22753}\right )}}\right )+\frac {4 \tanh ^{-1}\left (\frac {5+24 x}{\sqrt {217}}\right )}{3 \sqrt {217}}+\frac {\tanh ^{-1}\left (\frac {5+24 x}{4 \sqrt {3} \sqrt {-3+5 x+12 x^2}}\right )}{3 \sqrt {3}}-\frac {4 \tanh ^{-1}\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*}

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Mathematica [C]  time = 2.43, size = 727, normalized size = 3.13 \begin {gather*} \frac {434 \sqrt {68259} \text {RootSum}\left [144 \text {$\#$1}^4+120 \text {$\#$1}^3-35 \text {$\#$1}^2-25 \text {$\#$1}+7\&,\frac {12 \text {$\#$1}^2 \log (x-\text {$\#$1})+5 \text {$\#$1} \log (x-\text {$\#$1})-\log (x-\text {$\#$1})}{576 \text {$\#$1}^3+360 \text {$\#$1}^2-70 \text {$\#$1}-25}\&\right ]+2 \sqrt {14812203} \log \left (-48 \sqrt {217} \sqrt {12 x^2+5 x-3}+5208 x+169 \sqrt {217}+1085\right )-2 \sqrt {14812203} \log \left (48 \sqrt {217} \sqrt {12 x^2+5 x-3}+5208 x-169 \sqrt {217}+1085\right )+217 \sqrt {-1302+726 i \sqrt {3}} \log \left (24 \sqrt {-2-2 i \sqrt {3}} \sqrt {12 x^2+5 x-3}-24 \sqrt {145-24 i \sqrt {3}} x-5 \sqrt {145-24 i \sqrt {3}}-169\right )-217 \sqrt {-1302+726 i \sqrt {3}} \log \left (24 \sqrt {-2-2 i \sqrt {3}} \sqrt {12 x^2+5 x-3}+24 \sqrt {145-24 i \sqrt {3}} x+5 \sqrt {145-24 i \sqrt {3}}-169\right )+217 \sqrt {-1302-726 i \sqrt {3}} \log \left (24 \sqrt {-2+2 i \sqrt {3}} \sqrt {12 x^2+5 x-3}-24 \sqrt {145+24 i \sqrt {3}} x-5 \sqrt {145+24 i \sqrt {3}}-169\right )-217 \sqrt {-1302-726 i \sqrt {3}} \log \left (24 \sqrt {-2+2 i \sqrt {3}} \sqrt {12 x^2+5 x-3}+24 \sqrt {145+24 i \sqrt {3}} x+5 \sqrt {145+24 i \sqrt {3}}-169\right )-651 \sqrt {68259} x-2 \sqrt {14812203} \log \left (-24 x+\sqrt {217}-5\right )+2 \sqrt {14812203} \log \left (24 x-\sqrt {217}+5\right )+217 \sqrt {-1302+726 i \sqrt {3}} \log \left (24 x-\sqrt {145-24 i \sqrt {3}}+5\right )-217 \sqrt {-1302+726 i \sqrt {3}} \log \left (24 x+\sqrt {145-24 i \sqrt {3}}+5\right )+217 \sqrt {-1302-726 i \sqrt {3}} \log \left (24 x-\sqrt {145+24 i \sqrt {3}}+5\right )-217 \sqrt {-1302-726 i \sqrt {3}} \log \left (24 x+\sqrt {145+24 i \sqrt {3}}+5\right )}{651 \sqrt {68259}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-651*Sqrt[68259]*x - 2*Sqrt[14812203]*Log[-5 + Sqrt[217] - 24*x] + 2*Sqrt[14812203]*Log[5 - Sqrt[217] + 24*x]
+ 217*Sqrt[-1302 + (726*I)*Sqrt[3]]*Log[5 - Sqrt[145 - (24*I)*Sqrt[3]] + 24*x] - 217*Sqrt[-1302 + (726*I)*Sqr
t[3]]*Log[5 + Sqrt[145 - (24*I)*Sqrt[3]] + 24*x] + 217*Sqrt[-1302 - (726*I)*Sqrt[3]]*Log[5 - Sqrt[145 + (24*I)
*Sqrt[3]] + 24*x] - 217*Sqrt[-1302 - (726*I)*Sqrt[3]]*Log[5 + Sqrt[145 + (24*I)*Sqrt[3]] + 24*x] + 2*Sqrt[1481
2203]*Log[1085 + 169*Sqrt[217] + 5208*x - 48*Sqrt[217]*Sqrt[-3 + 5*x + 12*x^2]] - 2*Sqrt[14812203]*Log[1085 -
169*Sqrt[217] + 5208*x + 48*Sqrt[217]*Sqrt[-3 + 5*x + 12*x^2]] + 217*Sqrt[-1302 + (726*I)*Sqrt[3]]*Log[-169 -
5*Sqrt[145 - (24*I)*Sqrt[3]] - 24*Sqrt[145 - (24*I)*Sqrt[3]]*x + 24*Sqrt[-2 - (2*I)*Sqrt[3]]*Sqrt[-3 + 5*x + 1
2*x^2]] - 217*Sqrt[-1302 + (726*I)*Sqrt[3]]*Log[-169 + 5*Sqrt[145 - (24*I)*Sqrt[3]] + 24*Sqrt[145 - (24*I)*Sqr
t[3]]*x + 24*Sqrt[-2 - (2*I)*Sqrt[3]]*Sqrt[-3 + 5*x + 12*x^2]] + 217*Sqrt[-1302 - (726*I)*Sqrt[3]]*Log[-169 -
5*Sqrt[145 + (24*I)*Sqrt[3]] - 24*Sqrt[145 + (24*I)*Sqrt[3]]*x + 24*Sqrt[-2 + (2*I)*Sqrt[3]]*Sqrt[-3 + 5*x + 1
2*x^2]] - 217*Sqrt[-1302 - (726*I)*Sqrt[3]]*Log[-169 + 5*Sqrt[145 + (24*I)*Sqrt[3]] + 24*Sqrt[145 + (24*I)*Sqr
t[3]]*x + 24*Sqrt[-2 + (2*I)*Sqrt[3]]*Sqrt[-3 + 5*x + 12*x^2]] + 434*Sqrt[68259]*RootSum[7 - 25*#1 - 35*#1^2 +
120*#1^3 + 144*#1^4 & , (-Log[x - #1] + 5*Log[x - #1]*#1 + 12*Log[x - #1]*#1^2)/(-25 - 70*#1 + 360*#1^2 + 576
*#1^3) & ])/(651*Sqrt[68259])

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IntegrateAlgebraic [A]  time = 0.48, size = 232, normalized size = 1.00 \begin {gather*} -x-\frac {8 \tanh ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - (8*ArcTanh[(-39/Sqrt[217] - (52*x)/Sqrt[217] + (8*Sqrt[-3 + 5*x + 12*x^2])/Sqrt[217])/(3 + 4*x)])/(3*Sqrt
[217]) - (4*RootSum[144 - 156*#1 + 145*#1^2 + 13*#1^3 + #1^4 & , (12*Log[3 + 4*x] - 12*Log[4*Sqrt[-3 + 5*x + 1
2*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

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fricas [B]  time = 4.35, size = 4026, normalized size = 17.35

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24778017*22753^(1/4)*(2*sqrt(22753)*sqrt(3) - 217*sqrt(3))*sqrt(217*sqrt(22753) + 45506)*log(19084456267776
*x^2 + 1456192/363*22753^(1/4)*(73*sqrt(22753)*sqrt(3)*(24*x + 5) + 22753*sqrt(3)*(24*x + 5))*sqrt(217*sqrt(22
753) + 45506) + 7951856778240*x + 33132736576*sqrt(22753) + 828318414400) + 1/24778017*22753^(1/4)*(2*sqrt(227
53)*sqrt(3) - 217*sqrt(3))*sqrt(217*sqrt(22753) + 45506)*log(19084456267776*x^2 - 1456192/363*22753^(1/4)*(73*
sqrt(22753)*sqrt(3)*(24*x + 5) + 22753*sqrt(3)*(24*x + 5))*sqrt(217*sqrt(22753) + 45506) + 7951856778240*x + 3
3132736576*sqrt(22753) + 828318414400) + 1/49556034*22753^(1/4)*(2*sqrt(22753)*sqrt(3) - 217*sqrt(3))*sqrt(217
*sqrt(22753) + 45506)*log(2548336/363*(25633948819248*x^4 + 21361624016040*x^3 + 2*22753^(1/4)*(sqrt(22753)*sq
rt(3)*(2543904*x^3 + 1589940*x^2 - 1591391*x - 377545) - 22753*sqrt(3)*(69696*x^3 + 43560*x^2 - 15430*x - 4475
))*sqrt(12*x^2 + 5*x - 3)*sqrt(217*sqrt(22753) + 45506) - 7981387464633*x^2 + 33037356*sqrt(22753)*(6912*x^4 +
5760*x^3 - 228*x^2 - 595*x - 75) - 5179885750545*x + 1741704630303)/x^4) - 1/49556034*22753^(1/4)*(2*sqrt(227
53)*sqrt(3) - 217*sqrt(3))*sqrt(217*sqrt(22753) + 45506)*log(2548336/363*(25633948819248*x^4 + 21361624016040*
x^3 - 2*22753^(1/4)*(sqrt(22753)*sqrt(3)*(2543904*x^3 + 1589940*x^2 - 1591391*x - 377545) - 22753*sqrt(3)*(696
96*x^3 + 43560*x^2 - 15430*x - 4475))*sqrt(12*x^2 + 5*x - 3)*sqrt(217*sqrt(22753) + 45506) - 7981387464633*x^2
+ 33037356*sqrt(22753)*(6912*x^4 + 5760*x^3 - 228*x^2 - 595*x - 75) - 5179885750545*x + 1741704630303)/x^4) -
4/68259*22753^(1/4)*sqrt(217*sqrt(22753) + 45506)*arctan(-1/198224136*22753^(3/4)*(sqrt(22753)*(24*x + 5) + 1
752*x + 365)*sqrt(217*sqrt(22753) + 45506) + 1/6541396488*22753^(1/4)*sqrt(4757379264*x^2 + 22753^(1/4)*(73*sq
rt(22753)*sqrt(3)*(24*x + 5) + 22753*sqrt(3)*(24*x + 5))*sqrt(217*sqrt(22753) + 45506) + 1982241360*x + 825933
9*sqrt(22753) + 206483475)*(sqrt(22753)*sqrt(3) + 73*sqrt(3))*sqrt(217*sqrt(22753) + 45506) - 1/72*sqrt(22753)
*sqrt(3) - 145/72*sqrt(3)) - 4/68259*22753^(1/4)*sqrt(217*sqrt(22753) + 45506)*arctan(-1/198224136*22753^(3/4)
*(sqrt(22753)*(24*x + 5) + 1752*x + 365)*sqrt(217*sqrt(22753) + 45506) + 1/6541396488*22753^(1/4)*sqrt(4757379
264*x^2 - 22753^(1/4)*(73*sqrt(22753)*sqrt(3)*(24*x + 5) + 22753*sqrt(3)*(24*x + 5))*sqrt(217*sqrt(22753) + 45
506) + 1982241360*x + 8259339*sqrt(22753) + 206483475)*(sqrt(22753)*sqrt(3) + 73*sqrt(3))*sqrt(217*sqrt(22753)
+ 45506) + 1/72*sqrt(22753)*sqrt(3) + 145/72*sqrt(3)) - 2/68259*22753^(1/4)*sqrt(217*sqrt(22753) + 45506)*arc
tan(1/59040503574423657*(1356417723145733248*sqrt(22753)*sqrt(3)*(563469550783693848576*x^16 + 187823183594564
6161920*x^15 + 2271068175922858819584*x^14 + 917515317092920197120*x^13 - 397395114882246070272*x^12 - 4884481
94480130631680*x^11 - 102981130930246908672*x^10 + 50677635024881913600*x^9 + 28077563371310921808*x^8 + 19052
99283994534680*x^7 - 1783382469000052525*x^6 - 491153348417493500*x^5 - 13212596524478125*x^4 + 14755129740318
750*x^3 + 2937991251937500*x^2 + 238996760625000*x + 7491487500000) + 15396409028*sqrt(12*x^2 + 5*x - 3)*(8*22
753^(3/4)*(1457154908615709265231872*x^15 + 4553609089424091453849600*x^14 + 5102105388283684293083136*x^13 +
1828085567296311560540160*x^12 - 863574952880865256243200*x^11 - 890603170338272292000000*x^10 - 1492353735781
34499034752*x^9 + 85268363246378743209840*x^8 + 37350229493890469150352*x^7 + 1124904436500694490805*x^6 - 207
1578487513731147075*x^5 - 408393684914052252500*x^4 + 5113941819580075000*x^3 + 10286357222320650000*x^2 - sqr
t(22753)*(14563504231703447076864*x^15 + 45510950724073272115200*x^14 + 49683213355961339707392*x^13 + 1472373
4207966250987520*x^12 - 11905446987615605391360*x^11 - 9631257905511143827200*x^10 - 827225088416529429888*x^9
+ 1284470756168556018960*x^8 + 413130382691111915760*x^7 - 26586072949565917725*x^6 - 33494258600700079125*x^
5 - 4651259921729237500*x^4 + 503162532696125000*x^3 + 208409843265750000*x^2 + 22116943248750000*x + 82977142
5000000) + 1240755306257250000*x + 49031857035000000) + 22753^(1/4)*(226000863457782769580507136*x^15 + 706252
698305571154939084800*x^14 + 882047337231387757676101632*x^13 + 529242917217908536430161920*x^12 + 10387228495
6367546157232128*x^11 - 62388940077891611478455040*x^10 - 49563221657203570994499456*x^9 - 1273059589516798805
0106480*x^8 + 554064252652090432756800*x^7 + 1256928245962551637657875*x^6 + 328688538932058376022500*x^5 + 19
963652581720216112500*x^4 - 8191711857965818500000*x^3 - 2179114600304958750000*x^2 - sqrt(22753)*(20442858594
25092346183680*x^15 + 6388393310703413581824000*x^14 + 7841025005927637045473280*x^13 + 4414819134282875911756
800*x^12 + 538457555843790977376000*x^11 - 772314613728660371160000*x^10 - 481967552475778726993968*x^9 - 9265
5853015809690644940*x^8 + 21161605556499146054775*x^7 + 15688561753406924179875*x^6 + 3172766952980954255000*x
^5 - 45012653614268537500*x^4 - 151679297354024250000*x^3 - 32137635197688750000*x^2 - 3042642600900000000*x -
114444059625000000) - 221682523937175000000*x - 8642024551125000000))*sqrt(217*sqrt(22753) + 45506) - 2*sqrt(
159271/3)*(27134681728*sqrt(22753)*sqrt(3)*(139120804771135488*x^16 + 405769013915811840*x^15 + 49356791349549
4656*x^14 + 318121662053744640*x^13 + 100642264749831168*x^12 - 5111950591238400*x^11 - 19657015863638400*x^10
- 8715811943124000*x^9 - 1630769907877500*x^8 + 113368251221875*x^7 + 149750549812500*x^6 + 41899409296875*x^
5 + 6333082031250*x^4 + 527132812500*x^3 + 18984375000*x^2) + 2*sqrt(12*x^2 + 5*x - 3)*(16*22753^(3/4)*(154384
1518716120268800000*x^15 + 4181237446522825728000000*x^14 + 3860978610751616533929984*x^13 + 86307391496011375
6922880*x^12 - 779224547194529048780544*x^11 - 505412564215843004123520*x^10 - 44629598827645171524288*x^9 + 4
5140229707428827590580*x^8 + 15007263884546349529425*x^7 + 589773342093770225625*x^6 - 556680655243090300000*x
^5 - 128109070123096725000*x^4 - 11749611649025250000*x^3 - 410825582133750000*x^2 - sqrt(22753)*(212760276580
98433327104*x^15 + 57622574907349923594240*x^14 + 53119535706899223404544*x^13 + 11689157526079963438080*x^12
- 10919222381610157605120*x^11 - 7036738795771404969600*x^10 - 621131294598173087040*x^9 + 6279116437173149739
00*x^8 + 209401976660299863375*x^7 + 8625388473062034375*x^6 - 7622651689386500000*x^5 - 1763320756084875000*x
^4 - 161896223238750000*x^3 - 5660707106250000*x^2)) + 22753^(1/4)*(509656134754982980140171264*x^15 + 1380318
698294745571212963840*x^14 + 1486487472085635931644235776*x^13 + 770508498313922398326616320*x^12 + 1249630989
81444337336115328*x^11 - 82495543495022435187923760*x^10 - 58955528599267342238258400*x^9 - 154777675952433106
70483625*x^8 - 532651343778452286667500*x^7 + 892245329348370198662500*x^6 + 298063292793256165500000*x^5 + 47
370119073953276250000*x^4 + 3944120990334525000000*x^3 + 137906328333375000000*x^2 - sqrt(22753)*(583776555921
3345945255936*x^15 + 15810615056202811935068160*x^14 + 17009049955286781216534528*x^13 + 878518989436711471744
8960*x^12 + 1388383215845925452160384*x^11 - 972840468206549851019280*x^10 - 687499322224809431095200*x^9 - 18
1009235153441909725875*x^8 - 6892716223097226502500*x^7 + 10106730756524363487500*x^6 + 3404105226535396500000
*x^5 + 542149149604578750000*x^4 + 45171650098575000000*x^3 + 1579428325125000000*x^2)))*sqrt(217*sqrt(22753)
+ 45506) - 2753113*sqrt(3)*(769218282044368679190528*x^16 + 2243553322629408647639040*x^15 + 27220196197339954
20822528*x^14 + 1741473842011670423704320*x^13 + 535691716482687615998784*x^12 - 43747987364926664929200*x^11
- 116676997755758617916700*x^10 - 51170731794120633771375*x^9 - 9829950857470192751250*x^8 + 46621870668851221
2500*x^7 + 805909745662305375000*x^6 + 229732359820908750000*x^5 + 34928426088562500000*x^4 + 2913509442375000
000*x^3 + 104928311250000000*x^2) + 968*sqrt(22753)*(11*sqrt(22753)*sqrt(3)*(11133280894478707851264*x^16 + 32
472069275562897899520*x^15 + 34961846584368741912576*x^14 + 14116960109825036605440*x^13 - 2578559534967289865
472*x^12 - 4511715066433937606400*x^11 - 1398435875141700988944*x^10 + 107026713984510716760*x^9 + 18039331487
7965661225*x^8 + 42277231451151825375*x^7 - 570863512141243750*x^6 - 2156985962491425000*x^5 - 452241585490500
000*x^4 - 41499802811250000*x^3 - 1494590737500000*x^2) - 159271*sqrt(3)*(837463052302553972736*x^16 + 2442600
569215782420480*x^15 + 2614273634565545472000*x^14 + 1022870301725465856000*x^13 - 233881116511753062144*x^12
- 360426948847640812800*x^11 - 110377763294502560688*x^10 + 8394027644330120520*x^9 + 14260646520179433075*x^8
+ 3417531763495585125*x^7 - 307353208581250*x^6 - 158065693208475000*x^5 - 33827508643500000*x^4 - 3119304903
750000*x^3 - 112339912500000*x^2)))*sqrt((25633948819248*x^4 + 21361624016040*x^3 + 2*22753^(1/4)*(sqrt(22753)
*sqrt(3)*(2543904*x^3 + 1589940*x^2 - 1591391*x - 377545) - 22753*sqrt(3)*(69696*x^3 + 43560*x^2 - 15430*x - 4
475))*sqrt(12*x^2 + 5*x - 3)*sqrt(217*sqrt(22753) + 45506) - 7981387464633*x^2 + 33037356*sqrt(22753)*(6912*x^
4 + 5760*x^3 - 228*x^2 - 595*x - 75) - 5179885750545*x + 1741704630303)/x^4) - 1513859066010863*sqrt(3)*(13576
751558109854315642880*x^16 + 45255838527032847718809600*x^15 + 56292025222029827201384448*x^14 + 2668903637274
7614913904640*x^13 - 4717355119347430651415040*x^12 - 9964876712917755398169600*x^11 - 30043275148984233066848
64*x^10 + 551351931987193055323200*x^9 + 502630856461513673065557*x^8 + 72376829421365242444470*x^7 - 18222237
302931806775100*x^6 - 7057276971466800749000*x^5 - 514495325192847850000*x^4 + 108121429522446900000*x^3 + 234
67018179339000000*x^2 + 1564662500730000000*x + 29246767200000000) + 3725930984776*sqrt(22753)*(sqrt(22753)*sq
rt(3)*(217802415187974448742400*x^16 + 726008050626581495808000*x^15 + 942100751373820723445760*x^14 + 5420402
32138022912716800*x^13 + 42536156642286573911040*x^12 - 121361620564714749062400*x^11 - 6634649409687826861766
4*x^10 - 8423828365205592586800*x^9 + 4938327917753662627236*x^8 + 2356458373231956745185*x^7 + 30730033995654
2180475*x^6 - 53544679061058547750*x^5 - 25131460994000275000*x^4 - 3657330255837300000*x^3 - 2063342781742500
00*x^2 + 2324736382500000*x + 522263700000000) - 22753*sqrt(3)*(3686750823320524947456*x^16 + 1228916941106841
6491520*x^15 + 15098097987625062187008*x^14 + 6699232899028429885440*x^13 - 1933672081302751841280*x^12 - 3100
521086130742675200*x^11 - 894636158071089054528*x^10 + 219693320035867436400*x^9 + 195902528934273209964*x^8 +
32158375953240369315*x^7 - 8982246266755377975*x^6 - 4350253576672402250*x^5 - 439537219633475000*x^4 + 92790
568452300000*x^3 + 29873937449250000*x^2 + 3122149117500000*x + 119863800000000)))/(3273904261412608177668096*
x^16 + 10913014204708693925560320*x^15 + 13145192753445784215797760*x^14 + 5184286401994276237516800*x^13 - 24
70469615059733328056832*x^12 - 2910634018693345878382080*x^11 - 595321320958698206129472*x^10 + 31093509576194
7268053600*x^9 + 169217713571892642030717*x^8 + 11320598297034116794320*x^7 - 10774586517248316715600*x^6 - 29
72577173373156744000*x^5 - 85333313258014100000*x^4 + 87760768027346400000*x^3 + 17605709885484000000*x^2 + 14
39011964880000000*x + 45347015700000000)) - 2/68259*22753^(1/4)*sqrt(217*sqrt(22753) + 45506)*arctan(-1/590405
03574423657*(1356417723145733248*sqrt(22753)*sqrt(3)*(563469550783693848576*x^16 + 1878231835945646161920*x^15
+ 2271068175922858819584*x^14 + 917515317092920197120*x^13 - 397395114882246070272*x^12 - 4884481944801306316
80*x^11 - 102981130930246908672*x^10 + 50677635024881913600*x^9 + 28077563371310921808*x^8 + 19052992839945346
80*x^7 - 1783382469000052525*x^6 - 491153348417493500*x^5 - 13212596524478125*x^4 + 14755129740318750*x^3 + 29
37991251937500*x^2 + 238996760625000*x + 7491487500000) - 15396409028*sqrt(12*x^2 + 5*x - 3)*(8*22753^(3/4)*(1
457154908615709265231872*x^15 + 4553609089424091453849600*x^14 + 5102105388283684293083136*x^13 + 182808556729
6311560540160*x^12 - 863574952880865256243200*x^11 - 890603170338272292000000*x^10 - 149235373578134499034752*
x^9 + 85268363246378743209840*x^8 + 37350229493890469150352*x^7 + 1124904436500694490805*x^6 - 207157848751373
1147075*x^5 - 408393684914052252500*x^4 + 5113941819580075000*x^3 + 10286357222320650000*x^2 - sqrt(22753)*(14
563504231703447076864*x^15 + 45510950724073272115200*x^14 + 49683213355961339707392*x^13 + 1472373420796625098
7520*x^12 - 11905446987615605391360*x^11 - 9631257905511143827200*x^10 - 827225088416529429888*x^9 + 128447075
6168556018960*x^8 + 413130382691111915760*x^7 - 26586072949565917725*x^6 - 33494258600700079125*x^5 - 46512599
21729237500*x^4 + 503162532696125000*x^3 + 208409843265750000*x^2 + 22116943248750000*x + 829771425000000) + 1
240755306257250000*x + 49031857035000000) + 22753^(1/4)*(226000863457782769580507136*x^15 + 706252698305571154
939084800*x^14 + 882047337231387757676101632*x^13 + 529242917217908536430161920*x^12 + 10387228495636754615723
2128*x^11 - 62388940077891611478455040*x^10 - 49563221657203570994499456*x^9 - 12730595895167988050106480*x^8
+ 554064252652090432756800*x^7 + 1256928245962551637657875*x^6 + 328688538932058376022500*x^5 + 19963652581720
216112500*x^4 - 8191711857965818500000*x^3 - 2179114600304958750000*x^2 - sqrt(22753)*(20442858594250923461836
80*x^15 + 6388393310703413581824000*x^14 + 7841025005927637045473280*x^13 + 4414819134282875911756800*x^12 + 5
38457555843790977376000*x^11 - 772314613728660371160000*x^10 - 481967552475778726993968*x^9 - 9265585301580969
0644940*x^8 + 21161605556499146054775*x^7 + 15688561753406924179875*x^6 + 3172766952980954255000*x^5 - 4501265
3614268537500*x^4 - 151679297354024250000*x^3 - 32137635197688750000*x^2 - 3042642600900000000*x - 11444405962
5000000) - 221682523937175000000*x - 8642024551125000000))*sqrt(217*sqrt(22753) + 45506) - 2*sqrt(159271/3)*(2
7134681728*sqrt(22753)*sqrt(3)*(139120804771135488*x^16 + 405769013915811840*x^15 + 493567913495494656*x^14 +
318121662053744640*x^13 + 100642264749831168*x^12 - 5111950591238400*x^11 - 19657015863638400*x^10 - 871581194
3124000*x^9 - 1630769907877500*x^8 + 113368251221875*x^7 + 149750549812500*x^6 + 41899409296875*x^5 + 63330820
31250*x^4 + 527132812500*x^3 + 18984375000*x^2) - 2*sqrt(12*x^2 + 5*x - 3)*(16*22753^(3/4)*(154384151871612026
8800000*x^15 + 4181237446522825728000000*x^14 + 3860978610751616533929984*x^13 + 863073914960113756922880*x^12
- 779224547194529048780544*x^11 - 505412564215843004123520*x^10 - 44629598827645171524288*x^9 + 4514022970742
8827590580*x^8 + 15007263884546349529425*x^7 + 589773342093770225625*x^6 - 556680655243090300000*x^5 - 1281090
70123096725000*x^4 - 11749611649025250000*x^3 - 410825582133750000*x^2 - sqrt(22753)*(21276027658098433327104*
x^15 + 57622574907349923594240*x^14 + 53119535706899223404544*x^13 + 11689157526079963438080*x^12 - 1091922238
1610157605120*x^11 - 7036738795771404969600*x^10 - 621131294598173087040*x^9 + 627911643717314973900*x^8 + 209
401976660299863375*x^7 + 8625388473062034375*x^6 - 7622651689386500000*x^5 - 1763320756084875000*x^4 - 1618962
23238750000*x^3 - 5660707106250000*x^2)) + 22753^(1/4)*(509656134754982980140171264*x^15 + 1380318698294745571
212963840*x^14 + 1486487472085635931644235776*x^13 + 770508498313922398326616320*x^12 + 1249630989814443373361
15328*x^11 - 82495543495022435187923760*x^10 - 58955528599267342238258400*x^9 - 15477767595243310670483625*x^8
- 532651343778452286667500*x^7 + 892245329348370198662500*x^6 + 298063292793256165500000*x^5 + 47370119073953
276250000*x^4 + 3944120990334525000000*x^3 + 137906328333375000000*x^2 - sqrt(22753)*(583776555921334594525593
6*x^15 + 15810615056202811935068160*x^14 + 17009049955286781216534528*x^13 + 8785189894367114717448960*x^12 +
1388383215845925452160384*x^11 - 972840468206549851019280*x^10 - 687499322224809431095200*x^9 - 18100923515344
1909725875*x^8 - 6892716223097226502500*x^7 + 10106730756524363487500*x^6 + 3404105226535396500000*x^5 + 54214
9149604578750000*x^4 + 45171650098575000000*x^3 + 1579428325125000000*x^2)))*sqrt(217*sqrt(22753) + 45506) - 2
753113*sqrt(3)*(769218282044368679190528*x^16 + 2243553322629408647639040*x^15 + 2722019619733995420822528*x^1
4 + 1741473842011670423704320*x^13 + 535691716482687615998784*x^12 - 43747987364926664929200*x^11 - 1166769977
55758617916700*x^10 - 51170731794120633771375*x^9 - 9829950857470192751250*x^8 + 466218706688512212500*x^7 + 8
05909745662305375000*x^6 + 229732359820908750000*x^5 + 34928426088562500000*x^4 + 2913509442375000000*x^3 + 10
4928311250000000*x^2) + 968*sqrt(22753)*(11*sqrt(22753)*sqrt(3)*(11133280894478707851264*x^16 + 32472069275562
897899520*x^15 + 34961846584368741912576*x^14 + 14116960109825036605440*x^13 - 2578559534967289865472*x^12 - 4
511715066433937606400*x^11 - 1398435875141700988944*x^10 + 107026713984510716760*x^9 + 180393314877965661225*x
^8 + 42277231451151825375*x^7 - 570863512141243750*x^6 - 2156985962491425000*x^5 - 452241585490500000*x^4 - 41
499802811250000*x^3 - 1494590737500000*x^2) - 159271*sqrt(3)*(837463052302553972736*x^16 + 2442600569215782420
480*x^15 + 2614273634565545472000*x^14 + 1022870301725465856000*x^13 - 233881116511753062144*x^12 - 3604269488
47640812800*x^11 - 110377763294502560688*x^10 + 8394027644330120520*x^9 + 14260646520179433075*x^8 + 341753176
3495585125*x^7 - 307353208581250*x^6 - 158065693208475000*x^5 - 33827508643500000*x^4 - 3119304903750000*x^3 -
112339912500000*x^2)))*sqrt((25633948819248*x^4 + 21361624016040*x^3 - 2*22753^(1/4)*(sqrt(22753)*sqrt(3)*(25
43904*x^3 + 1589940*x^2 - 1591391*x - 377545) - 22753*sqrt(3)*(69696*x^3 + 43560*x^2 - 15430*x - 4475))*sqrt(1
2*x^2 + 5*x - 3)*sqrt(217*sqrt(22753) + 45506) - 7981387464633*x^2 + 33037356*sqrt(22753)*(6912*x^4 + 5760*x^3
- 228*x^2 - 595*x - 75) - 5179885750545*x + 1741704630303)/x^4) - 1513859066010863*sqrt(3)*(13576751558109854
315642880*x^16 + 45255838527032847718809600*x^15 + 56292025222029827201384448*x^14 + 2668903637274761491390464
0*x^13 - 4717355119347430651415040*x^12 - 9964876712917755398169600*x^11 - 3004327514898423306684864*x^10 + 55
1351931987193055323200*x^9 + 502630856461513673065557*x^8 + 72376829421365242444470*x^7 - 18222237302931806775
100*x^6 - 7057276971466800749000*x^5 - 514495325192847850000*x^4 + 108121429522446900000*x^3 + 234670181793390
00000*x^2 + 1564662500730000000*x + 29246767200000000) + 3725930984776*sqrt(22753)*(sqrt(22753)*sqrt(3)*(21780
2415187974448742400*x^16 + 726008050626581495808000*x^15 + 942100751373820723445760*x^14 + 5420402321380229127
16800*x^13 + 42536156642286573911040*x^12 - 121361620564714749062400*x^11 - 66346494096878268617664*x^10 - 842
3828365205592586800*x^9 + 4938327917753662627236*x^8 + 2356458373231956745185*x^7 + 307300339956542180475*x^6
- 53544679061058547750*x^5 - 25131460994000275000*x^4 - 3657330255837300000*x^3 - 206334278174250000*x^2 + 232
4736382500000*x + 522263700000000) - 22753*sqrt(3)*(3686750823320524947456*x^16 + 12289169411068416491520*x^15
+ 15098097987625062187008*x^14 + 6699232899028429885440*x^13 - 1933672081302751841280*x^12 - 3100521086130742
675200*x^11 - 894636158071089054528*x^10 + 219693320035867436400*x^9 + 195902528934273209964*x^8 + 32158375953
240369315*x^7 - 8982246266755377975*x^6 - 4350253576672402250*x^5 - 439537219633475000*x^4 + 92790568452300000
*x^3 + 29873937449250000*x^2 + 3122149117500000*x + 119863800000000)))/(3273904261412608177668096*x^16 + 10913
014204708693925560320*x^15 + 13145192753445784215797760*x^14 + 5184286401994276237516800*x^13 - 24704696150597
33328056832*x^12 - 2910634018693345878382080*x^11 - 595321320958698206129472*x^10 + 310935095761947268053600*x
^9 + 169217713571892642030717*x^8 + 11320598297034116794320*x^7 - 10774586517248316715600*x^6 - 29725771733731
56744000*x^5 - 85333313258014100000*x^4 + 87760768027346400000*x^3 + 17605709885484000000*x^2 + 14390119648800
00000*x + 45347015700000000)) + 1/651*sqrt(217)*log((16112016*x^4 + 13426680*x^3 - 4*sqrt(217)*(76320*x^3 + 47
700*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

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -2*1/3/sqrt(217)*ln(abs(24*x+5-sqrt(217)
)/abs(24*x+5+sqrt(217)))-(1/22753*sqrt(22753*(2*sqrt(22753)-217))*ln((9453972458735129516744113708531264794407
905484097024*x*sqrt(22753)+1370826006516593779927896487737033395189146295194068480*x+1969577595569818649321690
35594401349883498030918688*sqrt(2*(sqrt(22753)+145))*sqrt(22753)+285588751357623704151645101611881957331072144
83209760*sqrt(2*(sqrt(22753)+145))+1969577595569818649321690355944013498834980309186880*sqrt(22753)[phone]
57623704151645101611881957331072144832097600)*(9453972458735129516744113708531264794407905484097024*x*sqrt(227
53)+1370826006516593779927896487737033395189146295194068480*x+196957759556981864932169035594401349883498030918
688*sqrt(2*(sqrt(22753)+145))*sqrt(22753)+28558875135762370415164510161188195733107214483209760*sqrt(2*(sqrt(2
2753)+145))+1969577595569818649321690355944013498834980309186880*sqrt(22753)+285588751357623704151645101611881
957331072144832097600)+4726986229367564758372056854265632397203952742048512*sqrt(2*(sqrt(22753)+145))*sqrt(3)*
4726986229367564758372056854265632397203952742048512*sqrt(2*(sqrt(22753)+145))*sqrt(3))-2/22753*sqrt(22753*(2*
sqrt(22753)-217))*242/22753*sqrt(3)/(4/22753*sqrt(22753)-434/22753)*atan((822831873418741599297009319826345466
799848262527058008105795284587008*sqrt(2*(sqrt(22753)+145))*sqrt(22753)+11931062164571753189806635137482009268
5977998066423411175340316265116160*sqrt(2*(sqrt(22753)+145))+2266895950642754055197168817080842257148490796251
62795294394388798817152+8228318734187415992970093198263454667998482625270580081057952845870080*sqrt(22753)+966
416621392899913460946632040116701144930901039071316459008773852344448+(394959299240995967662564473516645824063
92716601298784389078173660176384*sqrt(22753)+57269098389944415311071848659913644489269439071883237364163351807
25575680)*x)*1/19747964962049798383128223675832291203196358300649392194539086830088192/sqrt(6*(sqrt(22753)+145
)))-1/22753*sqrt(22753*(2*sqrt(22753)-217))*ln((9453972458735129516744113708531264794407905484097024*x*sqrt(22
753)+1370826006516593779927896487737033395189146295194068480*x-19695775955698186493216903559440134988349803091
8688*sqrt(2*(sqrt(22753)+145))*sqrt(22753)-28558875135762370415164510161188195733107214483209760*sqrt(2*(sqrt(
22753)+145))+1969577595569818649321690355944013498834980309186880*sqrt(22753)+28558875135762370415164510161188
1957331072144832097600)*(9453972458735129516744113708531264794407905484097024*x*sqrt(22753)+137082600651659377
9927896487737033395189146295194068480*x-196957759556981864932169035594401349883498030918688*sqrt(2*(sqrt(22753
)+145))*sqrt(22753)-28558875135762370415164510161188195733107214483209760*sqrt(2*(sqrt(22753)+145))[phone]
569818649321690355944013498834980309186880*sqrt(22753)+285588751357623704151645101611881957331072144832097600)
+4726986229367564758372056854265632397203952742048512*sqrt(2*(sqrt(22753)+145))*sqrt(3)*4726986229367564758372
056854265632397203952742048512*sqrt(2*(sqrt(22753)+145))*sqrt(3))-2/22753*sqrt(22753*(2*sqrt(22753)-217))*242/
22753*sqrt(3)/(4/22753*sqrt(22753)-434/22753)*atan((-822831873418741599297009319826345466799848262527058008105
795284587008*sqrt(2*(sqrt(22753)+145))*sqrt(22753)-11931062164571753189806635137482009268597799806642341117534
0316265116160*sqrt(2*(sqrt(22753)+145))-2600263002207864945667340701945672000846798266288632063670994459751137
92+8228318734187415992970093198263454667998482625270580081057952845870080*sqrt(22753)+145313251667796181354739
7583942768126944459807293097318120502608626275392+(39495929924099596766256447351664582406392716601298784389078
173660176384*sqrt(22753)+5726909838994441531107184865991364448926943907188323736416335180725575680)*x)*1/19747
964962049798383128223675832291203196358300649392194539086830088192/sqrt(6*(sqrt(22753)+145))))/3-1728*x/1728-2
*(sqrt(3)/3/sqrt(651)*ln(abs(-5*sqrt(3)-12+12*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)-sqrt(651))/abs(-5*sqrt(3)-12+12
*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)+sqrt(651)))-sqrt(3)/3/sqrt(651)*ln(abs(-5*sqrt(3)+12+12*(sqrt(12*x^2+5*x-3)-
2*sqrt(3)*x)-sqrt(651))/(-5*sqrt(3)+12+12*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)+sqrt(651)))+(4877896607655611412062
65952561386871968325765396233687024774475214029707307340137482802624344555842785954943498861103126786933977133
4211907262284122307027711813484544/22753*sqrt(22753*(2*sqrt(22753)-217))*ln((sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)*(
sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)++infinity*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x))-4
87789660765561141206265952561386871968325765396233687024774475214029707307340137482802624344555842785954943498
8611031267869339771334211907262284122307027711813484544/22753*sqrt(22753*(2*sqrt(22753)-217))*ln((sqrt(12*x^2+
5*x-3)-2*sqrt(3)*x)*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)++infinity*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)*(sqrt(12*x^2+5
*x-3)-2*sqrt(3)*x))+undef)/29267379645933668472375957153683212318099545923774021221486468512841782438440408248
968157460673350567157296609931666187607216038628005271443573704733842166270880907264+(487789660765561141206265
95256138687196832576539623368702477447521402970730734013748280262434455584278595494349886110312678693397713342
11907262284122307027711813484544/22753*sqrt(22753*(2*sqrt(22753)-217))*ln((sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)*(sq
rt(12*x^2+5*x-3)-2*sqrt(3)*x)++infinity*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x))-487
78966076556114120626595256138687196832576539623368702477447521402970730734013748280262434455584278595494349886
11031267869339771334211907262284122307027711813484544/22753*sqrt(22753*(2*sqrt(22753)-217))*ln((sqrt(12*x^2+5*
x-3)-2*sqrt(3)*x)*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)++infinity*(sqrt(12*x^2+5*x-3)-2*sqrt(3)*x)*(sqrt(12*x^2+5*x
-3)-2*sqrt(3)*x))+undef)/2926737964593366847237595715368321231809954592377402122148646851284178243844040824896
8157460673350567157296609931666187607216038628005271443573704733842166270880907264)

________________________________________________________________________________________

maple [B]  time = 7.30, size = 544, normalized size = 2.34

method	result	size
trager	\(-x -\frac {4 \RootOf \left (\textit {\_Z}^{2}-217\right ) \ln \left (\frac {24 \RootOf \left (\textit {\_Z}^{2}-217\right ) x +217 \sqrt {12 x^{2}+5 x -3}+5 \RootOf \left (\textit {\_Z}^{2}-217\right )}{\RootOf \left (\textit {\_Z}^{2}-217\right ) x -5 x +8}\right )}{651}+\frac {4 \RootOf \left (\textit {\_Z}^{2}[phone] \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) \ln \left (-\frac {88776 x \RootOf \left (\textit {\_Z}^{2}[phone] \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+5119425 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2} \sqrt {12 x^{2}+5 x -3}+18495 \RootOf \left (\textit {\_Z}^{2}[phone] \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+24 \RootOf \left (\textit {\_Z}^{2}[phone] \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) x -45506 \sqrt {12 x^{2}+5 x -3}+5 \RootOf \left (\textit {\_Z}^{2}[phone] \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right )}{3474 x \RootOf \left (\textit {\_Z}^{2}[phone] \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+1023885 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{2}[phone] \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2}+4937401\right ) x -970 x +1694}\right )}{68259}+4 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {2019920328 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{3} x +1706475 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2} \sqrt {12 x^{2}+5 x -3}[phone] \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{3}+1594416 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right ) x +16977 \sqrt {12 x^{2}+5 x -3}+332170 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )}{237131766 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{3} x +1023885 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right )^{2} x +46509 \RootOf \left (1842993 \textit {\_Z}^{4}+1953 \textit {\_Z}^{2}+1\right ) x +2055 x -1694}\right )\)	\(544\)
default	\(\text {Expression too large to display}\)	\(2076\)

Verification of antiderivative is not currently implemented for this CAS.

[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))+4/68259*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+5119425*RootOf(
1842993*_Z^4+1953*_Z^2+1)^2*(12*x^2+5*x-3)^(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+24*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+49
37401)*x-45506*(12*x^2+5*x-3)^(1/2)+5*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+4937401))/(347
4*x*RootOf(_Z^2+4659291081*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+4937401)*RootOf(1842993*_Z^4+1953*_Z^2+1)^2+1023
885*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))+4*RootOf(1842993*_Z^4+1953*_Z^2+1)*ln((2019920328*RootOf(1842993*_Z^4+1953*_Z^2+1)^3*x+1706475*
RootOf(1842993*_Z^4+1953*_Z^2+1)^2*(12*x^2+5*x-3)^(1/2)[phone]*RootOf(1842993*_Z^4+1953*_Z^2+1)^3+1594416*R
ootOf(1842993*_Z^4+1953*_Z^2+1)*x+16977*(12*x^2+5*x-3)^(1/2)+332170*RootOf(1842993*_Z^4+1953*_Z^2+1))/(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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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