3.25.32 \(\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} (-12+52 x+9 x^2)} \, dx\)
Optimal. Leaf size=197 \[ \frac {\log \left (14 \sqrt [3]{3 x^2+4}+3 \sqrt [3]{14} x-10 \sqrt [3]{14}\right )}{14 \sqrt [3]{14}}-\frac {\log \left (9\ 14^{2/3} x^2+196 \left (3 x^2+4\right )^{2/3}+\left (140 \sqrt [3]{14}-42 \sqrt [3]{14} x\right ) \sqrt [3]{3 x^2+4}-60\ 14^{2/3} x+100\ 14^{2/3}\right )}{28 \sqrt [3]{14}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{3 x^2+4}}{\sqrt {3}}-\frac {\sqrt [3]{2} \sqrt {3} x}{7^{2/3}}+\frac {10 \sqrt [3]{2}}{\sqrt {3} 7^{2/3}}}{\sqrt [3]{3 x^2+4}}\right )}{14 \sqrt [3]{14}} \]
________________________________________________________________________________________
Rubi [F] time = 0.01, 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 {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
[Out]
Defer[Int][(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)), x]
Rubi steps
\begin {align*} \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx &=\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 1.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
[Out]
Integrate[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)), x]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.29, size = 197, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {10 \sqrt [3]{2}}{\sqrt {3} 7^{2/3}}-\frac {\sqrt [3]{2} \sqrt {3} x}{7^{2/3}}+\frac {\sqrt [3]{4+3 x^2}}{\sqrt {3}}}{\sqrt [3]{4+3 x^2}}\right )}{14 \sqrt [3]{14}}+\frac {\log \left (-10 \sqrt [3]{14}+3 \sqrt [3]{14} x+14 \sqrt [3]{4+3 x^2}\right )}{14 \sqrt [3]{14}}-\frac {\log \left (100\ 14^{2/3}-60\ 14^{2/3} x+9\ 14^{2/3} x^2+\left (140 \sqrt [3]{14}-42 \sqrt [3]{14} x\right ) \sqrt [3]{4+3 x^2}+196 \left (4+3 x^2\right )^{2/3}\right )}{28 \sqrt [3]{14}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
[Out]
-1/14*(Sqrt[3]*ArcTan[((10*2^(1/3))/(Sqrt[3]*7^(2/3)) - (2^(1/3)*Sqrt[3]*x)/7^(2/3) + (4 + 3*x^2)^(1/3)/Sqrt[3
])/(4 + 3*x^2)^(1/3)])/14^(1/3) + Log[-10*14^(1/3) + 3*14^(1/3)*x + 14*(4 + 3*x^2)^(1/3)]/(14*14^(1/3)) - Log[
100*14^(2/3) - 60*14^(2/3)*x + 9*14^(2/3)*x^2 + (140*14^(1/3) - 42*14^(1/3)*x)*(4 + 3*x^2)^(1/3) + 196*(4 + 3*
x^2)^(2/3)]/(28*14^(1/3))
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x + 2}{{\left (9 \, x^{2} + 52 \, x - 12\right )} {\left (3 \, x^{2} + 4\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x, algorithm="giac")
[Out]
integrate((3*x + 2)/((9*x^2 + 52*x - 12)*(3*x^2 + 4)^(1/3)), x)
________________________________________________________________________________________
maple [C] time = 18.48, size = 2122, normalized size = 10.77
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(2122\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x,method=_RETURNVERBOSE)
[Out]
RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*ln((788123448*RootOf(RootOf(_Z^3-196)^2+196*_Z*R
ootOf(_Z^3-196)+38416*_Z^2)^2*RootOf(_Z^3-196)^2*x^3+12185712*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196
)+38416*_Z^2)*RootOf(_Z^3-196)^3*x^3+14011083520*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)
^2*RootOf(_Z^3-196)^2*x^2+237846000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z
^2)*RootOf(_Z^3-196)^2*x+216634880*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-1
96)^3*x^2+26270781600*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)^2*RootOf(_Z^3-196)^2*x-792
820000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)^2-713538
000*(3*x^2+4)^(1/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)*x^2+4061904
00*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)^3*x-13674087*(3*x^2+4)^(1/3)
*RootOf(_Z^3-196)^2*x^2+4756920000*(3*x^2+4)^(1/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^
2)*RootOf(_Z^3-196)*x+1793382948*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*x^3+91160580*(3
*x^2+4)^(1/3)*RootOf(_Z^3-196)^2*x+27728712*RootOf(_Z^3-196)*x^3-7928200000*(3*x^2+4)^(1/3)*RootOf(RootOf(_Z^3
-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)-35097406792*RootOf(RootOf(_Z^3-196)^2+196*_Z*Root
Of(_Z^3-196)+38416*_Z^2)*x^2+893373684*(3*x^2+4)^(2/3)*x-151934300*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2-54266484
8*RootOf(_Z^3-196)*x^2+59779431600*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*x-2977912280*
(3*x^2+4)^(2/3)+924290400*RootOf(_Z^3-196)*x-89306360416*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+384
16*_Z^2)-1380826304*RootOf(_Z^3-196))/(9*x-2)/(6+x)^2)-1/196*ln((788123448*RootOf(RootOf(_Z^3-196)^2+196*_Z*Ro
otOf(_Z^3-196)+38416*_Z^2)^2*RootOf(_Z^3-196)^2*x^3-8164674*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+
38416*_Z^2)*RootOf(_Z^3-196)^3*x^3+14011083520*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)^2
*RootOf(_Z^3-196)^2*x^2-237846000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2
)*RootOf(_Z^3-196)^2*x-145149760*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196
)^3*x^2+26270781600*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)^2*RootOf(_Z^3-196)^2*x+79282
0000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)^2+71353800
0*(3*x^2+4)^(1/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)*x^2-272155800
*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)^3*x-10033587*(3*x^2+4)^(1/3)*R
ootOf(_Z^3-196)^2*x^2-4756920000*(3*x^2+4)^(1/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)
*RootOf(_Z^3-196)*x-1005259500*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*x^3+66890580*(3*x
^2+4)^(1/3)*RootOf(_Z^3-196)^2*x+10414125*RootOf(_Z^3-196)*x^3+7928200000*(3*x^2+4)^(1/3)*RootOf(RootOf(_Z^3-1
96)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)+49108490312*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf
(_Z^3-196)+38416*_Z^2)*x^2+655527684*(3*x^2+4)^(2/3)*x-111484300*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2-508746206*
RootOf(_Z^3-196)*x^2-33508650000*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*x-2185092280*(3
*x^2+4)^(2/3)+347137500*RootOf(_Z^3-196)*x+89306360416*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416
*_Z^2)-925181608*RootOf(_Z^3-196))/(9*x-2)/(6+x)^2)*RootOf(_Z^3-196)-ln((788123448*RootOf(RootOf(_Z^3-196)^2+1
96*_Z*RootOf(_Z^3-196)+38416*_Z^2)^2*RootOf(_Z^3-196)^2*x^3-8164674*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z
^3-196)+38416*_Z^2)*RootOf(_Z^3-196)^3*x^3+14011083520*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416
*_Z^2)^2*RootOf(_Z^3-196)^2*x^2-237846000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38
416*_Z^2)*RootOf(_Z^3-196)^2*x-145149760*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(
_Z^3-196)^3*x^2+26270781600*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)^2*RootOf(_Z^3-196)^2
*x+792820000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)^2+
713538000*(3*x^2+4)^(1/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)*x^2-2
72155800*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)^3*x-10033587*(3*x^2+4)
^(1/3)*RootOf(_Z^3-196)^2*x^2-4756920000*(3*x^2+4)^(1/3)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+384
16*_Z^2)*RootOf(_Z^3-196)*x-1005259500*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*x^3+66890
580*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2*x+10414125*RootOf(_Z^3-196)*x^3+7928200000*(3*x^2+4)^(1/3)*RootOf(RootO
f(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*RootOf(_Z^3-196)+49108490312*RootOf(RootOf(_Z^3-196)^2+196*_
Z*RootOf(_Z^3-196)+38416*_Z^2)*x^2+655527684*(3*x^2+4)^(2/3)*x-111484300*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2-50
8746206*RootOf(_Z^3-196)*x^2-33508650000*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+38416*_Z^2)*x-21850
92280*(3*x^2+4)^(2/3)+347137500*RootOf(_Z^3-196)*x+89306360416*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-19
6)+38416*_Z^2)-925181608*RootOf(_Z^3-196))/(9*x-2)/(6+x)^2)*RootOf(RootOf(_Z^3-196)^2+196*_Z*RootOf(_Z^3-196)+
38416*_Z^2)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x + 2}{{\left (9 \, x^{2} + 52 \, x - 12\right )} {\left (3 \, x^{2} + 4\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x, algorithm="maxima")
[Out]
integrate((3*x + 2)/((9*x^2 + 52*x - 12)*(3*x^2 + 4)^(1/3)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,x+2}{{\left (3\,x^2+4\right )}^{1/3}\,\left (9\,x^2+52\,x-12\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((3*x + 2)/((3*x^2 + 4)^(1/3)*(52*x + 9*x^2 - 12)),x)
[Out]
int((3*x + 2)/((3*x^2 + 4)^(1/3)*(52*x + 9*x^2 - 12)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x + 2}{\left (x + 6\right ) \left (9 x - 2\right ) \sqrt [3]{3 x^{2} + 4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+3*x)/(3*x**2+4)**(1/3)/(9*x**2+52*x-12),x)
[Out]
Integral((3*x + 2)/((x + 6)*(9*x - 2)*(3*x**2 + 4)**(1/3)), x)
________________________________________________________________________________________