3.22.10 \(\int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx\)
Optimal. Leaf size=108 \[ \frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} 10^{2/3}}-\frac {\log (3 x+2)}{6\ 10^{2/3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 108, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used =
{750} \begin {gather*} \frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} 10^{2/3}}-\frac {\log (3 x+2)}{6\ 10^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]
[Out]
-ArcTan[1/Sqrt[3] + (2^(2/3)*(8 - 3*x))/(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1/3))]/(3*Sqrt[3]*10^(2/3)) - L
og[2 + 3*x]/(6*10^(2/3)) + Log[216 - 81*x - 27*10^(1/3)*(52 - 54*x + 27*x^2)^(1/3)]/(6*10^(2/3))
Rule 750
Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[3*c*e^2*(2*c*
d - b*e), 3]}, -Simp[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*(c*d - b*e - c*e*x))/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/3
))])/q^2, x] + (-Simp[(3*c*e*Log[d + e*x])/(2*q^2), x] + Simp[(3*c*e*Log[c*d - b*e - c*e*x - q*(a + b*x + c*x^
2)^(1/3)])/(2*q^2), x])] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && EqQ[c^2*d^2 - b*c*d*e + b^2*e^
2 - 3*a*c*e^2, 0] && PosQ[c*e^2*(2*c*d - b*e)]
Rubi steps
\begin {align*} \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{3 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{6\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{6\ 10^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.07, size = 126, normalized size = 1.17 \begin {gather*} -\frac {\sqrt [3]{\frac {9 x-5 i \sqrt {3}-9}{3 x+2}} \sqrt [3]{\frac {9 x+5 i \sqrt {3}-9}{3 x+2}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {15-5 i \sqrt {3}}{9 x+6},\frac {15+5 i \sqrt {3}}{9 x+6}\right )}{2\ 3^{2/3} \sqrt [3]{27 x^2-54 x+52}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]
[Out]
-1/2*(((-9 - (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((-9 + (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3,
1/3, 1/3, 5/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)])/(3^(2/3)*(52 - 54*x + 27*x^2)^
(1/3))
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.28, size = 212, normalized size = 1.96 \begin {gather*} \frac {\log \left (10 \sqrt [3]{27 x^2-54 x+52}+3\ 10^{2/3} x-8\ 10^{2/3}\right )}{9\ 10^{2/3}}-\frac {\log \left (-9 \sqrt [3]{10} x^2-10 \left (27 x^2-54 x+52\right )^{2/3}+\left (3\ 10^{2/3} x-8\ 10^{2/3}\right ) \sqrt [3]{27 x^2-54 x+52}+48 \sqrt [3]{10} x-64 \sqrt [3]{10}\right )}{18\ 10^{2/3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{27 x^2-54 x+52}}{\sqrt {3}}-\frac {2^{2/3} \sqrt {3} x}{\sqrt [3]{5}}+\frac {8\ 2^{2/3}}{\sqrt {3} \sqrt [3]{5}}}{\sqrt [3]{27 x^2-54 x+52}}\right )}{3 \sqrt {3} 10^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]
[Out]
-1/3*ArcTan[((8*2^(2/3))/(Sqrt[3]*5^(1/3)) - (2^(2/3)*Sqrt[3]*x)/5^(1/3) + (52 - 54*x + 27*x^2)^(1/3)/Sqrt[3])
/(52 - 54*x + 27*x^2)^(1/3)]/(Sqrt[3]*10^(2/3)) + Log[-8*10^(2/3) + 3*10^(2/3)*x + 10*(52 - 54*x + 27*x^2)^(1/
3)]/(9*10^(2/3)) - Log[-64*10^(1/3) + 48*10^(1/3)*x - 9*10^(1/3)*x^2 + (-8*10^(2/3) + 3*10^(2/3)*x)*(52 - 54*x
+ 27*x^2)^(1/3) - 10*(52 - 54*x + 27*x^2)^(2/3)]/(18*10^(2/3))
________________________________________________________________________________________
fricas [B] time = 3.91, size = 214, normalized size = 1.98 \begin {gather*} -\frac {1}{90} \cdot 100^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {100^{\frac {1}{6}} {\left (2 \cdot 100^{\frac {2}{3}} \sqrt {3} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {2}{3}} {\left (3 \, x - 8\right )} + 100^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} + 20 \, \sqrt {3} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (9 \, x^{2} - 48 \, x + 64\right )}\right )}}{90 \, {\left (9 \, x^{3} - 162 \, x^{2} + 372 \, x - 344\right )}}\right ) - \frac {1}{1800} \cdot 100^{\frac {2}{3}} \log \left (\frac {100^{\frac {2}{3}} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {2}{3}} + 100^{\frac {1}{3}} {\left (9 \, x^{2} - 48 \, x + 64\right )} - 10 \, {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x - 8\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac {1}{900} \cdot 100^{\frac {2}{3}} \log \left (\frac {100^{\frac {1}{3}} {\left (3 \, x - 8\right )} + 10 \, {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}}{3 \, x + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)/(27*x^2-54*x+52)^(1/3),x, algorithm="fricas")
[Out]
-1/90*100^(1/6)*sqrt(3)*arctan(1/90*100^(1/6)*(2*100^(2/3)*sqrt(3)*(27*x^2 - 54*x + 52)^(2/3)*(3*x - 8) + 100^
(1/3)*sqrt(3)*(27*x^3 + 54*x^2 + 36*x + 8) + 20*sqrt(3)*(27*x^2 - 54*x + 52)^(1/3)*(9*x^2 - 48*x + 64))/(9*x^3
- 162*x^2 + 372*x - 344)) - 1/1800*100^(2/3)*log((100^(2/3)*(27*x^2 - 54*x + 52)^(2/3) + 100^(1/3)*(9*x^2 - 4
8*x + 64) - 10*(27*x^2 - 54*x + 52)^(1/3)*(3*x - 8))/(9*x^2 + 12*x + 4)) + 1/900*100^(2/3)*log((100^(1/3)*(3*x
- 8) + 10*(27*x^2 - 54*x + 52)^(1/3))/(3*x + 2))
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)/(27*x^2-54*x+52)^(1/3),x, algorithm="giac")
[Out]
integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)), x)
________________________________________________________________________________________
maple [C] time = 25.95, size = 2201, normalized size = 20.38 \begin {gather*} \text {Expression too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(3*x+2)/(27*x^2-54*x+52)^(1/3),x)
[Out]
1/3*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*ln(-(321317013531015450*RootOf(RootOf(_Z^3-10)^2+
30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^3+1201310549293725*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf
(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^3+593200640364951600*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+90
0*_Z^2)^2*RootOf(_Z^3-10)^2*x^2+2217804091003800*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*Root
Of(_Z^3-10)^3*x^2+35356252380606720*(27*x^2-54*x+52)^(2/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*
_Z^2)*RootOf(_Z^3-10)^2*x+527289458102179200*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf
(_Z^3-10)^2*x+1971381414225600*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x-94
283339681617920*(27*x^2-54*x+52)^(2/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10
)^2-106068757141820160*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(
_Z^3-10)*x^2-380074665718431*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x^2+565700038089707520*(27*x^2-54*x+52)^
(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)*x+212563562797440990*RootOf(Roo
tOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^3+2027064883831632*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x
+794713132609695*RootOf(_Z^3-10)*x^3-754266717452943360*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*
RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)-2771565214149579420*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+9
00*_Z^2)*x^2-2702753178442176*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2-10362073558523310*RootOf(_Z^3-10)*x^2+1
266915552394770*(27*x^2-54*x+52)^(2/3)*x+6676802763218844120*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+90
0*_Z^2)*x+24962617157631660*RootOf(_Z^3-10)*x-3378441473052720*(27*x^2-54*x+52)^(2/3)-6093610857938239440*Root
Of(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)-22782232806434920*RootOf(_Z^3-10))/(3*x+2)^3)-1/90*ln(-(3
21317013531015450*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^3+95092565684
06790*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^3+593200640364951600*RootOf
(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^2+17555550587827920*RootOf(RootOf(_Z^
3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^2-35356252380606720*(27*x^2-54*x+52)^(2/3)*RootOf(
RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^2*x+527289458102179200*RootOf(RootOf(_Z^3-10
)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x+15604933855847040*RootOf(RootOf(_Z^3-10)^2+30*_Z*Roo
tOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x+94283339681617920*(27*x^2-54*x+52)^(2/3)*RootOf(RootOf(_Z^3-10)^2+3
0*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^2+106068757141820160*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-
10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)*x^2+3155550572342241*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-
10)^2*x^2-565700038089707520*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*R
ootOf(_Z^3-10)*x-105457891620435840*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^3-1682960305249
1952*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x-3120986771169408*RootOf(_Z^3-10)*x^3+754266717452943360*(27*x^
2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)+2969298760937896620*
RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^2+22439470736655936*(27*x^2-54*x+52)^(1/3)*RootOf(_
Z^3-10)^2+87875283775738644*RootOf(_Z^3-10)*x^2-10518501907807470*(27*x^2-54*x+52)^(2/3)*x-6501039610518117720
*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x-192395830331047464*RootOf(_Z^3-10)*x+2804933842081
9920*(27*x^2-54*x+52)^(2/3)+6093610857938239440*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)+18033
8129124839728*RootOf(_Z^3-10))/(3*x+2)^3)*RootOf(_Z^3-10)-1/3*ln(-(321317013531015450*RootOf(RootOf(_Z^3-10)^2
+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^3+9509256568406790*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootO
f(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^3+593200640364951600*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+9
00*_Z^2)^2*RootOf(_Z^3-10)^2*x^2+17555550587827920*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*Ro
otOf(_Z^3-10)^3*x^2-35356252380606720*(27*x^2-54*x+52)^(2/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+90
0*_Z^2)*RootOf(_Z^3-10)^2*x+527289458102179200*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*Root
Of(_Z^3-10)^2*x+15604933855847040*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x
+94283339681617920*(27*x^2-54*x+52)^(2/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3
-10)^2+106068757141820160*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*Root
Of(_Z^3-10)*x^2+3155550572342241*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x^2-565700038089707520*(27*x^2-54*x+
52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)*x-105457891620435840*RootOf
(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^3-16829603052491952*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10
)^2*x-3120986771169408*RootOf(_Z^3-10)*x^3+754266717452943360*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+
30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)+2969298760937896620*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3
-10)+900*_Z^2)*x^2+22439470736655936*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2+87875283775738644*RootOf(_Z^3-10
)*x^2-10518501907807470*(27*x^2-54*x+52)^(2/3)*x-6501039610518117720*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^
3-10)+900*_Z^2)*x-192395830331047464*RootOf(_Z^3-10)*x+28049338420819920*(27*x^2-54*x+52)^(2/3)+60936108579382
39440*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)+180338129124839728*RootOf(_Z^3-10))/(3*x+2)^3)*
RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)/(27*x^2-54*x+52)^(1/3),x, algorithm="maxima")
[Out]
integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (3\,x+2\right )\,{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((3*x + 2)*(27*x^2 - 54*x + 52)^(1/3)),x)
[Out]
int(1/((3*x + 2)*(27*x^2 - 54*x + 52)^(1/3)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)/(27*x**2-54*x+52)**(1/3),x)
[Out]
Integral(1/((3*x + 2)*(27*x**2 - 54*x + 52)**(1/3)), x)
________________________________________________________________________________________