\(\int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx\) [2500]

Optimal result

Integrand size = 22, antiderivative size = 108 \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{3 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{6\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{6\ 10^{2/3}} \]

[Out]

-1/60*ln(2+3*x)*10^(1/3)+1/60*ln(216-81*x-27*10^(1/3)*(27*x^2-54*x+52)^(1/3))*10^(1/3)+1/90*arctan(-1/3*3^(1/2
)-1/15*2^(2/3)*(8-3*x)*5^(2/3)/(27*x^2-54*x+52)^(1/3)*3^(1/2))*10^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {764} \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {\arctan \left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} 10^{2/3}}+\frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac {\log (3 x+2)}{6\ 10^{2/3}} \]

[In]

Int[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]

[Out]

-1/3*ArcTan[1/Sqrt[3] + (2^(2/3)*(8 - 3*x))/(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1/3))]/(Sqrt[3]*10^(2/3)) -
Log[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 764

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*} \text {integral}& = -\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 [A] (verified)

Time = 0.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {8\ 10^{2/3}-3\ 10^{2/3} x+5 \sqrt [3]{52-54 x+27 x^2}}{5 \sqrt {3} \sqrt [3]{52-54 x+27 x^2}}\right )-2 \log \left (-8 10^{2/3}+3\ 10^{2/3} x+10 \sqrt [3]{52-54 x+27 x^2}\right )+\log \left (-64 \sqrt [3]{10}+48 \sqrt [3]{10} x-9 \sqrt [3]{10} x^2+10^{2/3} (-8+3 x) \sqrt [3]{52-54 x+27 x^2}-10 \left (52-54 x+27 x^2\right )^{2/3}\right )}{18\ 10^{2/3}} \]

[In]

Integrate[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]

[Out]

-1/18*(2*Sqrt[3]*ArcTan[(8*10^(2/3) - 3*10^(2/3)*x + 5*(52 - 54*x + 27*x^2)^(1/3))/(5*Sqrt[3]*(52 - 54*x + 27*
x^2)^(1/3))] - 2*Log[-8*10^(2/3) + 3*10^(2/3)*x + 10*(52 - 54*x + 27*x^2)^(1/3)] + Log[-64*10^(1/3) + 48*10^(1
/3)*x - 9*10^(1/3)*x^2 + 10^(2/3)*(-8 + 3*x)*(52 - 54*x + 27*x^2)^(1/3) - 10*(52 - 54*x + 27*x^2)^(2/3)])/10^(
2/3)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.76 (sec) , antiderivative size = 2182, normalized size of antiderivative = 20.20

method	result	size
trager	\(\text {Expression too large to display}\)	\(2182\)

[In]

int(1/(2+3*x)/(27*x^2-54*x+52)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/90*RootOf(_Z^3-10)*ln((10710567117700515*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^
3-10)^3*x^3+36039316478811750*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^3
+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+19773354678831720*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^2+6653
4122730114000*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^2-942833396816179
20*(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)^2-380074665
718431*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x^2+94666517170267230*(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+17576315270072640*RootOf(RootOf(_Z^3-10)^2+30*_Z*R
ootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x+59141442426768000*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+
900*_Z^2)^2*RootOf(_Z^3-10)^2*x+2027064883831632*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x-504888091574758560
*(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-35152630540
14528*RootOf(_Z^3-10)*x^3-11828288485353600*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^3+10518
501907807470*(27*x^2-54*x+52)^(2/3)*x-2702753178442176*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2+67318412209967
8080*(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)[phone]
64596554*RootOf(_Z^3-10)*x^2+333040247665737300*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^2-2
8049338420819920*(27*x^2-54*x+52)^(2/3)-216701320350603924*RootOf(_Z^3-10)*x-729164700586693800*RootOf(RootOf(
_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x+203120361931274648*RootOf(_Z^3-10)+683466984193047600*RootOf(Root
Of(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2))/(2+3*x)^3)-1/90*ln(-(-10710567117700515*RootOf(RootOf(_Z^3-10)^
2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^3-285277697052203700*RootOf(RootOf(_Z^3-10)^2+30*_Z*Root
Of(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^3+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-19773354678831720*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(
_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^2-526666517634837600*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900
*_Z^2)^2*RootOf(_Z^3-10)^2*x^2-94283339681617920*(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)^2-3155550572342241*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x^2+11402239971
552930*(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^2-175
76315270072640*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x-468148015675411200
*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x+16829603052491952*(27*x^2-54*x
+52)^(1/3)*RootOf(_Z^3-10)^2*x-60811946514948960*(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-7085452093248033*RootOf(_Z^3-10)*x^3-188722168819150140*RootOf(RootOf(_Z^
3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^3+1266915552394770*(27*x^2-54*x+52)^(2/3)*x-22439470736655936*(27*x^
2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2+81082595353265280*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*Roo
tOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)+92385507138319314*RootOf(_Z^3-10)*x^2+2460703007393880120*RootOf(RootOf
(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^2-3378441473052720*(27*x^2-54*x+52)^(2/3)-222560092107294804*Roo
tOf(_Z^3-10)*x-5927924248489894320*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x+2031203619312746
48*RootOf(_Z^3-10)+5410143873745191840*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2))/(2+3*x)^3)*Ro
otOf(_Z^3-10)-1/3*ln(-(-10710567117700515*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3
-10)^3*x^3-285277697052203700*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^3
+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-19773354678831720*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^2-5266
66517634837600*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^2-94283339681617
920*(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)^2-31555505
72342241*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x^2+11402239971552930*(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^2-17576315270072640*RootOf(RootOf(_Z^3-10)^2+30*_Z
*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x-468148015675411200*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-1
0)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x+16829603052491952*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x-60811946514948
960*(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-70854520
93248033*RootOf(_Z^3-10)*x^3-188722168819150140*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^3+1
266915552394770*(27*x^2-54*x+52)^(2/3)*x-22439470736655936*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2+8108259535
3265280*(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)+923855
07138319314*RootOf(_Z^3-10)*x^2+2460703007393880120*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x
^2-3378441473052720*(27*x^2-54*x+52)^(2/3)-222560092107294804*RootOf(_Z^3-10)*x-5927924248489894320*RootOf(Roo
tOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x+203120361931274648*RootOf(_Z^3-10)+5410143873745191840*RootOf
(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2))/(2+3*x)^3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+
900*_Z^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (83) = 166\).

Time = 2.41 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.98 \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {1}{90} \cdot 100^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {100^{\frac {1}{6}} {\left (2 \cdot 100^{\frac {2}{3}} \sqrt {3} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {2}{3}} {\left (3 \, x - 8\right )} + 100^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} + 20 \, \sqrt {3} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (9 \, x^{2} - 48 \, x + 64\right )}\right )}}{90 \, {\left (9 \, x^{3} - 162 \, x^{2} + 372 \, x - 344\right )}}\right ) - \frac {1}{1800} \cdot 100^{\frac {2}{3}} \log \left (\frac {100^{\frac {2}{3}} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {2}{3}} + 100^{\frac {1}{3}} {\left (9 \, x^{2} - 48 \, x + 64\right )} - 10 \, {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x - 8\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac {1}{900} \cdot 100^{\frac {2}{3}} \log \left (\frac {100^{\frac {1}{3}} {\left (3 \, x - 8\right )} + 10 \, {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}}{3 \, x + 2}\right ) \]

[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))

Sympy [F]

\[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {1}{\left (3\,x+2\right )\,{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \]

[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)