\(\int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx\) [232]

Optimal result

Integrand size = 17, antiderivative size = 75 \[ \int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {1+\frac {2 (-3+x)}{\sqrt [3]{9+3 x-5 x^2+x^3}}}{\sqrt {3}}\right )-\frac {1}{2} \log (1+x)-\frac {3}{2} \log \left (1-\frac {-3+x}{\sqrt [3]{9+3 x-5 x^2+x^3}}\right ) \] Output:

-1/2*ln(1+x)-3/2*ln(1+(3-x)/(x^3-5*x^2+3*x+9)^(1/3))+arctan(1/3*(1+2*(-3+x 
)/(x^3-5*x^2+3*x+9)^(1/3))*3^(1/2))*3^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx=\frac {(-3+x)^{2/3} \sqrt [3]{1+x} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+x}}{2 \sqrt [3]{-3+x}+\sqrt [3]{1+x}}\right )-2 \log \left (\sqrt [3]{-3+x}-\sqrt [3]{1+x}\right )+\log \left ((-3+x)^{2/3}+\sqrt [3]{-3+x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{2 \sqrt [3]{(-3+x)^2 (1+x)}} \] Input:

Integrate[(9 + 3*x - 5*x^2 + x^3)^(-1/3),x]
 

Output:

((-3 + x)^(2/3)*(1 + x)^(1/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(1 + x)^(1/3))/( 
2*(-3 + x)^(1/3) + (1 + x)^(1/3))] - 2*Log[(-3 + x)^(1/3) - (1 + x)^(1/3)] 
 + Log[(-3 + x)^(2/3) + (-3 + x)^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)]))/(2 
*((-3 + x)^2*(1 + x))^(1/3))
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2480, 27, 71}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\sqrt [3]{x^3-5 x^2+3 x+9}} \, dx\)

\(\Big \downarrow \) 2480

\(\displaystyle \frac {16\ 2^{2/3} (x-3)^{2/3} \sqrt [3]{x+1} \int \frac {1}{16\ 2^{2/3} (x-3)^{2/3} \sqrt [3]{x+1}}dx}{\sqrt [3]{x^3-5 x^2+3 x+9}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(x-3)^{2/3} \sqrt [3]{x+1} \int \frac {1}{(x-3)^{2/3} \sqrt [3]{x+1}}dx}{\sqrt [3]{x^3-5 x^2+3 x+9}}\)

\(\Big \downarrow \) 71

\(\displaystyle \frac {(x-3)^{2/3} \sqrt [3]{x+1} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x-3}}+\frac {1}{\sqrt {3}}\right )-\frac {1}{2} \log (x-3)-\frac {3}{2} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x-3}}-1\right )\right )}{\sqrt [3]{x^3-5 x^2+3 x+9}}\)

Input:

Int[(9 + 3*x - 5*x^2 + x^3)^(-1/3),x]
 

Output:

((-3 + x)^(2/3)*(1 + x)^(1/3)*(-(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + x)^(1/ 
3))/(Sqrt[3]*(-3 + x)^(1/3))]) - Log[-3 + x]/2 - (3*Log[-1 + (1 + x)^(1/3) 
/(-3 + x)^(1/3)])/2))/(9 + 3*x - 5*x^2 + x^3)^(1/3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 71

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> 
 With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( 
Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + 
 b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
 

rule 2480

Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] 
, c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Simp[Px^p/((c^3 - 4*b*c*d + 9* 
a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3*b*d)*x)^(2*p))   Int 
[(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3* 
b*d)*x)^(2*p), x], x] /; EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 27* 
a^2*d^2, 0] && NeQ[c^2 - 3*b*d, 0]] /; FreeQ[p, x] && PolyQ[Px, x, 3] &&  ! 
IntegerQ[p]
 

Maple [C] (verified)

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

Time = 0.36 (sec) , antiderivative size = 670, normalized size of antiderivative = 8.93

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \ln \left (-\frac {20 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x -60 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x -33 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}-216 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}-81 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}-216 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -36 x^{2}+648 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+315 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )+360 x -756}{-3+x}\right )}{3}-\frac {\ln \left (\frac {-20 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x +60 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x +87 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}+135 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}-81 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+135 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x -366 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -45 x^{2}-405 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+315 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )+198 x -189}{-3+x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )}{3}+\ln \left (\frac {-20 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x +60 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x +87 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}+135 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}-81 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+135 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}} x -366 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -45 x^{2}-405 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}+315 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )+198 x -189}{-3+x}\right )\)	\(670\)

Input:

int(1/(x^3-5*x^2+3*x+9)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

1/3*RootOf(_Z^2-3*_Z+9)*ln(-(20*RootOf(_Z^2-3*_Z+9)^2*x^2+27*RootOf(_Z^2-3 
*_Z+9)*(x^3-5*x^2+3*x+9)^(2/3)+27*RootOf(_Z^2-3*_Z+9)*(x^3-5*x^2+3*x+9)^(1 
/3)*x-60*RootOf(_Z^2-3*_Z+9)^2*x-33*RootOf(_Z^2-3*_Z+9)*x^2-216*(x^3-5*x^2 
+3*x+9)^(2/3)-81*RootOf(_Z^2-3*_Z+9)*(x^3-5*x^2+3*x+9)^(1/3)-216*(x^3-5*x^ 
2+3*x+9)^(1/3)*x-6*RootOf(_Z^2-3*_Z+9)*x-36*x^2+648*(x^3-5*x^2+3*x+9)^(1/3 
)+315*RootOf(_Z^2-3*_Z+9)+360*x-756)/(-3+x))-1/3*ln((-20*RootOf(_Z^2-3*_Z+ 
9)^2*x^2+27*RootOf(_Z^2-3*_Z+9)*(x^3-5*x^2+3*x+9)^(2/3)+27*RootOf(_Z^2-3*_ 
Z+9)*(x^3-5*x^2+3*x+9)^(1/3)*x+60*RootOf(_Z^2-3*_Z+9)^2*x+87*RootOf(_Z^2-3 
*_Z+9)*x^2+135*(x^3-5*x^2+3*x+9)^(2/3)-81*RootOf(_Z^2-3*_Z+9)*(x^3-5*x^2+3 
*x+9)^(1/3)+135*(x^3-5*x^2+3*x+9)^(1/3)*x-366*RootOf(_Z^2-3*_Z+9)*x-45*x^2 
-405*(x^3-5*x^2+3*x+9)^(1/3)+315*RootOf(_Z^2-3*_Z+9)+198*x-189)/(-3+x))*Ro 
otOf(_Z^2-3*_Z+9)+ln((-20*RootOf(_Z^2-3*_Z+9)^2*x^2+27*RootOf(_Z^2-3*_Z+9) 
*(x^3-5*x^2+3*x+9)^(2/3)+27*RootOf(_Z^2-3*_Z+9)*(x^3-5*x^2+3*x+9)^(1/3)*x+ 
60*RootOf(_Z^2-3*_Z+9)^2*x+87*RootOf(_Z^2-3*_Z+9)*x^2+135*(x^3-5*x^2+3*x+9 
)^(2/3)-81*RootOf(_Z^2-3*_Z+9)*(x^3-5*x^2+3*x+9)^(1/3)+135*(x^3-5*x^2+3*x+ 
9)^(1/3)*x-366*RootOf(_Z^2-3*_Z+9)*x-45*x^2-405*(x^3-5*x^2+3*x+9)^(1/3)+31 
5*RootOf(_Z^2-3*_Z+9)+198*x-189)/(-3+x))
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.71 \[ \int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x - 3\right )} + 2 \, \sqrt {3} {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}}}{3 \, {\left (x - 3\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}} {\left (x - 3\right )} - 6 \, x + {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {2}{3}} + 9}{x^{2} - 6 \, x + 9}\right ) - \log \left (-\frac {x - {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}} - 3}{x - 3}\right ) \] Input:

integrate(1/(x^3-5*x^2+3*x+9)^(1/3),x, algorithm="fricas")
 

Output:

-sqrt(3)*arctan(1/3*(sqrt(3)*(x - 3) + 2*sqrt(3)*(x^3 - 5*x^2 + 3*x + 9)^( 
1/3))/(x - 3)) + 1/2*log((x^2 + (x^3 - 5*x^2 + 3*x + 9)^(1/3)*(x - 3) - 6* 
x + (x^3 - 5*x^2 + 3*x + 9)^(2/3) + 9)/(x^2 - 6*x + 9)) - log(-(x - (x^3 - 
 5*x^2 + 3*x + 9)^(1/3) - 3)/(x - 3))
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{3} - 5 x^{2} + 3 x + 9}}\, dx \] Input:

integrate(1/(x**3-5*x**2+3*x+9)**(1/3),x)
 

Output:

Integral((x**3 - 5*x**2 + 3*x + 9)**(-1/3), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(x^3-5*x^2+3*x+9)^(1/3),x, algorithm="maxima")
 

Output:

integrate((x^3 - 5*x^2 + 3*x + 9)^(-1/3), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(x^3-5*x^2+3*x+9)^(1/3),x, algorithm="giac")
 

Output:

integrate((x^3 - 5*x^2 + 3*x + 9)^(-1/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx=\int \frac {1}{{\left (x^3-5\,x^2+3\,x+9\right )}^{1/3}} \,d x \] Input:

int(1/(3*x - 5*x^2 + x^3 + 9)^(1/3),x)
 

Output:

int(1/(3*x - 5*x^2 + x^3 + 9)^(1/3), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx=\int \frac {1}{\left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}}d x \] Input:

int(1/(x^3-5*x^2+3*x+9)^(1/3),x)
                                                                                    
                                                                                    
 

Output:

int(1/(x**3 - 5*x**2 + 3*x + 9)**(1/3),x)