3.1.46 \(\int \frac {a+b x+c x^2}{(1-x+x^2) \sqrt [3]{1-x^3}} \, dx\) [46]

3.1.46.1 Optimal result

Integrand size = 32, antiderivative size = 493 \[ \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {(a+b) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {c \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {(a-c) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(b+c) \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}-\frac {(a-c) \log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {(b+c) \log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {(a+b) \log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}-\frac {(a+b) \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}+\frac {(b+c) \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {(a-c) \log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )-\frac {(a+b) \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \]

output

1/24*(a+b)*ln((1-x)*(1+x)^2)*2^(2/3)-1/12*(a-c)*ln(x^3+1)*2^(2/3)-1/12*(b+ 
c)*ln(x^3+1)*2^(2/3)+1/12*(a+b)*ln(1+2^(2/3)*(1-x)^2/(-x^3+1)^(2/3)-2^(1/3 
)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)-1/6*(a+b)*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3 
))*2^(2/3)+1/4*(b+c)*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(2/3)+1/4*(a-c)*ln(-2^(1 
/3)*x-(-x^3+1)^(1/3))*2^(2/3)+1/2*c*ln(x+(-x^3+1)^(1/3))-1/8*(a+b)*ln(-1+x 
+2^(2/3)*(-x^3+1)^(1/3))*2^(2/3)+1/6*(a+b)*arctan(1/3*(1-2*2^(1/3)*(1-x)/( 
-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)+1/12*(a+b)*arctan(1/3*(1+2^(1/3)*( 
1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)-1/3*c*arctan(1/3*(1-2*x/(-x^ 
3+1)^(1/3))*3^(1/2))*3^(1/2)-1/6*(a-c)*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^ 
(1/3))*3^(1/2))*2^(2/3)*3^(1/2)+1/6*(b+c)*arctan(1/3*(1+2^(2/3)*(-x^3+1)^( 
1/3))*3^(1/2))*2^(2/3)*3^(1/2)
 

3.1.46.2 Mathematica [F]

\[ \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]

input

Integrate[(a + b*x + c*x^2)/((1 - x + x^2)*(1 - x^3)^(1/3)),x]
 

output

Integrate[(a + b*x + c*x^2)/((1 - x + x^2)*(1 - x^3)^(1/3)), x]
 

3.1.46.3 Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2583, 2009}

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.

\(\displaystyle \int \frac {a+b x+c x^2}{\left (x^2-x+1\right ) \sqrt [3]{1-x^3}} \, dx\)

\(\Big \downarrow \) 2583

\(\displaystyle \int \left (\frac {x (a+b)}{\sqrt [3]{1-x^3} \left (x^3+1\right )}+\frac {a}{\sqrt [3]{1-x^3} \left (x^3+1\right )}+\frac {x^2 (b+c)}{\sqrt [3]{1-x^3} \left (x^3+1\right )}+\frac {c x^3}{\sqrt [3]{1-x^3} \left (x^3+1\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(a+b) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {a \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}-\frac {(a+b) \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac {(a+b) \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac {(a+b) \log \left ((1-x) (x+1)^2\right )}{12 \sqrt [3]{2}}-\frac {a \log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {a \log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}+\frac {\arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right ) (b+c)}{\sqrt [3]{2} \sqrt {3}}-\frac {c \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {c \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {(b+c) \log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {(b+c) \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {c \log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {c \log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}+\frac {1}{2} c \log \left (\sqrt [3]{1-x^3}+x\right )\)

input

Int[(a + b*x + c*x^2)/((1 - x + x^2)*(1 - x^3)^(1/3)),x]
 

output

((a + b)*ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/(2^(1/ 
3)*Sqrt[3]) + ((a + b)*ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt 
[3]])/(2*2^(1/3)*Sqrt[3]) - (c*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]] 
)/Sqrt[3] - (a*ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(2^(1/ 
3)*Sqrt[3]) + (c*ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(2^( 
1/3)*Sqrt[3]) + ((b + c)*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]])/(2 
^(1/3)*Sqrt[3]) + ((a + b)*Log[(1 - x)*(1 + x)^2])/(12*2^(1/3)) - (a*Log[1 
 + x^3])/(6*2^(1/3)) + (c*Log[1 + x^3])/(6*2^(1/3)) - ((b + c)*Log[1 + x^3 
])/(6*2^(1/3)) + ((a + b)*Log[1 + (2^(2/3)*(1 - x)^2)/(1 - x^3)^(2/3) - (2 
^(1/3)*(1 - x))/(1 - x^3)^(1/3)])/(6*2^(1/3)) - ((a + b)*Log[1 + (2^(1/3)* 
(1 - x))/(1 - x^3)^(1/3)])/(3*2^(1/3)) + ((b + c)*Log[2^(1/3) - (1 - x^3)^ 
(1/3)])/(2*2^(1/3)) + (a*Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)])/(2*2^(1/3)) 
- (c*Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)])/(2*2^(1/3)) + (c*Log[x + (1 - x^ 
3)^(1/3)])/2 - ((a + b)*Log[-1 + x + 2^(2/3)*(1 - x^3)^(1/3)])/(4*2^(1/3))
 

3.1.46.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2583

Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p 
_.), x_Symbol] :> Simp[1/c^q   Int[ExpandIntegrand[(c^3 - d^3*x^3)^q*(a + b 
*x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Poly 
Q[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denomina 
tor[p], 3]
 

3.1.46.4 Maple [F]

\[\int \frac {c \,x^{2}+b x +a}{\left (x^{2}-x +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}d x\]

input

int((c*x^2+b*x+a)/(x^2-x+1)/(-x^3+1)^(1/3),x)
 

output

int((c*x^2+b*x+a)/(x^2-x+1)/(-x^3+1)^(1/3),x)
 

3.1.46.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (residue poly has multiple non-linear fac 
tors)
 

3.1.46.6 Sympy [F]

\[ \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int \frac {a + b x + c x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - x + 1\right )}\, dx \]

input

integrate((c*x**2+b*x+a)/(x**2-x+1)/(-x**3+1)**(1/3),x)
 

output

Integral((a + b*x + c*x**2)/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x**2 - x + 
1)), x)
 

3.1.46.7 Maxima [F]

\[ \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int { \frac {c x^{2} + b x + a}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}} \,d x } \]

input

integrate((c*x^2+b*x+a)/(x^2-x+1)/(-x^3+1)^(1/3),x, algorithm="maxima")
 

output

integrate((c*x^2 + b*x + a)/((-x^3 + 1)^(1/3)*(x^2 - x + 1)), x)
 

3.1.46.8 Giac [F]

\[ \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int { \frac {c x^{2} + b x + a}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}} \,d x } \]

input

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

output

integrate((c*x^2 + b*x + a)/((-x^3 + 1)^(1/3)*(x^2 - x + 1)), x)
 

3.1.46.9 Mupad [F(-1)]

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

input

int((a + b*x + c*x^2)/((1 - x^3)^(1/3)*(x^2 - x + 1)),x)
 

output

int((a + b*x + c*x^2)/((1 - x^3)^(1/3)*(x^2 - x + 1)), x)
 

3.1.46.10 Reduce [F]

\[ \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\left (\int \frac {x^{2}}{\left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}d x \right ) c +\left (\int \frac {x}{\left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}d x \right ) b +\left (\int \frac {1}{\left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}d x \right ) a \]

input

int((a + b*x + c*x**2)/(( - x**3 + 1)**(1/3)*(x**2 - x + 1)),x)
 

output

int(x**2/(( - x**3 + 1)**(1/3)*x**2 - ( - x**3 + 1)**(1/3)*x + ( - x**3 + 
1)**(1/3)),x)*c + int(x/(( - x**3 + 1)**(1/3)*x**2 - ( - x**3 + 1)**(1/3)* 
x + ( - x**3 + 1)**(1/3)),x)*b + int(1/(( - x**3 + 1)**(1/3)*x**2 - ( - x* 
*3 + 1)**(1/3)*x + ( - x**3 + 1)**(1/3)),x)*a