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

Optimal. Leaf size=493 \[ \frac{(a+b) \log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}-\frac{(a+b) \log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac{(a+b) \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac{(a+b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{(a+b) \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{(a+b) \log \left ((1-x) (x+1)^2\right )}{12 \sqrt [3]{2}}-\frac{(a-c) \log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{(a-c) \log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}-\frac{(a-c) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{(b+c) \log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{(b+c) \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{(b+c) \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{1}{2} c \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{c \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

((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 - c)*ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]) + ((b + c)*ArcTa
n[(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 - 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 - 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))

________________________________________________________________________________________

Rubi [C]  time = 0.846166, antiderivative size = 576, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {6728, 239, 2148} \[ -\frac{\log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x-i \sqrt{3}+1\right ) \left (3 i b-\sqrt{3} \left (2 a+b-i \sqrt{3} c-c\right )\right )}{4 \sqrt [3]{2} \left (\sqrt{3}+i\right )}-\frac{\log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x+i \sqrt{3}+1\right ) \left (\sqrt{3} \left (2 a+b+i \sqrt{3} c-c\right )+3 i b\right )}{4 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}-\frac{\tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (2 x-i \sqrt{3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right ) \left (2 a-i \sqrt{3} b+b-\left (1+i \sqrt{3}\right ) c\right )}{2 \sqrt [3]{2} \left (\sqrt{3}+i\right )}+\frac{\tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (2 x+i \sqrt{3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right ) \left (2 a+i \sqrt{3} b+b+i \sqrt{3} c-c\right )}{2 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}+\frac{\log \left (-\left (-2 x-i \sqrt{3}+1\right )^2 \left (2 x-i \sqrt{3}+1\right )\right ) \left (3 i b-\sqrt{3} \left (2 a+b-i \sqrt{3} c-c\right )\right )}{12 \sqrt [3]{2} \left (\sqrt{3}+i\right )}+\frac{\log \left (-\left (-2 x+i \sqrt{3}+1\right )^2 \left (2 x+i \sqrt{3}+1\right )\right ) \left (\sqrt{3} \left (2 a+b+i \sqrt{3} c-c\right )+3 i b\right )}{12 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}+\frac{1}{2} c \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{c \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 2148

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*ArcTan[(1 - (2^(1/3)*Rt[b,
3]*(c - d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(2^(4/3)*Rt[b, 3]*c), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx &=\int \left (\frac{c}{\sqrt [3]{1-x^3}}+\frac{a-c+(b+c) x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=c \int \frac{1}{\sqrt [3]{1-x^3}} \, dx+\int \frac{a-c+(b+c) x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx\\ &=-\frac{c \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )+\int \left (\frac{b-\frac{i (2 a+b-c)}{\sqrt{3}}+c}{\left (-1-i \sqrt{3}+2 x\right ) \sqrt [3]{1-x^3}}+\frac{b+\frac{i (2 a+b-c)}{\sqrt{3}}+c}{\left (-1+i \sqrt{3}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=-\frac{c \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )+\frac{1}{3} \left (3 b-i \sqrt{3} (2 a+b-c)+3 c\right ) \int \frac{1}{\left (-1-i \sqrt{3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\frac{1}{3} \left (3 b+i \sqrt{3} (2 a+b-c)+3 c\right ) \int \frac{1}{\left (-1+i \sqrt{3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx\\ &=-\frac{c \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\left (2 a+b-i \sqrt{3} b-c-i \sqrt{3} c\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (1-i \sqrt{3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (i+\sqrt{3}\right )}+\frac{\left (2 a+b+i \sqrt{3} b-c+i \sqrt{3} c\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (1+i \sqrt{3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (i-\sqrt{3}\right )}+\frac{\left (3 i b-\sqrt{3} \left (2 a+b-c-i \sqrt{3} c\right )\right ) \log \left (-\left (1-i \sqrt{3}-2 x\right )^2 \left (1-i \sqrt{3}+2 x\right )\right )}{12 \sqrt [3]{2} \left (i+\sqrt{3}\right )}+\frac{\left (3 i b+\sqrt{3} \left (2 a+b-c+i \sqrt{3} c\right )\right ) \log \left (-\left (1+i \sqrt{3}-2 x\right )^2 \left (1+i \sqrt{3}+2 x\right )\right )}{12 \sqrt [3]{2} \left (i-\sqrt{3}\right )}+\frac{1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )-\frac{\left (3 i b-\sqrt{3} \left (2 a+b-c-i \sqrt{3} c\right )\right ) \log \left (1-i \sqrt{3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i+\sqrt{3}\right )}-\frac{\left (3 i b+\sqrt{3} \left (2 a+b-c+i \sqrt{3} c\right )\right ) \log \left (1+i \sqrt{3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i-\sqrt{3}\right )}\\ \end{align*}

Mathematica [F]  time = 0.382454, size = 0, normalized size = 0. \[ \int \frac{a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.177, size = 0, normalized size = 0. \begin{align*} \int{\frac{c{x}^{2}+bx+a}{{x}^{2}-x+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c x^{2} + b x + a}{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - x + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c x^{2} + b x + a}{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - x + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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