### 3.2588 $$\int \frac{1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx$$

Optimal. Leaf size=196 $\frac{\sqrt [3]{b^3 e-c^3 e x^3} \log \left (\sqrt [3]{b^3 e-c^3 e x^3}+c \sqrt [3]{e} x\right )}{2 c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}-\frac{\sqrt [3]{b^3 e-c^3 e x^3} \tan ^{-1}\left (\frac{1-\frac{2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt{3}}\right )}{\sqrt{3} c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}$

[Out]

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

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Rubi [A]  time = 0.0585334, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.061, Rules used = {713, 239} $\frac{\sqrt [3]{b^3 e-c^3 e x^3} \log \left (\sqrt [3]{b^3 e-c^3 e x^3}+c \sqrt [3]{e} x\right )}{2 c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}-\frac{\sqrt [3]{b^3 e-c^3 e x^3} \tan ^{-1}\left (\frac{1-\frac{2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt{3}}\right )}{\sqrt{3} c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 713

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((d + e*x)^FracPart[p
]*(a + b*x + c*x^2)^FracPart[p])/(a*d + c*e*x^3)^FracPart[p], Int[(d + e*x)^(m - p)*(a*d + c*e*x^3)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && IGtQ[m - p + 1, 0] &&  !Intege
rQ[p]

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]

Rubi steps

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

Mathematica [C]  time = 0.190699, size = 241, normalized size = 1.23 $-\frac{3 \sqrt [3]{\frac{-\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}} \sqrt [3]{\frac{\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{\sqrt{3} \sqrt{-b^2 c^2}+3 b c}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 c (b-c x)}{3 b c+\sqrt{3} \sqrt{-b^2 c^2}},\frac{2 c (b-c x)}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}\right ) (e (b-c x))^{2/3}}{2 c e \sqrt [3]{b^2+b c x+c^2 x^2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(e*(b - c*x))^(2/3)*((b*c - Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c - Sqrt[3]*Sqrt[-(b^2*c^2)]))^(1/3)*
((b*c + Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]))^(1/3)*AppellF1[2/3, 1/3, 1/3,
5/3, (2*c*(b - c*x))/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]), (2*c*(b - c*x))/(3*b*c - Sqrt[3]*Sqrt[-(b^2*c^2)])])/
(2*c*e*(b^2 + b*c*x + c^2*x^2)^(1/3))

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Maple [F]  time = 3.069, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{-cex+be}}}{\frac{1}{\sqrt [3]{{c}^{2}{x}^{2}+bcx+{b}^{2}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out