### 3.2496 $$\int \frac{1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx$$

Optimal. Leaf size=242 $\frac{\log \left (-3 c e^2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}+3 c e^2 (c d-b e)-3 c^2 e^3 x\right )}{2 e (2 c d-b e)^{2/3}}-\frac{\tan ^{-1}\left (\frac{2 (-b e+c d-c e x)}{\sqrt{3} \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} e (2 c d-b e)^{2/3}}-\frac{\log (d+e x)}{2 e (2 c d-b e)^{2/3}}$

[Out]

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

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Rubi [A]  time = 0.139311, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 52, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.019, Rules used = {750} $\frac{\log \left (-3 c e^2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}+3 c e^2 (c d-b e)-3 c^2 e^3 x\right )}{2 e (2 c d-b e)^{2/3}}-\frac{\tan ^{-1}\left (\frac{2 (-b e+c d-c e x)}{\sqrt{3} \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} e (2 c d-b e)^{2/3}}-\frac{\log (d+e x)}{2 e (2 c d-b e)^{2/3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 750

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

Mathematica [C]  time = 0.491314, size = 317, normalized size = 1.31 $-\frac{\sqrt [3]{3} \sqrt [3]{\frac{-\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} \sqrt [3]{\frac{\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{-6 d e c^2+3 b e^2 c+\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)},\frac{6 d e c^2-3 b e^2 c+\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)}\right )}{2\ 2^{2/3} e \sqrt [3]{b^2 e^2+b c e (3 e x-d)+c^2 \left (d^2+3 e^2 x^2\right )}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(3^(1/3)*((3*b*c*e^2 - Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)] + 6*c^2*e^2*x)/(c^2*e*(d + e*x)))^(1/3)*((3*
b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)] + 6*c^2*e^2*x)/(c^2*e*(d + e*x)))^(1/3)*AppellF1[2/3, 1/3,
1/3, 5/3, -(-6*c^2*d*e + 3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)])/(6*c^2*e*(d + e*x)), (6*c^2*d
*e - 3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)])/(6*c^2*e*(d + e*x))])/(2*2^(2/3)*e*(b^2*e^2 + b*c*
e*(-d + 3*e*x) + c^2*(d^2 + 3*e^2*x^2))^(1/3))

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Maple [F]  time = 1.477, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ex+d}{\frac{1}{\sqrt [3]{3\,{c}^{2}{e}^{2}{x}^{2}+3\,bc{e}^{2}x+{b}^{2}{e}^{2}-bcde+{c}^{2}{d}^{2}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((3*c^2*e^2*x^2 + 3*b*c*e^2*x + c^2*d^2 - b*c*d*e + b^2*e^2)^(1/3)*(e*x + d)), 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/(e*x+d)/(3*c^2*e^2*x^2+3*b*c*e^2*x+b^2*e^2-b*c*d*e+c^2*d^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}{\left (d + e x\right ) \sqrt [3]{b^{2} e^{2} - b c d e + 3 b c e^{2} x + c^{2} d^{2} + 3 c^{2} e^{2} x^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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