### 3.699 $$\int \frac{1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx$$

Optimal. Leaf size=151 $\frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (d-e x)}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} d^{2/3} e}-\frac{\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}$

[Out]

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

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Rubi [A]  time = 0.0391441, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {751} $\frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (d-e x)}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} d^{2/3} e}-\frac{\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 751

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[(6*c^2*e^2)/d^2, 3]}, -Simp
[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*c*(d - e*x))/(Sqrt[3]*d*q*(a + c*x^2)^(1/3))])/(d^2*q^2), x] + (-Simp[(3*c
*e*Log[d + e*x])/(2*d^2*q^2), x] + Simp[(3*c*e*Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)])/(2*d^2*q^2), x])] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]

Rubi steps

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

Mathematica [C]  time = 0.134442, size = 176, normalized size = 1.17 $-\frac{\sqrt [3]{\frac{e \left (\sqrt{3} \sqrt{-\frac{d^2}{e^2}}+3 x\right )}{d+e x}} \sqrt [3]{\frac{e \left (9 x-3 \sqrt{3} \sqrt{-\frac{d^2}{e^2}}\right )}{d+e x}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3 d-\sqrt{3} \sqrt{-\frac{d^2}{e^2}} e}{3 d+3 e x},\frac{3 d+\sqrt{3} \sqrt{-\frac{d^2}{e^2}} e}{3 d+3 e x}\right )}{2 e \sqrt [3]{d^2+3 e^2 x^2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((e*(Sqrt[3]*Sqrt[-(d^2/e^2)] + 3*x))/(d + e*x))^(1/3)*((e*(-3*Sqrt[3]*Sqrt[-(d^2/e^2)] + 9*x))/(d + e*x))^(
1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (3*d - Sqrt[3]*Sqrt[-(d^2/e^2)]*e)/(3*d + 3*e*x), (3*d + Sqrt[3]*Sqrt[-(d^2/
e^2)]*e)/(3*d + 3*e*x)])/(2*e*(d^2 + 3*e^2*x^2)^(1/3))

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Maple [F]  time = 0.542, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ex+d}{\frac{1}{\sqrt [3]{3\,{e}^{2}{x}^{2}+{d}^{2}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(e*x+d)/(3*e^2*x^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 \, e^{2} x^{2} + d^{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*e^2*x^2+d^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((3*e^2*x^2 + d^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*e^2*x^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]{d^{2} + 3 e^{2} x^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((d + e*x)*(d**2 + 3*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 \, e^{2} x^{2} + d^{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*e^2*x^2+d^2)^(1/3),x, algorithm="giac")

[Out]

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