3.3036 \(\int \frac{1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx\)

Optimal. Leaf size=120 \[ \frac{\log (d+e x)}{4 d^{2/3} e}-\frac{3 \log \left (-\frac{(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(d-3 e x)^{2/3}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e} \]

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] - (d - 3*e*x)^(2/3)/(Sqrt[3]*d^(1/3)*(d + 3*e*x)^(1/3))])/(4*d^(2/3)*e) + Log[d + e*
x]/(4*d^(2/3)*e) - (3*Log[-(d - 3*e*x)^(2/3)/(2*d^(1/3)) - (d + 3*e*x)^(1/3)])/(8*d^(2/3)*e)

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Rubi [A]  time = 0.0261856, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {123} \[ \frac{\log (d+e x)}{4 d^{2/3} e}-\frac{3 \log \left (-\frac{(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(d-3 e x)^{2/3}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] - (d - 3*e*x)^(2/3)/(Sqrt[3]*d^(1/3)*(d + 3*e*x)^(1/3))])/(4*d^(2/3)*e) + Log[d + e*
x]/(4*d^(2/3)*e) - (3*Log[-(d - 3*e*x)^(2/3)/(2*d^(1/3)) - (d + 3*e*x)^(1/3)])/(8*d^(2/3)*e)

Rule 123

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx &=\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(d-3 e x)^{2/3}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac{\log (d+e x)}{4 d^{2/3} e}-\frac{3 \log \left (-\frac{(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}\\ \end{align*}

Mathematica [C]  time = 0.0786628, size = 100, normalized size = 0.83 \[ -\frac{3 \sqrt [3]{1-\frac{4 d}{3 (d+e x)}} \sqrt [3]{1-\frac{2 d}{3 (d+e x)}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{4 d}{3 (d+e x)},\frac{2 d}{3 (d+e x)}\right )}{2 e \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(1 - (4*d)/(3*(d + e*x)))^(1/3)*(1 - (2*d)/(3*(d + e*x)))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (4*d)/(3*(d +
e*x)), (2*d)/(3*(d + e*x))])/(2*e*(d - 3*e*x)^(1/3)*(d + 3*e*x)^(1/3))

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ex+d}{\frac{1}{\sqrt [3]{-3\,ex+d}}}{\frac{1}{\sqrt [3]{3\,ex+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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