3.716 \(\int \frac{1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx\)

Optimal. Leaf size=206 \[ \frac{\sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \log (d+e x)}{4 e \sqrt [3]{d^2-9 e^2 x^2}}-\frac{3 \sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \log \left (-\frac{1}{2} \left (1-\frac{3 e x}{d}\right )^{2/3}-\sqrt [3]{\frac{3 e x}{d}+1}\right )}{8 e \sqrt [3]{d^2-9 e^2 x^2}}+\frac{\sqrt{3} \sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{\left (1-\frac{3 e x}{d}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 e x}{d}+1}}\right )}{4 e \sqrt [3]{d^2-9 e^2 x^2}} \]

[Out]

(Sqrt[3]*(1 - (9*e^2*x^2)/d^2)^(1/3)*ArcTan[1/Sqrt[3] - (1 - (3*e*x)/d)^(2/3)/(Sqrt[3]*(1 + (3*e*x)/d)^(1/3))]
)/(4*e*(d^2 - 9*e^2*x^2)^(1/3)) + ((1 - (9*e^2*x^2)/d^2)^(1/3)*Log[d + e*x])/(4*e*(d^2 - 9*e^2*x^2)^(1/3)) - (
3*(1 - (9*e^2*x^2)/d^2)^(1/3)*Log[-(1 - (3*e*x)/d)^(2/3)/2 - (1 + (3*e*x)/d)^(1/3)])/(8*e*(d^2 - 9*e^2*x^2)^(1
/3))

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Rubi [A]  time = 0.0651082, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {754, 753, 123} \[ \frac{\sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \log (d+e x)}{4 e \sqrt [3]{d^2-9 e^2 x^2}}-\frac{3 \sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \log \left (-\frac{1}{2} \left (1-\frac{3 e x}{d}\right )^{2/3}-\sqrt [3]{\frac{3 e x}{d}+1}\right )}{8 e \sqrt [3]{d^2-9 e^2 x^2}}+\frac{\sqrt{3} \sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{\left (1-\frac{3 e x}{d}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 e x}{d}+1}}\right )}{4 e \sqrt [3]{d^2-9 e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3]*(1 - (9*e^2*x^2)/d^2)^(1/3)*ArcTan[1/Sqrt[3] - (1 - (3*e*x)/d)^(2/3)/(Sqrt[3]*(1 + (3*e*x)/d)^(1/3))]
)/(4*e*(d^2 - 9*e^2*x^2)^(1/3)) + ((1 - (9*e^2*x^2)/d^2)^(1/3)*Log[d + e*x])/(4*e*(d^2 - 9*e^2*x^2)^(1/3)) - (
3*(1 - (9*e^2*x^2)/d^2)^(1/3)*Log[-(1 - (3*e*x)/d)^(2/3)/2 - (1 + (3*e*x)/d)^(1/3)])/(8*e*(d^2 - 9*e^2*x^2)^(1
/3))

Rule 754

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> Dist[(1 + (c*x^2)/a)^(1/3)/(a + c*x^2)^(
1/3), Int[1/((d + e*x)*(1 + (c*x^2)/a)^(1/3)), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] &&
!GtQ[a, 0]

Rule 753

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> Dist[a^(1/3), Int[1/((d + e*x)*(1 - (3*e
*x)/d)^(1/3)*(1 + (3*e*x)/d)^(1/3)), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] && GtQ[a, 0]

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}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx &=\frac{\sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \int \frac{1}{(d+e x) \sqrt [3]{1-\frac{9 e^2 x^2}{d^2}}} \, dx}{\sqrt [3]{d^2-9 e^2 x^2}}\\ &=\frac{\sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \int \frac{1}{(d+e x) \sqrt [3]{1-\frac{3 e x}{d}} \sqrt [3]{1+\frac{3 e x}{d}}} \, dx}{\sqrt [3]{d^2-9 e^2 x^2}}\\ &=\frac{\sqrt{3} \sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{\left (1-\frac{3 e x}{d}\right )^{2/3}}{\sqrt{3} \sqrt [3]{1+\frac{3 e x}{d}}}\right )}{4 e \sqrt [3]{d^2-9 e^2 x^2}}+\frac{\sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \log (d+e x)}{4 e \sqrt [3]{d^2-9 e^2 x^2}}-\frac{3 \sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \log \left (-\frac{1}{2} \left (1-\frac{3 e x}{d}\right )^{2/3}-\sqrt [3]{1+\frac{3 e x}{d}}\right )}{8 e \sqrt [3]{d^2-9 e^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.105073, size = 155, normalized size = 0.75 \[ -\frac{\sqrt [3]{3} \sqrt [3]{-\frac{e \left (\sqrt{\frac{d^2}{e^2}}-3 x\right )}{d+e x}} \sqrt [3]{\frac{e \left (\sqrt{\frac{d^2}{e^2}}+3 x\right )}{d+e x}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3 d-\sqrt{\frac{d^2}{e^2}} e}{3 d+3 e x},\frac{3 d+\sqrt{\frac{d^2}{e^2}} e}{3 d+3 e x}\right )}{2 e \sqrt [3]{d^2-9 e^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.521, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ex+d}{\frac{1}{\sqrt [3]{-9\,{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)/(-9*e^2*x^2+d^2)^(1/3),x)

[Out]

int(1/(e*x+d)/(-9*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 (-9 \, 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)/(-9*e^2*x^2+d^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((-9*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)/(-9*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}{\sqrt [3]{- \left (- d + 3 e x\right ) \left (d + 3 e x\right )} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(-9*e**2*x**2+d**2)**(1/3),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-9 \, 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)/(-9*e^2*x^2+d^2)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((-9*e^2*x^2 + d^2)^(1/3)*(e*x + d)), x)