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

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)/(S
qrt[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))

_______________________________________________________________________________________

Rubi [A]  time = 0.26507, 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 \[ \frac{\sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \log (d+e x)}{4 e \sqrt [3]{d^2-9 e^2 x^2}}-\frac{3 \sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \log \left (-\frac{1}{2} \left (1-\frac{3 e x}{d}\right )^{2/3}-\sqrt [3]{\frac{3 e x}{d}+1}\right )}{8 e \sqrt [3]{d^2-9 e^2 x^2}}+\frac{\sqrt{3} \sqrt [3]{1-\frac{9 e^2 x^2}{d^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{\left (1-\frac{3 e x}{d}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 e x}{d}+1}}\right )}{4 e \sqrt [3]{d^2-9 e^2 x^2}} \]

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)/(S
qrt[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))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.2095, size = 90, normalized size = 0.44 \[ - \frac{3^{\frac{2}{3}} \sqrt [3]{\frac{3 d + 9 e x}{d + e x}} \sqrt [3]{- \frac{3 d - 9 e x}{d + e x}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{2 d}{3 \left (d + e x\right )},\frac{4 d}{3 \left (d + e x\right )} \right )}}{6 e \sqrt [3]{d^{2} - 9 e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3**(2/3)*((3*d + 9*e*x)/(d + e*x))**(1/3)*(-(3*d - 9*e*x)/(d + e*x))**(1/3)*app
ellf1(2/3, 1/3, 1/3, 5/3, 2*d/(3*(d + e*x)), 4*d/(3*(d + e*x)))/(6*e*(d**2 - 9*e
**2*x**2)**(1/3))

_______________________________________________________________________________________

Mathematica [C]  time = 0.148195, 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))

_______________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d}{\frac{1}{\sqrt [3]{-9\,{e}^{2}{x}^{2}+{d}^{2}}}}}\, dx \]

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)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-9 \, e^{2} x^{2} + d^{2}\right )}^{\frac{1}{3}}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-9*e^2*x^2 + d^2)^(1/3)*(e*x + d)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-9*e^2*x^2 + d^2)^(1/3)*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{- \left (- d + 3 e x\right ) \left (d + 3 e x\right )} \left (d + e x\right )}\, dx \]

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)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-9 \, e^{2} x^{2} + d^{2}\right )}^{\frac{1}{3}}{\left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-9*e^2*x^2 + d^2)^(1/3)*(e*x + d)),x, algorithm="giac")

[Out]

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