3.3019 \(\int \frac{1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx\)

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 = 0.117647, 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 \[ \frac{\log (d+e x)}{4 d^{2/3} e}-\frac{3 \log \left (-\frac{(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(d-3 e x)^{2/3}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e} \]

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)

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Rubi in Sympy [A]  time = 10.0373, size = 109, normalized size = 0.91 \[ \frac{\log{\left (d + e x \right )}}{4 d^{\frac{2}{3}} e} - \frac{3 \log{\left (- \sqrt [3]{d + 3 e x} - \frac{\left (d - 3 e x\right )^{\frac{2}{3}}}{2 \sqrt [3]{d}} \right )}}{8 d^{\frac{2}{3}} e} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{\sqrt{3} \left (d - 3 e x\right )^{\frac{2}{3}}}{3 \sqrt [3]{d} \sqrt [3]{d + 3 e x}} \right )}}{4 d^{\frac{2}{3}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(d + e*x)/(4*d**(2/3)*e) - 3*log(-(d + 3*e*x)**(1/3) - (d - 3*e*x)**(2/3)/(2*
d**(1/3)))/(8*d**(2/3)*e) + sqrt(3)*atan(sqrt(3)/3 - sqrt(3)*(d - 3*e*x)**(2/3)/
(3*d**(1/3)*(d + 3*e*x)**(1/3)))/(4*d**(2/3)*e)

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Mathematica [C]  time = 0.489244, size = 196, normalized size = 1.63 \[ -\frac{45 (d+e x) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{4 d}{3 (d+e x)},\frac{2 d}{3 (d+e x)}\right )}{2 e \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x} \left (15 (d+e x) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{4 d}{3 (d+e x)},\frac{2 d}{3 (d+e x)}\right )+2 d \left (F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{4 d}{3 (d+e x)},\frac{2 d}{3 (d+e x)}\right )+2 F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{4 d}{3 (d+e x)},\frac{2 d}{3 (d+e x)}\right )\right )\right )} \]

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]

(-45*(d + e*x)*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)*(15*(d + e*x)*AppellF1[2/3, 1/3,
1/3, 5/3, (4*d)/(3*(d + e*x)), (2*d)/(3*(d + e*x))] + 2*d*(AppellF1[5/3, 1/3, 4
/3, 8/3, (4*d)/(3*(d + e*x)), (2*d)/(3*(d + e*x))] + 2*AppellF1[5/3, 4/3, 1/3, 8
/3, (4*d)/(3*(d + e*x)), (2*d)/(3*(d + e*x))])))

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d}{\frac{1}{\sqrt [3]{-3\,ex+d}}}{\frac{1}{\sqrt [3]{3\,ex+d}}}}\, dx \]

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. \[ \int \frac{1}{{\left (3 \, e x + d\right )}^{\frac{1}{3}}{\left (e x + d\right )}{\left (-3 \, e x + d\right )}^{\frac{1}{3}}}\,{d x} \]

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. \[ \text{Timed out} \]

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

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

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)