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