Optimal. Leaf size=151 \[ -\frac{\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (d-e x)}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} d^{2/3} e}+\frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e} \]
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Rubi [A] time = 0.121569, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (d-e x)}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} d^{2/3} e}+\frac{\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*(d^2 + 3*e^2*x^2)^(1/3)),x]
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Rubi in Sympy [A] time = 9.79553, size = 143, normalized size = 0.95 \[ - \frac{\sqrt [3]{2} \log{\left (d + e x \right )}}{4 d^{\frac{2}{3}} e} + \frac{\sqrt [3]{2} \log{\left (- 3 \sqrt [3]{2} \sqrt [3]{d} e^{2} \sqrt [3]{d^{2} + 3 e^{2} x^{2}} + 3 d e^{2} - 3 e^{3} x \right )}}{4 d^{\frac{2}{3}} e} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \left (d - e x\right )}{3 \sqrt [3]{d} \sqrt [3]{d^{2} + 3 e^{2} x^{2}}} \right )}}{6 d^{\frac{2}{3}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(3*e**2*x**2+d**2)**(1/3),x)
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Mathematica [C] time = 0.193166, size = 176, normalized size = 1.17 \[ -\frac{\sqrt [3]{\frac{e \left (\sqrt{3} \sqrt{-\frac{d^2}{e^2}}+3 x\right )}{d+e x}} \sqrt [3]{\frac{e \left (9 x-3 \sqrt{3} \sqrt{-\frac{d^2}{e^2}}\right )}{d+e x}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3 d-\sqrt{3} \sqrt{-\frac{d^2}{e^2}} e}{3 d+3 e x},\frac{3 d+\sqrt{3} \sqrt{-\frac{d^2}{e^2}} e}{3 d+3 e x}\right )}{2 e \sqrt [3]{d^2+3 e^2 x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((d + e*x)*(d^2 + 3*e^2*x^2)^(1/3)),x]
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Maple [F] time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d}{\frac{1}{\sqrt [3]{3\,{e}^{2}{x}^{2}+{d}^{2}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac{1}{3}}{\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*e^2*x^2 + d^2)^(1/3)*(e*x + d)),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^2*x^2 + d^2)^(1/3)*(e*x + d)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right ) \sqrt [3]{d^{2} + 3 e^{2} x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(3*e**2*x**2+d**2)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac{1}{3}}{\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*e^2*x^2 + d^2)^(1/3)*(e*x + d)),x, algorithm="giac")
[Out]