Optimal. Leaf size=76 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt{a} \left (1-x \sqrt [3]{\frac{b}{a}}\right )}{\sqrt{b x^3-a}}\right )}{\sqrt{2 \sqrt{3}-3} \sqrt{a} \sqrt [3]{\frac{b}{a}}} \]
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Rubi [A] time = 0.329729, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt{a} \left (1-x \sqrt [3]{\frac{b}{a}}\right )}{\sqrt{b x^3-a}}\right )}{\sqrt{2 \sqrt{3}-3} \sqrt{a} \sqrt [3]{\frac{b}{a}}} \]
Antiderivative was successfully verified.
[In] Int[(1 + Sqrt[3] - (b/a)^(1/3)*x)/((1 - Sqrt[3] - (b/a)^(1/3)*x)*Sqrt[-a + b*x^3]),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-(b/a)**(1/3)*x+3**(1/2))/(1-(b/a)**(1/3)*x-3**(1/2))/(b*x**3-a)**(1/2),x)
[Out]
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Mathematica [C] time = 8.34384, size = 1492, normalized size = 19.63 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + Sqrt[3] - (b/a)^(1/3)*x)/((1 - Sqrt[3] - (b/a)^(1/3)*x)*Sqrt[-a + b*x^3]),x]
[Out]
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Maple [F] time = 0.111, size = 0, normalized size = 0. \[ \int{1 \left ( 1-\sqrt [3]{{\frac{b}{a}}}x+\sqrt{3} \right ) \left ( 1-\sqrt [3]{{\frac{b}{a}}}x-\sqrt{3} \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}-a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-(b/a)^(1/3)*x+3^(1/2))/(1-(b/a)^(1/3)*x-3^(1/2))/(b*x^3-a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \sqrt{3} - 1}{\sqrt{b x^{3} - a}{\left (x \left (\frac{b}{a}\right )^{\frac{1}{3}} + \sqrt{3} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x*(b/a)^(1/3) - sqrt(3) - 1)/(sqrt(b*x^3 - a)*(x*(b/a)^(1/3) + sqrt(3) - 1)),x, algorithm="maxima")
[Out]
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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((x*(b/a)^(1/3) - sqrt(3) - 1)/(sqrt(b*x^3 - a)*(x*(b/a)^(1/3) + sqrt(3) - 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.5137, size = 0, normalized size = 0. \[ \mathrm{NaN} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-(b/a)**(1/3)*x+3**(1/2))/(1-(b/a)**(1/3)*x-3**(1/2))/(b*x**3-a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.606431, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x*(b/a)^(1/3) - sqrt(3) - 1)/(sqrt(b*x^3 - a)*(x*(b/a)^(1/3) + sqrt(3) - 1)),x, algorithm="giac")
[Out]