Optimal. Leaf size=277 \[ \frac{4 \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{b x^3-a}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}} \]
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Rubi [A] time = 0.569637, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{4 \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{b x^3-a}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}} \]
Antiderivative was successfully verified.
[In] Int[x/((2*a^(1/3) + b^(1/3)*x)*Sqrt[-a + b*x^3]),x]
[Out]
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Rubi in Sympy [A] time = 170.524, size = 687, normalized size = 2.48 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(2*a**(1/3)+b**(1/3)*x)/(b*x**3-a)**(1/2),x)
[Out]
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Mathematica [C] time = 1.46143, size = 372, normalized size = 1.34 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\left (\sqrt [3]{-1}-2\right ) \left (\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )+\frac{2 \sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{3}}\right )}{\left (\sqrt [3]{-1}-2\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3-a}} \]
Antiderivative was successfully verified.
[In] Integrate[x/((2*a^(1/3) + b^(1/3)*x)*Sqrt[-a + b*x^3]),x]
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Maple [F] time = 0.086, size = 0, normalized size = 0. \[ \int{x \left ( 2\,\sqrt [3]{a}+\sqrt [3]{b}x \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}-a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(2*a^(1/3)+b^(1/3)*x)/(b*x^3-a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{b x^{3} - a}{\left (b^{\frac{1}{3}} x + 2 \, a^{\frac{1}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(b*x^3 - a)*(b^(1/3)*x + 2*a^(1/3))),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/(sqrt(b*x^3 - a)*(b^(1/3)*x + 2*a^(1/3))),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (2 \sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{- a + b x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(2*a**(1/3)+b**(1/3)*x)/(b*x**3-a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{b x^{3} - a}{\left (b^{\frac{1}{3}} x + 2 \, a^{\frac{1}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(b*x^3 - a)*(b^(1/3)*x + 2*a^(1/3))),x, algorithm="giac")
[Out]