Optimal. Leaf size=260 \[ \frac{4 \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{a+b x^3}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.52817, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{4 \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{a+b x^3}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In] Int[x/((2*a^(1/3) - b^(1/3)*x)*Sqrt[a + b*x^3]),x]
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Rubi in Sympy [A] time = 170.311, size = 687, normalized size = 2.64 \[ \text{result too large to display} \]
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)
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Mathematica [C] time = 2.72808, size = 407, normalized size = 1.57 \[ \frac{\sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (8 i \sqrt [3]{a} \sqrt{\frac{\left (\sqrt{3}+i\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-\sqrt{2} \sqrt [4]{3} \left (\left (\sqrt{3}+i\right ) \sqrt [3]{a}-\left (\sqrt{3}-i\right ) \sqrt [3]{b} x\right ) \sqrt{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )\right )}{2 \left (\sqrt [3]{-1}-2\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x/((2*a^(1/3) - b^(1/3)*x)*Sqrt[a + b*x^3]),x]
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Maple [F] time = 0.095, 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)
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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")
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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")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{- 2 \sqrt [3]{a} \sqrt{a + b x^{3}} + \sqrt [3]{b} x \sqrt{a + b x^{3}}}\, dx \]
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]