Optimal. Leaf size=50 \[ \frac{2 \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
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Rubi [A] time = 0.229679, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.047 \[ \frac{2 \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In] Int[(a^(1/3) + b^(1/3)*x)/((2*a^(1/3) - b^(1/3)*x)*Sqrt[a + b*x^3]),x]
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Rubi in Sympy [A] time = 171.741, size = 677, normalized size = 13.54 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a**(1/3)+b**(1/3)*x)/(2*a**(1/3)-b**(1/3)*x)/(b*x**3+a)**(1/2),x)
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Mathematica [C] time = 2.31001, size = 407, normalized size = 8.14 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (3 i \sqrt [3]{a} \sqrt{\frac{\left (\sqrt{3}+i\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-\frac{\sqrt [4]{3} \left (\left (\sqrt{3}+i\right ) \sqrt [3]{a}-\left (\sqrt{3}-i\right ) \sqrt [3]{b} x\right ) \sqrt{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )}{2 \sqrt{2}}\right )}{\left (\sqrt [3]{-1}-2\right ) \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a^(1/3) + b^(1/3)*x)/((2*a^(1/3) - b^(1/3)*x)*Sqrt[a + b*x^3]),x]
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Maple [F] time = 0.135, size = 0, normalized size = 0. \[ \int{1 \left ( \sqrt [3]{a}+\sqrt [3]{b}x \right ) \left ( 2\,\sqrt [3]{a}-\sqrt [3]{b}x \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a^(1/3)+b^(1/3)*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. \[ -\int \frac{b^{\frac{1}{3}} x + a^{\frac{1}{3}}}{\sqrt{b x^{3} + a}{\left (b^{\frac{1}{3}} x - 2 \, a^{\frac{1}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b^(1/3)*x + a^(1/3))/(sqrt(b*x^3 + a)*(b^(1/3)*x - 2*a^(1/3))),x, algorithm="maxima")
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Fricas [A] time = 0.718822, size = 1, normalized size = 0.02 \[ \left [\frac{1}{6} \, a^{\frac{1}{3}} \sqrt{\frac{1}{a b^{\frac{2}{3}}}} \log \left (\frac{{\left (b^{2} x^{6} - 88 \, a b x^{3} + 136 \, a^{2}\right )} a^{\frac{2}{3}} b^{\frac{2}{3}} + 12 \,{\left (6 \, a^{2} b^{\frac{5}{3}} x^{2} +{\left (13 \, a b^{2} x^{3} + 10 \, a^{2} b\right )} a^{\frac{2}{3}} +{\left (a b^{2} x^{4} + 4 \, a^{2} b x\right )} a^{\frac{1}{3}} b^{\frac{1}{3}}\right )} \sqrt{b x^{3} + a} \sqrt{\frac{1}{a b^{\frac{2}{3}}}} + 12 \,{\left (17 \, a b^{2} x^{4} - 4 \, a^{2} b x\right )} a^{\frac{1}{3}} + 12 \,{\left (5 \, a b^{2} x^{5} + 26 \, a^{2} b x^{2}\right )} b^{\frac{1}{3}}}{{\left (b^{2} x^{6} - 160 \, a b x^{3} + 64 \, a^{2}\right )} a^{\frac{2}{3}} b^{\frac{2}{3}} + 12 \,{\left (5 \, a b^{2} x^{4} - 16 \, a^{2} b x\right )} a^{\frac{1}{3}} - 12 \,{\left (a b^{2} x^{5} - 20 \, a^{2} b x^{2}\right )} b^{\frac{1}{3}}}\right ), -\frac{1}{3} \, a^{\frac{1}{3}} \sqrt{-\frac{1}{a b^{\frac{2}{3}}}} \arctan \left (-\frac{12 \, a^{\frac{2}{3}} b x^{2} - 6 \, a b^{\frac{2}{3}} x +{\left (b x^{3} + 10 \, a\right )} a^{\frac{1}{3}} b^{\frac{1}{3}}}{6 \, \sqrt{b x^{3} + a}{\left (a b x + a^{\frac{4}{3}} b^{\frac{2}{3}}\right )} \sqrt{-\frac{1}{a b^{\frac{2}{3}}}}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b^(1/3)*x + a^(1/3))/(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. \[ - \int \frac{\sqrt [3]{a}}{- 2 \sqrt [3]{a} \sqrt{a + b x^{3}} + \sqrt [3]{b} x \sqrt{a + b x^{3}}}\, dx - \int \frac{\sqrt [3]{b} x}{- 2 \sqrt [3]{a} \sqrt{a + b x^{3}} + \sqrt [3]{b} x \sqrt{a + b x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a**(1/3)+b**(1/3)*x)/(2*a**(1/3)-b**(1/3)*x)/(b*x**3+a)**(1/2),x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b^(1/3)*x + a^(1/3))/(sqrt(b*x^3 + a)*(b^(1/3)*x - 2*a^(1/3))),x, algorithm="giac")
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