Optimal. Leaf size=768 \[ \frac{2 \sqrt{2} \sqrt [3]{a} \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 )}{\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}}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \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}} E\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 )}{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}}+\frac{2 \sqrt{-a-b x^3}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{3^{3/4} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}} \]
[Out]
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Rubi [A] time = 0.645224, antiderivative size = 768, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139 \[ \frac{2 \sqrt{2} \sqrt [3]{a} \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 )}{\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}}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \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}} E\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 )}{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}}+\frac{2 \sqrt{-a-b x^3}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{3^{3/4} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{\sqrt [4]{3} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{-a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}} \]
Warning: Unable to verify antiderivative.
[In] Int[(x*Sqrt[-a - b*x^3])/(2*(5 - 3*Sqrt[3])*a + b*x^3),x]
[Out]
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Rubi in Sympy [A] time = 33.9893, size = 71, normalized size = 0.09 \[ \frac{x^{2} \sqrt{- a - b x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},- \frac{1}{2},1,\frac{5}{3},- \frac{b x^{3}}{a},- \frac{b x^{3}}{2 a \left (- 3 \sqrt{3} + 5\right )} \right )}}{4 a \sqrt{1 + \frac{b x^{3}}{a}} \left (- 3 \sqrt{3} + 5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(-b*x**3-a)**(1/2)/(b*x**3+2*a*(5-3*3**(1/2))),x)
[Out]
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Mathematica [C] time = 0.634986, size = 253, normalized size = 0.33 \[ \frac{10 \left (15 \sqrt{3}-26\right ) a x^2 \sqrt{-a-b x^3} F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )}{\left (3 \sqrt{3}-5\right ) \left (2 \left (3 \sqrt{3}-5\right ) a-b x^3\right ) \left (3 b x^3 \left (F_1\left (\frac{5}{3};-\frac{1}{2},2;\frac{8}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )+\left (3 \sqrt{3}-5\right ) F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )\right )+10 \left (3 \sqrt{3}-5\right ) a F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x*Sqrt[-a - b*x^3])/(2*(5 - 3*Sqrt[3])*a + b*x^3),x]
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Maple [C] time = 0.103, size = 983, normalized size = 1.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(-b*x^3-a)^(1/2)/(b*x^3+2*a*(5-3*3^(1/2))),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-b x^{3} - a} x}{b x^{3} - 2 \, a{\left (3 \, \sqrt{3} - 5\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x^3 - a)*x/(b*x^3 - 2*a*(3*sqrt(3) - 5)),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(sqrt(-b*x^3 - a)*x/(b*x^3 - 2*a*(3*sqrt(3) - 5)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sqrt{- a - b x^{3}}}{- 6 \sqrt{3} a + 10 a + b x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(-b*x**3-a)**(1/2)/(b*x**3+2*a*(5-3*3**(1/2))),x)
[Out]
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GIAC/XCAS [A] time = 0.552624, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x^3 - a)*x/(b*x^3 - 2*a*(3*sqrt(3) - 5)),x, algorithm="giac")
[Out]