Optimal. Leaf size=738 \[ \frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a+b x^3}}\right )}{\sqrt{2} b^{2/3}}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a+b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a+b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a+b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}} \]
[Out]
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Rubi [A] time = 0.642932, antiderivative size = 738, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ \frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a+b x^3}}\right )}{\sqrt{2} b^{2/3}}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a+b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a+b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a+b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}} \]
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 = 32.525, size = 70, 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 = 0.65162, size = 250, normalized size = 0.34 \[ \frac{10 \left (15 \sqrt{3}-26\right ) a x^2 \sqrt{a+b x^3} F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )}{\left (3 \sqrt{3}-5\right ) \left (2 \left (3 \sqrt{3}-5\right ) a-b x^3\right ) \left (3 b x^3 \left (F_1\left (\frac{5}{3};-\frac{1}{2},2;\frac{8}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )+\left (3 \sqrt{3}-5\right ) F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )\right )+10 \left (3 \sqrt{3}-5\right ) a F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right )\right )} \]
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 = 0.233, size = 977, 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 = 0.549616, 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]