Optimal. Leaf size=345 \[ -\frac{\sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\sqrt [3]{b} e-\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}-\frac{\left (\sqrt [3]{b} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{-a-b x^3}}\right )}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{a} b^{2/3}} \]
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Rubi [A] time = 0.996539, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089 \[ -\frac{\sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\sqrt [3]{b} e-\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}-\frac{\left (\sqrt [3]{b} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{-a-b x^3}}\right )}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{a} b^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[(e + f*x)/(((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]
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Rubi in Sympy [A] time = 38.4549, size = 177, normalized size = 0.51 \[ \frac{2 \tilde{\infty } \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (\frac{f}{\sqrt [3]{b}} + \frac{e}{\sqrt [3]{a}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- a - b x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x+e)/(b**(1/3)*x+a**(1/3)*(1-3**(1/2)))/(-b*x**3-a)**(1/2),x)
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Mathematica [C] time = 3.08967, size = 441, normalized size = 1.28 \[ -\frac{4 \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (i \sqrt{\frac{\left (\sqrt{3}+i\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \left (\left (\sqrt{3}-1\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-\frac{i \sqrt [4]{3} f \left (\left (\sqrt{3}+(-2-i)\right ) \sqrt [3]{a}+\left ((1+2 i)-i \sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )}{2 \sqrt{2}}\right )}{\left (3-(2-i) \sqrt{3}\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{-a-b x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(e + f*x)/(((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]
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Maple [F] time = 0.105, size = 0, normalized size = 0. \[ \int{(fx+e) \left ( \sqrt [3]{b}x+\sqrt [3]{a} \left ( -\sqrt{3}+1 \right ) \right ) ^{-1}{\frac{1}{\sqrt{-b{x}^{3}-a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x+e)/(b^(1/3)*x+a^(1/3)*(-3^(1/2)+1))/(-b*x^3-a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{\sqrt{-b x^{3} - a}{\left (b^{\frac{1}{3}} x - a^{\frac{1}{3}}{\left (\sqrt{3} - 1\right )}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-b*x^3 - a)*(b^(1/3)*x - a^(1/3)*(sqrt(3) - 1))),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((f*x + e)/(sqrt(-b*x^3 - a)*(b^(1/3)*x - a^(1/3)*(sqrt(3) - 1))),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e + f x}{\sqrt{- a - b x^{3}} \left (- \sqrt{3} \sqrt [3]{a} + \sqrt [3]{a} + \sqrt [3]{b} x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x+e)/(b**(1/3)*x+a**(1/3)*(1-3**(1/2)))/(-b*x**3-a)**(1/2),x)
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GIAC/XCAS [A] time = 0.596571, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-b*x^3 - a)*(b^(1/3)*x - a^(1/3)*(sqrt(3) - 1))),x, algorithm="giac")
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