Optimal. Leaf size=271 \[ \frac{2 \sqrt{b x^3-a}}{\sqrt [3]{b} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{b} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.160202, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028 \[ \frac{2 \sqrt{b x^3-a}}{\sqrt [3]{b} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{b} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}} \]
Antiderivative was successfully verified.
[In] Int[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/Sqrt[-a + b*x^3],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.7682, size = 224, normalized size = 0.83 \[ - \frac{\sqrt [4]{3} \sqrt [3]{a} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt [3]{b} \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- a + b x^{3}}} - \frac{2 \sqrt{- a + b x^{3}}}{\sqrt [3]{b} \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b**(1/3)*x+a**(1/3)*(1+3**(1/2)))/(b*x**3-a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.598886, size = 257, normalized size = 0.95 \[ \frac{2 \sqrt [3]{-a} \sqrt{-\frac{(-1)^{5/6} \left ((-a)^{2/3} \sqrt [3]{-b} x+a\right )}{a}} \sqrt{\frac{\sqrt [3]{-b} x \left (\sqrt [3]{-a}+\sqrt [3]{-b} x\right )}{(-a)^{2/3}}+1} \left (i \left (\left (3+\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{-b}-\sqrt{3} \sqrt [3]{-a} \sqrt [3]{b}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{-a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+3 (-1)^{2/3} \sqrt [3]{-a} \sqrt [3]{b} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{-a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} (-b)^{2/3} \sqrt{b x^3-a}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/Sqrt[-a + b*x^3],x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.06, size = 952, normalized size = 3.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b^(1/3)*x+a^(1/3)*(1+3^(1/2)))/(b*x^3-a)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{b^{\frac{1}{3}} x - a^{\frac{1}{3}}{\left (\sqrt{3} + 1\right )}}{\sqrt{b x^{3} - a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b^(1/3)*x - a^(1/3)*(sqrt(3) + 1))/sqrt(b*x^3 - a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{b^{\frac{1}{3}} x - a^{\frac{1}{3}}{\left (\sqrt{3} + 1\right )}}{\sqrt{b x^{3} - a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b^(1/3)*x - a^(1/3)*(sqrt(3) + 1))/sqrt(b*x^3 - a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.04426, size = 112, normalized size = 0.41 \[ \frac{i \sqrt [3]{b} x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3}}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{5}{3}\right )} - \frac{\sqrt{3} i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3}}{a}} \right )}}{3 \sqrt [6]{a} \Gamma \left (\frac{4}{3}\right )} - \frac{i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3}}{a}} \right )}}{3 \sqrt [6]{a} \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b**(1/3)*x+a**(1/3)*(1+3**(1/2)))/(b*x**3-a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{b^{\frac{1}{3}} x - a^{\frac{1}{3}}{\left (\sqrt{3} + 1\right )}}{\sqrt{b x^{3} - a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b^(1/3)*x - a^(1/3)*(sqrt(3) + 1))/sqrt(b*x^3 - a),x, algorithm="giac")
[Out]