3.25 \(\int \frac{2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{a-b x^3}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3} \sqrt [6]{a} \sqrt [3]{b}} \]

[Out]

(-2*2^(2/3)*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/Sqrt[a - b*x^
3]])/(Sqrt[3]*a^(1/6)*b^(1/3))

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Rubi [A]  time = 0.292361, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3} \sqrt [6]{a} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(2^(2/3)*a^(1/3) + 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]

[Out]

(-2*2^(2/3)*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/Sqrt[a - b*x^
3]])/(Sqrt[3]*a^(1/6)*b^(1/3))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2**(2/3)*a**(1/3)+2*b**(1/3)*x)/(2**(2/3)*a**(1/3)-b**(1/3)*x)/(-b*x**3+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.06184, size = 336, normalized size = 5.17 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (-\frac{2 \left (\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}}+\frac{\sqrt [3]{-1} 2^{2/3} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt{\frac{3 b^{2/3} x^2}{a^{2/3}}+\frac{3 \sqrt [3]{b} x}{\sqrt [3]{a}}+3} \Pi \left (\frac{i \sqrt{3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}+2^{2/3}}\right )}{\sqrt [3]{b} \sqrt{a-b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(2^(2/3)*a^(1/3) + 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]

[Out]

(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-2*((-1)^(1/3)*a^(1/
3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1
/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)
^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1
)^(1/3))*a^(1/3))] + ((-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[3 + (3*b^
(1/3)*x)/a^(1/3) + (3*b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) +
2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3
))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/(b^(1/3)*Sqrt[a - b*x^3])

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Maple [F]  time = 0.27, size = 0, normalized size = 0. \[ \int{1 \left ({2}^{{\frac{2}{3}}}\sqrt [3]{a}+2\,\sqrt [3]{b}x \right ) \left ({2}^{{\frac{2}{3}}}\sqrt [3]{a}-\sqrt [3]{b}x \right ) ^{-1}{\frac{1}{\sqrt{-b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2^(2/3)*a^(1/3)+2*b^(1/3)*x)/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(-b*x^3+a)^(1/2),x)

[Out]

int((2^(2/3)*a^(1/3)+2*b^(1/3)*x)/(2^(2/3)*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. \[ -\int \frac{2 \, b^{\frac{1}{3}} x + 2^{\frac{2}{3}} a^{\frac{1}{3}}}{\sqrt{-b x^{3} + a}{\left (b^{\frac{1}{3}} x - 2^{\frac{2}{3}} a^{\frac{1}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*b^(1/3)*x + 2^(2/3)*a^(1/3))/(sqrt(-b*x^3 + a)*(b^(1/3)*x - 2^(2/3)*a^(1/3))),x, algorithm="maxima")

[Out]

-integrate((2*b^(1/3)*x + 2^(2/3)*a^(1/3))/(sqrt(-b*x^3 + a)*(b^(1/3)*x - 2^(2/3
)*a^(1/3))), x)

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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(-(2*b^(1/3)*x + 2^(2/3)*a^(1/3))/(sqrt(-b*x^3 + a)*(b^(1/3)*x - 2^(2/3)*a^(1/3))),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{2^{\frac{2}{3}} \sqrt [3]{a}}{- 2^{\frac{2}{3}} \sqrt [3]{a} \sqrt{a - b x^{3}} + \sqrt [3]{b} x \sqrt{a - b x^{3}}}\, dx - \int \frac{2 \sqrt [3]{b} x}{- 2^{\frac{2}{3}} \sqrt [3]{a} \sqrt{a - b x^{3}} + \sqrt [3]{b} x \sqrt{a - b x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2**(2/3)*a**(1/3)+2*b**(1/3)*x)/(2**(2/3)*a**(1/3)-b**(1/3)*x)/(-b*x**3+a)**(1/2),x)

[Out]

-Integral(2**(2/3)*a**(1/3)/(-2**(2/3)*a**(1/3)*sqrt(a - b*x**3) + b**(1/3)*x*sq
rt(a - b*x**3)), x) - Integral(2*b**(1/3)*x/(-2**(2/3)*a**(1/3)*sqrt(a - b*x**3)
+ b**(1/3)*x*sqrt(a - b*x**3)), 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(-(2*b^(1/3)*x + 2^(2/3)*a^(1/3))/(sqrt(-b*x^3 + a)*(b^(1/3)*x - 2^(2/3)*a^(1/3))),x, algorithm="giac")

[Out]

Exception raised: TypeError