3.95 \(\int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{a-b x^3}} \, dx\)

Optimal. Leaf size=71 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3+2 \sqrt{3}} \sqrt [6]{a} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.323023, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 61, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3+2 \sqrt{3}} \sqrt [6]{a} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.3561, size = 162, normalized size = 2.28 \[ \frac{2 \tilde{\infty } \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \sqrt{a - b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*zoo*sqrt((a**(2/3) + a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3)*(1 + sqrt(
3)) - b**(1/3)*x)**2)*(a**(1/3) - b**(1/3)*x)*elliptic_f(asin((a**(1/3)*(-1 + sq
rt(3)) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) - b**(1/3)*x)), -7 - 4*sqrt(3))/(sq
rt(a**(1/3)*(a**(1/3) - b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) - b**(1/3)*x)**2)*sq
rt(a - b*x**3))

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Mathematica [C]  time = 1.35331, size = 329, normalized size = 4.63 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac{\left (\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}}-\frac{4 \sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{3+(2+i) \sqrt{3}}\right )}{\sqrt [3]{b} \sqrt{a-b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)/(((1 + Sqrt[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))]*((((-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))] - (4*(-1)^(1/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 + (b^(1/3)*x)/a^
(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]),
ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)
^(1/3)])/(3 + (2 + I)*Sqrt[3])))/(b^(1/3)*Sqrt[a - b*x^3])

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Maple [F]  time = 0.167, size = 0, normalized size = 0. \[ \int{1 \left ( -\sqrt [3]{b}x+\sqrt [3]{a} \left ( -\sqrt{3}+1 \right ) \right ) \left ( -\sqrt [3]{b}x+\sqrt [3]{a} \left ( 1+\sqrt{3} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-a**(1/3) + sqrt(3)*a**(1/3) + b**(1/3)*x)/(sqrt(a - b*x**3)*(-sqrt(3)
*a**(1/3) - a**(1/3) + b**(1/3)*x)), x)

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GIAC/XCAS [A]  time = 0.601401, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]

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

[Out]

sage0*x