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

Optimal. Leaf size=72 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{-a-b x^3}}\right )}{\sqrt{3+2 \sqrt{3}} \sqrt [6]{a} \sqrt [3]{b}} \]

[Out]

(-2*ArcTanh[(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 = 0.297422, antiderivative size = 72, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{-a-b x^3}}\right )}{\sqrt{3+2 \sqrt{3}} \sqrt [6]{a} \sqrt [3]{b}} \]

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*ArcTanh[(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 = 59.654, size = 253, normalized size = 3.51 \[ - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} \left (1 - \frac{\sqrt [3]{b} x}{\sqrt [3]{a}} + \frac{b^{\frac{2}{3}} x^{2}}{a^{\frac{2}{3}}}\right )}{\left (\sqrt [3]{a} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{b} x\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \operatorname{atanh}{\left (\frac{\sqrt{1 - \frac{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \left (\sqrt{3} + 2\right )}{\sqrt{4 \sqrt{3} + 7 + \frac{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}}} \right )}}{3 \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a} \left (- \sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \sqrt{- a - b x^{3}}} \]

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*3**(3/4)*sqrt(a**(2/3)*(1 - b**(1/3)*x/a**(1/3) + b**(2/3)*x**2/a**(2/3))/(a*
*(1/3)*(-sqrt(3) + 1) + b**(1/3)*x)**2)*(a**(1/3) + b**(1/3)*x)*atanh(sqrt(1 - (
a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2/(a**(1/3)*(-1 + sqrt(3)) - b**(1/3)*x)**
2)*(sqrt(3) + 2)/sqrt(4*sqrt(3) + 7 + (a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2/(
a**(1/3)*(-1 + sqrt(3)) - b**(1/3)*x)**2))/(3*b**(1/3)*sqrt(a**(1/3)*(-a**(1/3)
- b**(1/3)*x)/(a**(1/3)*(-sqrt(3) + 1) + b**(1/3)*x)**2)*sqrt(sqrt(3) + 2)*sqrt(
-a - b*x**3))

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Mathematica [C]  time = 1.04142, size = 323, normalized size = 4.49 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac{4 \sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\left (3+(2+i) \sqrt{3}\right ) \sqrt [3]{b}}-\frac{\left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}}\right )}{\sqrt{-a-b x^3}} \]

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)^(1/6) - (I*b^(1/3)*x)/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)])/(3^(1/
4)*b^(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 = 0.119, 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. \[ \int \frac{- \sqrt{3} \sqrt [3]{a} + \sqrt [3]{a} + \sqrt [3]{b} x}{\sqrt{- a - b x^{3}} \left (\sqrt [3]{a} + \sqrt{3} \sqrt [3]{a} + \sqrt [3]{b} x\right )}\, dx \]

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

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GIAC/XCAS [A]  time = 0.615311, 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