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

Optimal. Leaf size=282 \[ \frac{\sqrt{2} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{b x^3-a}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}} \]

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*(a^(1/3) - b^(1/3)*x))/Sqrt[-a +
b*x^3]])/(3^(3/4)*a^(1/6)*b^(2/3))) + (Sqrt[2]*(a^(1/3) - b^(1/3)*x)*Sqrt[(a^(2
/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2]*El
lipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/((1 - Sqrt[3])*a^(1/3) - b^(1
/3)*x)], -7 + 4*Sqrt[3]])/(3^(3/4)*b^(2/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1/3)*x)
)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2)]*Sqrt[-a + b*x^3])

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Rubi [A]  time = 0.855942, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098 \[ \frac{\sqrt{2} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{b x^3-a}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*(a^(1/3) - b^(1/3)*x))/Sqrt[-a +
b*x^3]])/(3^(3/4)*a^(1/6)*b^(2/3))) + (Sqrt[2]*(a^(1/3) - b^(1/3)*x)*Sqrt[(a^(2
/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2]*El
lipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/((1 - Sqrt[3])*a^(1/3) - b^(1
/3)*x)], -7 + 4*Sqrt[3]])/(3^(3/4)*b^(2/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1/3)*x)
)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2)]*Sqrt[-a + b*x^3])

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Rubi in Sympy [A]  time = 33.8402, size = 168, normalized size = 0.6 \[ \frac{2 \tilde{\infty } \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt [3]{b} \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [C]  time = 2.33455, size = 455, normalized size = 1.61 \[ -\frac{4 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (i \left (\sqrt{3}-1\right ) \sqrt [3]{a} \sqrt{-\frac{i \left (2 \sqrt [3]{a}+\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x\right )}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )+\frac{1}{2} \left (i \left (-3+(2+i) \sqrt{3}\right ) \sqrt [3]{a}+\left (3-(2-i) \sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{\frac{\left (\sqrt{3}-i\right ) \sqrt [3]{a}+\left (\sqrt{3}+i\right ) \sqrt [3]{b} x}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )\right )}{\left (3-(2-i) \sqrt{3}\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3-a}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \[ \int{x \left ( -\sqrt [3]{b}x+\sqrt [3]{a} \left ( -\sqrt{3}+1 \right ) \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}-a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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