3.104 \(\int \frac{1-\sqrt{3}-\sqrt [3]{\frac{b}{a}} x}{\sqrt{a-b x^3}} \, dx\)

Optimal. Leaf size=248 \[ \frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (1-x \sqrt [3]{\frac{b}{a}}\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}+x \sqrt [3]{\frac{b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}{-\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{\frac{1-x \sqrt [3]{\frac{b}{a}}}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )+\sqrt{3}+1\right )^2}} \sqrt{a-b x^3}}-\frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{a-b x^3}}{b \left (x \left (-\sqrt [3]{\frac{b}{a}}\right )+\sqrt{3}+1\right )} \]

[Out]

(-2*(b/a)^(2/3)*Sqrt[a - b*x^3])/(b*(1 + Sqrt[3] - (b/a)^(1/3)*x)) + (3^(1/4)*Sq
rt[2 - Sqrt[3]]*(1 - (b/a)^(1/3)*x)*Sqrt[(1 + (b/a)^(1/3)*x + (b/a)^(2/3)*x^2)/(
1 + Sqrt[3] - (b/a)^(1/3)*x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] - (b/a)^(1/3)*x)/(
1 + Sqrt[3] - (b/a)^(1/3)*x)], -7 - 4*Sqrt[3]])/((b/a)^(1/3)*Sqrt[(1 - (b/a)^(1/
3)*x)/(1 + Sqrt[3] - (b/a)^(1/3)*x)^2]*Sqrt[a - b*x^3])

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Rubi [A]  time = 0.152683, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029 \[ \frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (1-x \sqrt [3]{\frac{b}{a}}\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}+x \sqrt [3]{\frac{b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}{-\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{\frac{1-x \sqrt [3]{\frac{b}{a}}}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )+\sqrt{3}+1\right )^2}} \sqrt{a-b x^3}}-\frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{a-b x^3}}{b \left (x \left (-\sqrt [3]{\frac{b}{a}}\right )+\sqrt{3}+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b/a)^(2/3)*Sqrt[a - b*x^3])/(b*(1 + Sqrt[3] - (b/a)^(1/3)*x)) + (3^(1/4)*Sq
rt[2 - Sqrt[3]]*(1 - (b/a)^(1/3)*x)*Sqrt[(1 + (b/a)^(1/3)*x + (b/a)^(2/3)*x^2)/(
1 + Sqrt[3] - (b/a)^(1/3)*x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] - (b/a)^(1/3)*x)/(
1 + Sqrt[3] - (b/a)^(1/3)*x)], -7 - 4*Sqrt[3]])/((b/a)^(1/3)*Sqrt[(1 - (b/a)^(1/
3)*x)/(1 + Sqrt[3] - (b/a)^(1/3)*x)^2]*Sqrt[a - b*x^3])

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Rubi in Sympy [A]  time = 35.7004, size = 444, normalized size = 1.79 \[ - \frac{\sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{b^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \sqrt{a - b x^{3}}} - \frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{a - b x^{3}}}{b^{\frac{2}{3}} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \left (- \sqrt{3} + 1\right ) \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) \left (- \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} + \sqrt [3]{b}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 b^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \sqrt{a - b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3**(1/4)*a**(1/3)*(b/a)**(1/3)*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)*sqrt(-sqrt(3) + 2)*(a**(1/3) - b
**(1/3)*x)*elliptic_e(asin((a**(1/3)*(-1 + sqrt(3)) + b**(1/3)*x)/(a**(1/3)*(1 +
sqrt(3)) - b**(1/3)*x)), -7 - 4*sqrt(3))/(b**(2/3)*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)) - 2*(b/a)*
*(1/3)*sqrt(a - b*x**3)/(b**(2/3)*(a**(1/3)*(1 + sqrt(3)) - b**(1/3)*x)) + 2*3**
(3/4)*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)*(-sqrt(3) + 1)*sqrt(sqrt(3) + 2)*(a**(1/3) - b**(1/3)*x)*(
-a**(1/3)*(b/a)**(1/3) + b**(1/3))*elliptic_f(asin((a**(1/3)*(-1 + sqrt(3)) + b*
*(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) - b**(1/3)*x)), -7 - 4*sqrt(3))/(3*b**(2/3)*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 = 0.397025, size = 232, normalized size = 0.94 \[ \frac{2 \sqrt [3]{a} \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{b} x-\sqrt [3]{a}\right )}{\sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \left (i \left (\left (\sqrt{3}-3\right ) \sqrt [3]{b}-\sqrt{3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+3 (-1)^{2/3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} b^{2/3} \sqrt{a-b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*a^(1/3)*Sqrt[((-1)^(5/6)*(-a^(1/3) + b^(1/3)*x))/a^(1/3)]*Sqrt[1 + (b^(1/3)*x
)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*(3*(-1)^(2/3)*a^(1/3)*(b/a)^(1/3)*EllipticE[A
rcSin[Sqrt[-(-1)^(5/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)] + I*((-3 +
Sqrt[3])*b^(1/3) - Sqrt[3]*a^(1/3)*(b/a)^(1/3))*EllipticF[ArcSin[Sqrt[-(-1)^(5/
6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)]))/(3^(3/4)*b^(2/3)*Sqrt[a - b*
x^3])

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Maple [B]  time = 0.05, size = 950, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*I/b*(a*b^2)^(1/3)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3
^(1/2)*b/(a*b^2)^(1/3))^(1/2)*((x-1/b*(a*b^2)^(1/3))/(-3/2/b*(a*b^2)^(1/3)-1/2*I
*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2)*(I*(x+1/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^
2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)/(-b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*
(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3
))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^
2)^(1/3)))^(1/2))-2/3*I*(b/a)^(1/3)*3^(1/2)/b*(a*b^2)^(1/3)*(-I*(x+1/2/b*(a*b^2)
^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)*((x-1/b*(a*
b^2)^(1/3))/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2)*(I*(x+1/
2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)/
(-b*x^3+a)^(1/2)*((-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*EllipticE
(1/3*3^(1/2)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b
/(a*b^2)^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^
(1/2)/b*(a*b^2)^(1/3)))^(1/2))+1/b*(a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(-I*(x+1/
2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2),
(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3))
)^(1/2)))+2/3*I*3^(1/2)/b*(a*b^2)^(1/3)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)
/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)*((x-1/b*(a*b^2)^(1/3))/(-3/2/b*
(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2)*(I*(x+1/2/b*(a*b^2)^(1/3)-1/
2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)/(-b*x^3+a)^(1/2)*Ell
ipticF(1/3*3^(1/2)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(
1/2)*b/(a*b^2)^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/
2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.75645, size = 0, normalized size = 0. \[ \mathrm{NaN} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

nan

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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