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

Optimal. Leaf size=549 \[ -\frac{2 \sqrt{2-\sqrt{3}} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b}-\left (1+\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{\frac{b}{a}}\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}+\frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{b x^3-a}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

[Out]

(2*(b/a)^(1/3)*Sqrt[-a + b*x^3])/(b^(2/3)*((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)) -
(3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(b/a)^(1/3)*(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
lipticE[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]])/(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]) - (2*Sqrt[2 - Sqrt[3]]*((1 -
Sqrt[3])*b^(1/3) - (1 + Sqrt[3])*a^(1/3)*(b/a)^(1/3))*(a^(1/3) - b^(1/3)*x)*Sqr
t[(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]*EllipticF[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^(1/4)*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 = 0.495688, antiderivative size = 549, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{2 \sqrt{2-\sqrt{3}} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b}-\left (1+\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{\frac{b}{a}}\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}+\frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{b x^3-a}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b/a)^(1/3)*Sqrt[-a + b*x^3])/(b^(2/3)*((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)) -
(3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(b/a)^(1/3)*(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
lipticE[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]])/(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]) - (2*Sqrt[2 - Sqrt[3]]*((1 -
Sqrt[3])*b^(1/3) - (1 + Sqrt[3])*a^(1/3)*(b/a)^(1/3))*(a^(1/3) - b^(1/3)*x)*Sqr
t[(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]*EllipticF[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^(1/4)*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 = 37.3886, size = 454, normalized size = 0.83 \[ - \frac{\sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 + 4 \sqrt{3}\right )}{b^{\frac{2}{3}} \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- a + b x^{3}}} - \frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{- a + b x^{3}}}{b^{\frac{2}{3}} \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) \left (\sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} \left (- \sqrt{3} + 1\right )\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 b^{\frac{2}{3}} \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- a + b x^{3}}} \]

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(-sqrt(3) + 2)*(a**(1/3) - b**(1/3)*x)*(a**(1/3
)*(b/a)**(1/3)*(1 + sqrt(3)) - b**(1/3)*(-sqrt(3) + 1))*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*s
qrt(3))/(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))

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Mathematica [C]  time = 0.570496, size = 267, normalized size = 0.49 \[ \frac{2 \sqrt [3]{-a} \sqrt{-\frac{(-1)^{5/6} \left ((-a)^{2/3} \sqrt [3]{-b} x+a\right )}{a}} \sqrt{\frac{\sqrt [3]{-b} x \left (\sqrt [3]{-a}+\sqrt [3]{-b} x\right )}{(-a)^{2/3}}+1} \left (i \left (-\sqrt{3} \sqrt [3]{-a} \sqrt [3]{\frac{b}{a}}+\sqrt{3} \sqrt [3]{-b}-3 \sqrt [3]{-b}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{-a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+3 (-1)^{2/3} \sqrt [3]{-a} \sqrt [3]{\frac{b}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{-a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} (-b)^{2/3} \sqrt{b x^3-a}} \]

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 + (-a)^(2/3)*(-b)^(1/3)*x))/a)]*Sqrt[1 + ((-
b)^(1/3)*x*((-a)^(1/3) + (-b)^(1/3)*x))/(-a)^(2/3)]*(3*(-1)^(2/3)*(-a)^(1/3)*(b/
a)^(1/3)*EllipticE[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/3)*x)/(-a)^(1/3)]/3^(1/4
)], (-1)^(1/3)] + I*(-3*(-b)^(1/3) + 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 = 0.02, size = 953, normalized size = 1.7 \[ \text{result too large to display} \]

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)*Ellipt
icF(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 = 1.7375, 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)