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

Optimal. Leaf size=533 \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{b}-\left (1-\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{\frac{b}{a}}\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}-\frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

[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 + Sq
rt[3])*a^(1/3) - b^(1/3)*x)^2]*Sqrt[a - b*x^3]) - (2*Sqrt[2 + Sqrt[3]]*((1 + Sqr
t[3])*b^(1/3) - (1 - 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]
*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 = 0.397507, antiderivative size = 533, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{b}-\left (1-\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{\frac{b}{a}}\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}-\frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

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 + Sq
rt[3])*a^(1/3) - b^(1/3)*x)^2]*Sqrt[a - b*x^3]) - (2*Sqrt[2 + Sqrt[3]]*((1 + Sqr
t[3])*b^(1/3) - (1 - 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]
*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 = 35.8613, size = 450, normalized size = 0.84 \[ - \frac{\sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{b^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \sqrt{a - b x^{3}}} - \frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{a - b x^{3}}}{b^{\frac{2}{3}} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) \left (- \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \left (- \sqrt{3} + 1\right ) + \sqrt [3]{b} \left (1 + \sqrt{3}\right )\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 b^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}} \sqrt{a - b x^{3}}} \]

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)*(-sqrt(3) + 1) + b**(1/3)*(1 + sqrt(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)*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 = 0.389734, size = 232, normalized size = 0.44 \[ \frac{2 \sqrt [3]{a} \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{b} x-\sqrt [3]{a}\right )}{\sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \left (i \left (\left (3+\sqrt{3}\right ) \sqrt [3]{b}-\sqrt{3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+3 (-1)^{2/3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} b^{2/3} \sqrt{a-b x^3}} \]

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 = 0.056, size = 950, normalized size = 1.8 \[ \text{result too large to display} \]

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]

2/3*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)*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*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)*Ellip
ticF(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 = 1.69432, 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)