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

Optimal. Leaf size=256 \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{b x^3-a}}{b \left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-x \sqrt [3]{\frac{b}{a}}\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}+x \sqrt [3]{\frac{b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}{-\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{1-x \sqrt [3]{\frac{b}{a}}}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} \sqrt{b x^3-a}} \]

[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 = 0.179784, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{b x^3-a}}{b \left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-x \sqrt [3]{\frac{b}{a}}\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}+x \sqrt [3]{\frac{b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}{-\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{1-x \sqrt [3]{\frac{b}{a}}}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} \sqrt{b x^3-a}} \]

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 = 35.6471, size = 449, normalized size = 1.75 \[ - \frac{\sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 + 4 \sqrt{3}\right )}{b^{\frac{2}{3}} \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- a + b x^{3}}} - \frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{- a + b x^{3}}}{b^{\frac{2}{3}} \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (1 + \sqrt{3}\right ) \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right ) \left (- \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} + \sqrt [3]{b}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} x}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 b^{\frac{2}{3}} \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- a + b x^{3}}} \]

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)*(1 + sqrt(3))*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)*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 = 0.559792, size = 267, normalized size = 1.04 \[ \frac{2 \sqrt [3]{-a} \sqrt{-\frac{(-1)^{5/6} \left ((-a)^{2/3} \sqrt [3]{-b} x+a\right )}{a}} \sqrt{\frac{\sqrt [3]{-b} x \left (\sqrt [3]{-a}+\sqrt [3]{-b} x\right )}{(-a)^{2/3}}+1} \left (i \left (-\sqrt{3} \sqrt [3]{-a} \sqrt [3]{\frac{b}{a}}+\sqrt{3} \sqrt [3]{-b}+3 \sqrt [3]{-b}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{-a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+3 (-1)^{2/3} \sqrt [3]{-a} \sqrt [3]{\frac{b}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{-a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} (-b)^{2/3} \sqrt{b x^3-a}} \]

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 = 0.029, size = 953, normalized size = 3.7 \[ \text{result too large to display} \]

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]

2/3*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)*Elliptic
F(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 = 1.72006, 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)