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

Optimal. Leaf size=540 \[ \frac{2 \sqrt{2-\sqrt{3}} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b}-\left (1+\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{\frac{b}{a}}\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}-\frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{-a-b x^3}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )} \]

[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]*E
llipticE[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)*Sq
rt[(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])

_______________________________________________________________________________________

Rubi [A]  time = 0.388655, antiderivative size = 540, 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 \[ \frac{2 \sqrt{2-\sqrt{3}} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b}-\left (1+\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{\frac{b}{a}}\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}}-\frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{-a-b x^3}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )} \]

Warning: Unable to verify antiderivative.

[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]*E
llipticE[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)*Sq
rt[(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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.4923, size = 456, normalized size = 0.84 \[ \frac{\sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 + 4 \sqrt{3}\right )}{b^{\frac{2}{3}} \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- a - b x^{3}}} + \frac{2 \sqrt [3]{\frac{b}{a}} \sqrt{- a - b x^{3}}}{b^{\frac{2}{3}} \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (\sqrt [3]{a} \sqrt [3]{\frac{b}{a}} \left (1 + \sqrt{3}\right ) - \sqrt [3]{b} \left (- \sqrt{3} + 1\right )\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 b^{\frac{2}{3}} \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- a - b x^{3}}} \]

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
*sqrt(3))/(3*b**(2/3)*sqrt(-a**(1/3)*(a**(1/3) + b**(1/3)*x)/(-a**(1/3)*(-1 + sq
rt(3)) + b**(1/3)*x)**2)*sqrt(-a - b*x**3))

_______________________________________________________________________________________

Mathematica [C]  time = 0.484302, size = 245, normalized size = 0.45 \[ \frac{2 i \sqrt [3]{-a} \sqrt{-\frac{(-1)^{5/6} \left ((-a)^{2/3} \sqrt [3]{b} x+a\right )}{a}} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{-a}+\sqrt [3]{b} x\right )}{(-a)^{2/3}}+1} \left (\left (\sqrt{3} \sqrt [3]{-a} \sqrt [3]{\frac{b}{a}}+\left (\sqrt{3}-3\right ) \sqrt [3]{b}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{b} x}{\sqrt [3]{-a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-3 \sqrt [6]{-1} \sqrt [3]{-a} \sqrt [3]{\frac{b}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{b} x}{\sqrt [3]{-a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} b^{2/3} \sqrt{-a-b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((2*I)*(-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)^(1/6)*(-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)] + ((-3 + Sqrt[3])*b^(1/3) + Sqrt[3]*(-a)^(1/3)*(b/a)^(1/3))*EllipticF[A
rcSin[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])

_______________________________________________________________________________________

Maple [B]  time = 0.028, size = 1013, normalized size = 1.9 \[ \text{result too large to display} \]

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*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))-2/3*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)))

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

Sympy [A]  time = 1.50767, 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

_______________________________________________________________________________________

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)