3.76 \(\int \frac{x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{-a-b x^3}} \, dx\)

Optimal. Leaf size=273 \[ \frac{4 \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{-a-b x^3}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}} \]

[Out]

(4*ArcTan[(a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*Sqrt[-a - b*x^3])])/(9*a^(1/6)*b^(2
/3)) - (2*Sqrt[2 - Sqrt[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*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 = 0.589524, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{4 \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{-a-b x^3}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{-a-b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(4*ArcTan[(a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*Sqrt[-a - b*x^3])])/(9*a^(1/6)*b^(2
/3)) - (2*Sqrt[2 - Sqrt[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*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 2.71778, size = 410, normalized size = 1.5 \[ \frac{\sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (8 i \sqrt [3]{a} \sqrt{\frac{\left (\sqrt{3}+i\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-\sqrt{2} \sqrt [4]{3} \left (\left (\sqrt{3}+i\right ) \sqrt [3]{a}-\left (\sqrt{3}-i\right ) \sqrt [3]{b} x\right ) \sqrt{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )\right )}{2 \left (\sqrt [3]{-1}-2\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{-a-b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-(Sqrt[2]*3^(1/4)*((I +
Sqrt[3])*a^(1/3) - (-I + Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*
x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-
3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]) + (8*I)*a^(1/3)*Sqrt[((-2*I)*a^(1
/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/
a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[
Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1
+ I*Sqrt[3])/2]))/(2*(-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/
3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a - b*x^3])

_______________________________________________________________________________________

Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{x \left ( 2\,\sqrt [3]{a}-\sqrt [3]{b}x \right ) ^{-1}{\frac{1}{\sqrt{-b{x}^{3}-a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x/(2*a^(1/3)-b^(1/3)*x)/(-b*x^3-a)^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x}{\sqrt{-b x^{3} - a}{\left (b^{\frac{1}{3}} x - 2 \, a^{\frac{1}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(sqrt(-b*x^3 - a)*(b^(1/3)*x - 2*a^(1/3))),x, algorithm="maxima")

[Out]

-integrate(x/(sqrt(-b*x^3 - a)*(b^(1/3)*x - 2*a^(1/3))), x)

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(sqrt(-b*x^3 - a)*(b^(1/3)*x - 2*a^(1/3))),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{- 2 \sqrt [3]{a} \sqrt{- a - b x^{3}} + \sqrt [3]{b} x \sqrt{- a - b x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(-2*a**(1/3)*sqrt(-a - b*x**3) + b**(1/3)*x*sqrt(-a - b*x**3)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x}{\sqrt{-b x^{3} - a}{\left (b^{\frac{1}{3}} x - 2 \, a^{\frac{1}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(sqrt(-b*x^3 - a)*(b^(1/3)*x - 2*a^(1/3))),x, algorithm="giac")

[Out]

integrate(-x/(sqrt(-b*x^3 - a)*(b^(1/3)*x - 2*a^(1/3))), x)