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

Optimal. Leaf size=53 \[ -\frac{2 \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{b x^3-a}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]

[Out]

(-2*ArcTan[(a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*Sqrt[-a + b*x^3])])/(3*a^(1/6)*b^(
1/3))

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Rubi [A]  time = 0.243995, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.044 \[ -\frac{2 \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{b x^3-a}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcTan[(a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*Sqrt[-a + b*x^3])])/(3*a^(1/6)*b^(
1/3))

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

[Out]

Timed out

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Mathematica [C]  time = 1.23532, size = 371, normalized size = 7. \[ -\frac{2 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\left (\sqrt [3]{-1}-2\right ) \left (\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )+\sqrt [3]{-1} \sqrt{3} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )\right )}{\left (\sqrt [3]{-1}-2\right ) \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3-a}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-2 + (-1)^(1/3))*((
-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x)
)/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3
)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] + (-1)^(1/3)*Sqrt[3]*(1 + (-1)^(1
/3))*a^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*S
qrt[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[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1
/3))]], (-1)^(1/3)]))/((-2 + (-1)^(1/3))*b^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1
/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])

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Maple [F]  time = 0.124, size = 0, normalized size = 0. \[ \int{1 \left ( \sqrt [3]{a}-\sqrt [3]{b}x \right ) \left ( 2\,\sqrt [3]{a}+\sqrt [3]{b}x \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}-a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.711295, size = 1, normalized size = 0.02 \[ \left [\frac{1}{6} \, a^{\frac{1}{3}} \sqrt{-\frac{1}{a b^{\frac{2}{3}}}} \log \left (\frac{{\left (b^{2} x^{6} + 88 \, a b x^{3} + 136 \, a^{2}\right )} a^{\frac{2}{3}} b^{\frac{2}{3}} - 12 \,{\left (6 \, a^{2} b^{\frac{5}{3}} x^{2} -{\left (13 \, a b^{2} x^{3} - 10 \, a^{2} b\right )} a^{\frac{2}{3}} +{\left (a b^{2} x^{4} - 4 \, a^{2} b x\right )} a^{\frac{1}{3}} b^{\frac{1}{3}}\right )} \sqrt{b x^{3} - a} \sqrt{-\frac{1}{a b^{\frac{2}{3}}}} + 12 \,{\left (17 \, a b^{2} x^{4} + 4 \, a^{2} b x\right )} a^{\frac{1}{3}} - 12 \,{\left (5 \, a b^{2} x^{5} - 26 \, a^{2} b x^{2}\right )} b^{\frac{1}{3}}}{{\left (b^{2} x^{6} + 160 \, a b x^{3} + 64 \, a^{2}\right )} a^{\frac{2}{3}} b^{\frac{2}{3}} + 12 \,{\left (5 \, a b^{2} x^{4} + 16 \, a^{2} b x\right )} a^{\frac{1}{3}} + 12 \,{\left (a b^{2} x^{5} + 20 \, a^{2} b x^{2}\right )} b^{\frac{1}{3}}}\right ), -\frac{1}{3} \, a^{\frac{1}{3}} \sqrt{\frac{1}{a b^{\frac{2}{3}}}} \arctan \left (-\frac{12 \, a^{\frac{2}{3}} b x^{2} + 6 \, a b^{\frac{2}{3}} x -{\left (b x^{3} - 10 \, a\right )} a^{\frac{1}{3}} b^{\frac{1}{3}}}{6 \, \sqrt{b x^{3} - a}{\left (a b x - a^{\frac{4}{3}} b^{\frac{2}{3}}\right )} \sqrt{\frac{1}{a b^{\frac{2}{3}}}}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*a^(1/3)*sqrt(-1/(a*b^(2/3)))*log(((b^2*x^6 + 88*a*b*x^3 + 136*a^2)*a^(2/3)*
b^(2/3) - 12*(6*a^2*b^(5/3)*x^2 - (13*a*b^2*x^3 - 10*a^2*b)*a^(2/3) + (a*b^2*x^4
- 4*a^2*b*x)*a^(1/3)*b^(1/3))*sqrt(b*x^3 - a)*sqrt(-1/(a*b^(2/3))) + 12*(17*a*b
^2*x^4 + 4*a^2*b*x)*a^(1/3) - 12*(5*a*b^2*x^5 - 26*a^2*b*x^2)*b^(1/3))/((b^2*x^6
+ 160*a*b*x^3 + 64*a^2)*a^(2/3)*b^(2/3) + 12*(5*a*b^2*x^4 + 16*a^2*b*x)*a^(1/3)
+ 12*(a*b^2*x^5 + 20*a^2*b*x^2)*b^(1/3))), -1/3*a^(1/3)*sqrt(1/(a*b^(2/3)))*arc
tan(-1/6*(12*a^(2/3)*b*x^2 + 6*a*b^(2/3)*x - (b*x^3 - 10*a)*a^(1/3)*b^(1/3))/(sq
rt(b*x^3 - a)*(a*b*x - a^(4/3)*b^(2/3))*sqrt(1/(a*b^(2/3)))))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError