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

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

[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 = 0.244809, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{2 \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{-a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]

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 = 2.31099, size = 410, normalized size = 7.74 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (3 i \sqrt [3]{a} \sqrt{\frac{\left (\sqrt{3}+i\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-\frac{\sqrt [4]{3} \left (\left (\sqrt{3}+i\right ) \sqrt [3]{a}-\left (\sqrt{3}-i\right ) \sqrt [3]{b} x\right ) \sqrt{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )}{2 \sqrt{2}}\right )}{\left (\sqrt [3]{-1}-2\right ) \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{-a-b x^3}} \]

Warning: Unable to verify antiderivative.

[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))]*(-(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])/(2*Sqrt[2]) + (3*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]), A
rcSin[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 + (-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 = 0.119, 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 = 0.706053, size = 1, normalized size = 0.02 \[ \left [\frac{1}{6} \, a^{\frac{1}{3}} \sqrt{-\frac{1}{a b^{\frac{2}{3}}}} \log \left (\frac{{\left (b^{2} x^{6} - 88 \, a b x^{3} + 136 \, a^{2}\right )} a^{\frac{2}{3}} b^{\frac{2}{3}} + 12 \,{\left (6 \, \sqrt{-b x^{3} - a} a^{2} b^{\frac{5}{3}} x^{2} +{\left (13 \, a b^{2} x^{3} + 10 \, a^{2} b\right )} \sqrt{-b x^{3} - a} a^{\frac{2}{3}} +{\left (a b^{2} x^{4} + 4 \, a^{2} b x\right )} \sqrt{-b x^{3} - a} a^{\frac{1}{3}} b^{\frac{1}{3}}\right )} \sqrt{-\frac{1}{a b^{\frac{2}{3}}}} + 12 \,{\left (17 \, a b^{2} x^{4} - 4 \, a^{2} b x\right )} a^{\frac{1}{3}} + 12 \,{\left (5 \, a b^{2} x^{5} + 26 \, a^{2} b x^{2}\right )} b^{\frac{1}{3}}}{{\left (b^{2} x^{6} - 160 \, a b x^{3} + 64 \, a^{2}\right )} a^{\frac{2}{3}} b^{\frac{2}{3}} + 12 \,{\left (5 \, a b^{2} x^{4} - 16 \, a^{2} b x\right )} a^{\frac{1}{3}} - 12 \,{\left (a b^{2} x^{5} - 20 \, a^{2} b x^{2}\right )} b^{\frac{1}{3}}}\right ), -\frac{1}{3} \, a^{\frac{1}{3}} \sqrt{\frac{1}{a b^{\frac{2}{3}}}} \arctan \left (-\frac{12 \, a^{\frac{2}{3}} b x^{2} - 6 \, a b^{\frac{2}{3}} x +{\left (b x^{3} + 10 \, a\right )} a^{\frac{1}{3}} b^{\frac{1}{3}}}{6 \,{\left (\sqrt{-b x^{3} - a} a b x + \sqrt{-b x^{3} - a} a^{\frac{4}{3}} b^{\frac{2}{3}}\right )} \sqrt{\frac{1}{a b^{\frac{2}{3}}}}}\right )\right ] \]

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*sqrt(-b*x^3 - a)*a^2*b^(5/3)*x^2 + (13*a*b^2*x^3 + 10*a^2*b)*sqr
t(-b*x^3 - a)*a^(2/3) + (a*b^2*x^4 + 4*a^2*b*x)*sqrt(-b*x^3 - a)*a^(1/3)*b^(1/3)
)*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)))*arctan(-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))/((sqrt(-b*x^3 - a)*a*b*x + sqrt(-b*x^3 - a)*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. \[ - \int \frac{\sqrt [3]{a}}{- 2 \sqrt [3]{a} \sqrt{- a - b x^{3}} + \sqrt [3]{b} x \sqrt{- a - b x^{3}}}\, dx - \int \frac{\sqrt [3]{b} x}{- 2 \sqrt [3]{a} \sqrt{- a - b x^{3}} + \sqrt [3]{b} x \sqrt{- a - b x^{3}}}\, dx \]

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