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

Optimal. Leaf size=278 \[ \frac{2 \sqrt{\frac{7}{6}+\frac{2}{\sqrt{3}}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{a+b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}} \]

[Out]

-((Sqrt[2]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*(a^(1/3) + b^(1/3)*x))/Sqrt[a +
b*x^3]])/(3^(3/4)*a^(1/6)*b^(2/3))) + (2*Sqrt[7/6 + 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^(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])

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Rubi [A]  time = 0.877481, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\sqrt{2 \left (7+4 \sqrt{3}\right )} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt{a+b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*(a^(1/3) + b^(1/3)*x))/Sqrt[a +
b*x^3]])/(3^(3/4)*a^(1/6)*b^(2/3))) + (Sqrt[2*(7 + 4*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^(3/4)*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])

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Rubi in Sympy [A]  time = 76.9876, size = 444, normalized size = 1.6 \[ \frac{\sqrt [4]{3} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 b^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{\sqrt [4]{3} \sqrt{\frac{a^{\frac{2}{3}} \left (1 - \frac{\sqrt [3]{b} x}{\sqrt [3]{a}} + \frac{b^{\frac{2}{3}} x^{2}}{a^{\frac{2}{3}}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (- \sqrt{3} + 1\right ) \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \operatorname{atanh}{\left (\frac{\left (- \sqrt{3} + 2\right ) \sqrt{- \frac{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}} + 1}}{\sqrt{\frac{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{b} x\right )^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{3 b^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3**(1/4)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3)*(1 + sq
rt(3)) + b**(1/3)*x)**2)*(1 + sqrt(3))*sqrt(sqrt(3) + 2)*(a**(1/3) + b**(1/3)*x)
*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)) + 3**(1/4)*sqrt(a
**(2/3)*(1 - b**(1/3)*x/a**(1/3) + b**(2/3)*x**2/a**(2/3))/(a**(1/3)*(1 + sqrt(3
)) + b**(1/3)*x)**2)*(-sqrt(3) + 1)*(a**(1/3) + b**(1/3)*x)*atanh((-sqrt(3) + 2)
*sqrt(-(a**(1/3)*(-1 + sqrt(3)) - b**(1/3)*x)**2/(a**(1/3)*(1 + sqrt(3)) + b**(1
/3)*x)**2 + 1)/sqrt((a**(1/3)*(-1 + sqrt(3)) - b**(1/3)*x)**2/(a**(1/3)*(1 + sqr
t(3)) + b**(1/3)*x)**2 - 4*sqrt(3) + 7))/(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(-sqrt(3) + 2)*sqrt(a +
b*x**3))

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Mathematica [C]  time = 2.05345, size = 427, normalized size = 1.54 \[ -\frac{4 \sqrt{\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (i \left (\sqrt{3}-1\right ) \sqrt [3]{a} \sqrt{\frac{\left (\sqrt{3}+i\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-\frac{i \sqrt [4]{3} \left (\left (\sqrt{3}+(-2-i)\right ) \sqrt [3]{a}+\left ((1+2 i)-i \sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (i+\sqrt{3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )}{2 \sqrt{2}}\right )}{\left (3-(2-i) \sqrt{3}\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-4*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((-I/2)*3^(1/4)*(((-
2 - I) + Sqrt[3])*a^(1/3) + ((1 + 2*I) - 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])/Sqrt[2] + I*(-1
+ Sqrt[3])*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 + (1 + 2*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]))/((3 - (2 - I)*
Sqrt[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])

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \[ \int{x \left ( \sqrt [3]{b}x+\sqrt [3]{a} \left ( -\sqrt{3}+1 \right ) \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{b x^{3} + a}{\left (b^{\frac{1}{3}} x - a^{\frac{1}{3}}{\left (\sqrt{3} - 1\right )}\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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