3.347 \(\int \frac{x \sqrt{a-b x^3}}{2 \left (5-3 \sqrt{3}\right ) a-b x^3} \, dx\)

Optimal. Leaf size=758 \[ \frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{2 \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}} \]

[Out]

(2*Sqrt[a - b*x^3])/(b^(2/3)*((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)) - (3^(1/4)*a^(
1/6)*ArcTan[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqrt[2]*Sqrt[
a - b*x^3])])/(2*Sqrt[2]*b^(2/3)) - (3^(1/4)*a^(1/6)*ArcTan[(3^(1/4)*a^(1/6)*((1
- Sqrt[3])*a^(1/3) + 2*b^(1/3)*x))/(Sqrt[2]*Sqrt[a - b*x^3])])/(Sqrt[2]*b^(2/3)
) + (3^(3/4)*a^(1/6)*ArcTanh[(3^(1/4)*(1 - Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x
))/(Sqrt[2]*Sqrt[a - b*x^3])])/(2*Sqrt[2]*b^(2/3)) + (a^(1/6)*ArcTanh[((1 + Sqrt
[3])*Sqrt[a - b*x^3])/(Sqrt[2]*3^(3/4)*Sqrt[a])])/(Sqrt[2]*3^(1/4)*b^(2/3)) - (3
^(1/4)*Sqrt[2 - Sqrt[3]]*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]*EllipticE[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]*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]*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 = 0.657208, antiderivative size = 758, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{2 \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt [4]{3} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{2} \sqrt{a-b x^3}}\right )}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt [6]{a} \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right )}{\sqrt{2} \sqrt [4]{3} b^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(x*Sqrt[a - b*x^3])/(2*(5 - 3*Sqrt[3])*a - b*x^3),x]

[Out]

(2*Sqrt[a - b*x^3])/(b^(2/3)*((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)) - (3^(1/4)*a^(
1/6)*ArcTan[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqrt[2]*Sqrt[
a - b*x^3])])/(2*Sqrt[2]*b^(2/3)) - (3^(1/4)*a^(1/6)*ArcTan[(3^(1/4)*a^(1/6)*((1
- Sqrt[3])*a^(1/3) + 2*b^(1/3)*x))/(Sqrt[2]*Sqrt[a - b*x^3])])/(Sqrt[2]*b^(2/3)
) + (3^(3/4)*a^(1/6)*ArcTanh[(3^(1/4)*(1 - Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x
))/(Sqrt[2]*Sqrt[a - b*x^3])])/(2*Sqrt[2]*b^(2/3)) + (a^(1/6)*ArcTanh[((1 + Sqrt
[3])*Sqrt[a - b*x^3])/(Sqrt[2]*3^(3/4)*Sqrt[a])])/(Sqrt[2]*3^(1/4)*b^(2/3)) - (3
^(1/4)*Sqrt[2 - Sqrt[3]]*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]*EllipticE[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]*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]*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 in Sympy [A]  time = 35.4711, size = 66, normalized size = 0.09 \[ \frac{x^{2} \sqrt{a - b x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},- \frac{1}{2},1,\frac{5}{3},\frac{b x^{3}}{a},\frac{b x^{3}}{2 a \left (- 3 \sqrt{3} + 5\right )} \right )}}{4 a \sqrt{1 - \frac{b x^{3}}{a}} \left (- 3 \sqrt{3} + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(-b*x**3+a)**(1/2)/(-b*x**3+2*a*(5-3*3**(1/2))),x)

[Out]

x**2*sqrt(a - b*x**3)*appellf1(2/3, -1/2, 1, 5/3, b*x**3/a, b*x**3/(2*a*(-3*sqrt
(3) + 5)))/(4*a*sqrt(1 - b*x**3/a)*(-3*sqrt(3) + 5))

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Mathematica [C]  time = 0.597577, size = 242, normalized size = 0.32 \[ -\frac{10 \left (26-15 \sqrt{3}\right ) a x^2 \sqrt{a-b x^3} F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )}{\left (3 \sqrt{3}-5\right ) \left (2 \left (3 \sqrt{3}-5\right ) a+b x^3\right ) \left (10 \left (3 \sqrt{3}-5\right ) a F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )-3 b x^3 \left (F_1\left (\frac{5}{3};-\frac{1}{2},2;\frac{8}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )+\left (3 \sqrt{3}-5\right ) F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x*Sqrt[a - b*x^3])/(2*(5 - 3*Sqrt[3])*a - b*x^3),x]

[Out]

(-10*(26 - 15*Sqrt[3])*a*x^2*Sqrt[a - b*x^3]*AppellF1[2/3, -1/2, 1, 5/3, (b*x^3)
/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)])/((-5 + 3*Sqrt[3])*(2*(-5 + 3*Sqrt[3])*a + b*x
^3)*(10*(-5 + 3*Sqrt[3])*a*AppellF1[2/3, -1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a
- 6*Sqrt[3]*a)] - 3*b*x^3*(AppellF1[5/3, -1/2, 2, 8/3, (b*x^3)/a, (b*x^3)/(10*a
- 6*Sqrt[3]*a)] + (-5 + 3*Sqrt[3])*AppellF1[5/3, 1/2, 1, 8/3, (b*x^3)/a, (b*x^3)
/(10*a - 6*Sqrt[3]*a)])))

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Maple [C]  time = 0.214, size = 924, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-b*x^3+a)^(1/2)/(-b*x^3+2*a*(5-3*3^(1/2))),x)

[Out]

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)*((-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)))-1/9*I/b^3*2^(1/2)*su
m(1/_alpha*(2*3^(1/2)-3)*(a*b^2)^(1/3)*(-1/2*I*b*(2*x+1/b*(I*3^(1/2)*(a*b^2)^(1/
3)+(a*b^2)^(1/3)))/(a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(a*b^2)^(1/3))/(-3*(a*b^2)^(1/
3)-I*3^(1/2)*(a*b^2)^(1/3)))^(1/2)*(1/2*I*b*(2*x+1/b*(-I*3^(1/2)*(a*b^2)^(1/3)+(
a*b^2)^(1/3)))/(a*b^2)^(1/3))^(1/2)/(-b*x^3+a)^(1/2)*(-3*I*(a*b^2)^(1/3)*_alpha*
3^(1/2)*b+4*b^2*_alpha^2*3^(1/2)+3*I*(a*b^2)^(2/3)*3^(1/2)-2*(a*b^2)^(1/3)*_alph
a*3^(1/2)*b-6*I*(a*b^2)^(1/3)*_alpha*b+6*b^2*_alpha^2-2*(a*b^2)^(2/3)*3^(1/2)+6*
I*(a*b^2)^(2/3)-3*(a*b^2)^(1/3)*_alpha*b-3*(a*b^2)^(2/3))*EllipticPi(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),1/6/b*(-2*I*(a*b^2)^(1/3)*_alpha^2*3^(1/2)*b+I*(a*b^2)^(2/3)*_alpha*3^
(1/2)-4*I*(a*b^2)^(1/3)*_alpha^2*b-2*(a*b^2)^(2/3)*_alpha*3^(1/2)+2*I*(a*b^2)^(2
/3)*_alpha+I*3^(1/2)*a*b-3*(a*b^2)^(2/3)*_alpha+2*3^(1/2)*a*b+2*I*a*b+3*a*b)/a,(
-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)),_alpha=RootOf(b*_Z^3+6*a*3^(1/2)-10*a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{-b x^{3} + a} x}{b x^{3} + 2 \, a{\left (3 \, \sqrt{3} - 5\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-b*x^3 + a)*x/(b*x^3 + 2*a*(3*sqrt(3) - 5)),x, algorithm="maxima")

[Out]

-integrate(sqrt(-b*x^3 + a)*x/(b*x^3 + 2*((3*sqrt(3)) - 5)*a), x)

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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(-sqrt(-b*x^3 + a)*x/(b*x^3 + 2*a*(3*sqrt(3) - 5)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x \sqrt{a - b x^{3}}}{- 10 a + 6 \sqrt{3} a + b x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(-b*x**3+a)**(1/2)/(-b*x**3+2*a*(5-3*3**(1/2))),x)

[Out]

-Integral(x*sqrt(a - b*x**3)/(-10*a + 6*sqrt(3)*a + b*x**3), x)

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GIAC/XCAS [A]  time = 0.579187, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-b*x^3 + a)*x/(b*x^3 + 2*a*(3*sqrt(3) - 5)),x, algorithm="giac")

[Out]

sage0*x