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

Optimal. Leaf size=768 \[ -\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{\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 [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{\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 )}{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} \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{\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{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)*a^(1/6)*((1 + Sqrt[3])*a^(1/3) - 2*b^(1/3)*x))/(Sqrt[2]*S
qrt[-a - b*x^3])])/(Sqrt[2]*b^(2/3)) + (3^(1/4)*a^(1/6)*ArcTan[(3^(1/4)*(1 - Sqr
t[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^(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[A
rcSin[((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 = 0.729289, antiderivative size = 768, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\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{\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 [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{\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 )}{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} \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{\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{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}} \]

Warning: Unable to verify antiderivative.

[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)*a^(1/6)*((1 + Sqrt[3])*a^(1/3) - 2*b^(1/3)*x))/(Sqrt[2]*S
qrt[-a - b*x^3])])/(Sqrt[2]*b^(2/3)) + (3^(1/4)*a^(1/6)*ArcTan[(3^(1/4)*(1 - Sqr
t[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^(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[A
rcSin[((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 = 35.5355, size = 73, normalized size = 0.1 \[ - \frac{x^{2} \sqrt{- a - b x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},- \frac{1}{2},1,\frac{5}{3},- \frac{b x^{3}}{a},- \frac{b x^{3}}{2 a \left (5 + 3 \sqrt{3}\right )} \right )}}{4 a \sqrt{1 + \frac{b x^{3}}{a}} \left (5 + 3 \sqrt{3}\right )} \]

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*(5 +
3*sqrt(3))))/(4*a*sqrt(1 + b*x**3/a)*(5 + 3*sqrt(3)))

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Mathematica [C]  time = 0.522787, size = 253, normalized size = 0.33 \[ -\frac{10 \left (26+15 \sqrt{3}\right ) a x^2 \sqrt{-a-b x^3} F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}{\left (5+3 \sqrt{3}\right ) \left (2 \left (5+3 \sqrt{3}\right ) a+b x^3\right ) \left (10 \left (5+3 \sqrt{3}\right ) a F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )-3 b x^3 \left (F_1\left (\frac{5}{3};-\frac{1}{2},2;\frac{8}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )-\left (5+3 \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )\right )\right )} \]

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 = 0.138, size = 983, normalized size = 1.3 \[ \text{result too large to display} \]

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)*sum(1/_alpha*(2*3^(1/2)+3)*(-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/b*
((-a*b^2)^(1/3)-I*3^(1/2)*(-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*((-a*b^2)^(1/3)+I*3^(1/2)*(-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*3^(1/2)*(-a*b^2)^(1/3)*_alpha*b+6*I*(-a*b^2)^(1/3)*_alpha*b-6*b^2*
_alpha^2-2*3^(1/2)*(-a*b^2)^(2/3)-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*3^(1/2)*(-a*b^2)^
(1/3)*_alpha^2*b-I*3^(1/2)*(-a*b^2)^(2/3)*_alpha-4*I*(-a*b^2)^(1/3)*_alpha^2*b+I
*3^(1/2)*a*b+2*3^(1/2)*(-a*b^2)^(2/3)*_alpha+2*I*(-a*b^2)^(2/3)*_alpha+2*3^(1/2)
*a*b-2*I*a*b-3*(-a*b^2)^(2/3)*_alpha-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 = 0.557379, 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