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

Optimal. Leaf size=497 \[ \frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),4 \sqrt{3}-7\right )}{\sqrt [3]{b} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{b} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}+\frac{2 \sqrt{b x^3-a}}{\sqrt [3]{b} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

[Out]

(2*Sqrt[-a + b*x^3])/(b^(1/3)*((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)) - (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]*Ellipt
icE[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^(1/3)
*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]) + (4*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]*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]])/(b^(1/3)*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 = 0.132815, antiderivative size = 497, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {1880, 219, 1879} \[ \frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{b} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{b} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}+\frac{2 \sqrt{b x^3-a}}{\sqrt [3]{b} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1880

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a,
3]]}, Dist[(c*r - (1 + Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 + Sqrt[3])*s + r*x)
/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && NeQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x
] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt{-a+b x^3}} \, dx &=-\left (\left (2 \sqrt{3} \sqrt [3]{a}\right ) \int \frac{1}{\sqrt{-a+b x^3}} \, dx\right )+\int \frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt{-a+b x^3}} \, dx\\ &=\frac{2 \sqrt{-a+b x^3}}{\sqrt [3]{b} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\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 )}{\sqrt [3]{b} \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{4 \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}} 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 [3]{b} \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}}\\ \end{align*}

Mathematica [C]  time = 0.0393337, size = 91, normalized size = 0.18 \[ -\frac{x \sqrt{1-\frac{b x^3}{a}} \left (2 \left (\sqrt{3}-1\right ) \sqrt [3]{a} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right )+\sqrt [3]{b} x \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right )\right )}{2 \sqrt{b x^3-a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*Sqrt[1 - (b*x^3)/a]*(2*(-1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + b^(1/3)*x*Hype
rgeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/(2*Sqrt[-a + b*x^3])

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Maple [B]  time = 0.014, size = 952, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*I/b^(2/3)*3^(1/2)*(b^2*a)^(1/3)*(-I*(x+1/2/b*(b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(b^2*a)^(1/3))*3^(1/2)*b/(b^2*
a)^(1/3))^(1/2)*((x-1/b*(b^2*a)^(1/3))/(-3/2/b*(b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(b^2*a)^(1/3)))^(1/2)*(I*(x+1/2/b
*(b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(b^2*a)^(1/3))*3^(1/2)*b/(b^2*a)^(1/3))^(1/2)/(b*x^3-a)^(1/2)*((-3/2/b*(b^2*a)^
(1/3)-1/2*I*3^(1/2)/b*(b^2*a)^(1/3))*EllipticE(1/3*3^(1/2)*(-I*(x+1/2/b*(b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(b^2*a)^
(1/3))*3^(1/2)*b/(b^2*a)^(1/3))^(1/2),(-I*3^(1/2)/b*(b^2*a)^(1/3)/(-3/2/b*(b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(b^2*a
)^(1/3)))^(1/2))+1/b*(b^2*a)^(1/3)*EllipticF(1/3*3^(1/2)*(-I*(x+1/2/b*(b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(b^2*a)^(1
/3))*3^(1/2)*b/(b^2*a)^(1/3))^(1/2),(-I*3^(1/2)/b*(b^2*a)^(1/3)/(-3/2/b*(b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(b^2*a)^
(1/3)))^(1/2)))-2*I*a^(1/3)/b*(b^2*a)^(1/3)*(-I*(x+1/2/b*(b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(b^2*a)^(1/3))*3^(1/2)*
b/(b^2*a)^(1/3))^(1/2)*((x-1/b*(b^2*a)^(1/3))/(-3/2/b*(b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(b^2*a)^(1/3)))^(1/2)*(I*(
x+1/2/b*(b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(b^2*a)^(1/3))*3^(1/2)*b/(b^2*a)^(1/3))^(1/2)/(b*x^3-a)^(1/2)*EllipticF(
1/3*3^(1/2)*(-I*(x+1/2/b*(b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(b^2*a)^(1/3))*3^(1/2)*b/(b^2*a)^(1/3))^(1/2),(-I*3^(1/
2)/b*(b^2*a)^(1/3)/(-3/2/b*(b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(b^2*a)^(1/3)))^(1/2))+2/3*I*a^(1/3)*3^(1/2)/b*(b^2*a
)^(1/3)*(-I*(x+1/2/b*(b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(b^2*a)^(1/3))*3^(1/2)*b/(b^2*a)^(1/3))^(1/2)*((x-1/b*(b^2*
a)^(1/3))/(-3/2/b*(b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(b^2*a)^(1/3)))^(1/2)*(I*(x+1/2/b*(b^2*a)^(1/3)-1/2*I*3^(1/2)/
b*(b^2*a)^(1/3))*3^(1/2)*b/(b^2*a)^(1/3))^(1/2)/(b*x^3-a)^(1/2)*EllipticF(1/3*3^(1/2)*(-I*(x+1/2/b*(b^2*a)^(1/
3)+1/2*I*3^(1/2)/b*(b^2*a)^(1/3))*3^(1/2)*b/(b^2*a)^(1/3))^(1/2),(-I*3^(1/2)/b*(b^2*a)^(1/3)/(-3/2/b*(b^2*a)^(
1/3)-1/2*I*3^(1/2)/b*(b^2*a)^(1/3)))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.54467, size = 112, normalized size = 0.23 \begin{align*} \frac{i \sqrt [3]{b} x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3}}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{5}{3}\right )} - \frac{i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3}}{a}} \right )}}{3 \sqrt [6]{a} \Gamma \left (\frac{4}{3}\right )} + \frac{\sqrt{3} i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3}}{a}} \right )}}{3 \sqrt [6]{a} \Gamma \left (\frac{4}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*b**(1/3)*x**2*gamma(2/3)*hyper((1/2, 2/3), (5/3,), b*x**3/a)/(3*sqrt(a)*gamma(5/3)) - I*x*gamma(1/3)*hyper((
1/3, 1/2), (4/3,), b*x**3/a)/(3*a**(1/6)*gamma(4/3)) + sqrt(3)*I*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), b*x**3
/a)/(3*a**(1/6)*gamma(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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