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

Optimal. Leaf size=256 \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{b x^3-a}}{b \left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-x \sqrt [3]{\frac{b}{a}}\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}+x \sqrt [3]{\frac{b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}{-\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{1-x \sqrt [3]{\frac{b}{a}}}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} \sqrt{b x^3-a}} \]

[Out]

(2*(b/a)^(2/3)*Sqrt[-a + b*x^3])/(b*(1 - Sqrt[3] - (b/a)^(1/3)*x)) - (3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (b/a)^(1/
3)*x)*Sqrt[(1 + (b/a)^(1/3)*x + (b/a)^(2/3)*x^2)/(1 - Sqrt[3] - (b/a)^(1/3)*x)^2]*EllipticE[ArcSin[(1 + Sqrt[3
] - (b/a)^(1/3)*x)/(1 - Sqrt[3] - (b/a)^(1/3)*x)], -7 + 4*Sqrt[3]])/((b/a)^(1/3)*Sqrt[-((1 - (b/a)^(1/3)*x)/(1
- Sqrt[3] - (b/a)^(1/3)*x)^2)]*Sqrt[-a + b*x^3])

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Rubi [A]  time = 0.0831561, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {1879} \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{b x^3-a}}{b \left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-x \sqrt [3]{\frac{b}{a}}\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}+x \sqrt [3]{\frac{b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}{-\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{1-x \sqrt [3]{\frac{b}{a}}}{\left (x \left (-\sqrt [3]{\frac{b}{a}}\right )-\sqrt{3}+1\right )^2}} \sqrt{b x^3-a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b/a)^(2/3)*Sqrt[-a + b*x^3])/(b*(1 - Sqrt[3] - (b/a)^(1/3)*x)) - (3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (b/a)^(1/
3)*x)*Sqrt[(1 + (b/a)^(1/3)*x + (b/a)^(2/3)*x^2)/(1 - Sqrt[3] - (b/a)^(1/3)*x)^2]*EllipticE[ArcSin[(1 + Sqrt[3
] - (b/a)^(1/3)*x)/(1 - Sqrt[3] - (b/a)^(1/3)*x)], -7 + 4*Sqrt[3]])/((b/a)^(1/3)*Sqrt[-((1 - (b/a)^(1/3)*x)/(1
- Sqrt[3] - (b/a)^(1/3)*x)^2)]*Sqrt[-a + b*x^3])

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{1+\sqrt{3}-\sqrt [3]{\frac{b}{a}} x}{\sqrt{-a+b x^3}} \, dx &=\frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{-a+b x^3}}{b \left (1-\sqrt{3}-\sqrt [3]{\frac{b}{a}} x\right )}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{\frac{b}{a}} x\right ) \sqrt{\frac{1+\sqrt [3]{\frac{b}{a}} x+\left (\frac{b}{a}\right )^{2/3} x^2}{\left (1-\sqrt{3}-\sqrt [3]{\frac{b}{a}} x\right )^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{\frac{b}{a}} x}{1-\sqrt{3}-\sqrt [3]{\frac{b}{a}} x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{1-\sqrt [3]{\frac{b}{a}} x}{\left (1-\sqrt{3}-\sqrt [3]{\frac{b}{a}} x\right )^2}} \sqrt{-a+b x^3}}\\ \end{align*}

Mathematica [C]  time = 0.032436, size = 90, normalized size = 0.35 \[ -\frac{x \sqrt{1-\frac{b x^3}{a}} \left (x \sqrt [3]{\frac{b}{a}} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right )-2 \left (1+\sqrt{3}\right ) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right )\right )}{2 \sqrt{b x^3-a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*Sqrt[1 - (b*x^3)/a]*(-2*(1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + (b/a)^(1/3)*x*Hypergeo
metric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/(2*Sqrt[-a + b*x^3])

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Maple [B]  time = 0.019, size = 953, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*I*(b/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)*((-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/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*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{x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \sqrt{3} - 1}{\sqrt{b x^{3} - a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((x*(b/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{x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \sqrt{3} - 1}{\sqrt{b x^{3} - a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.75303, size = 114, normalized size = 0.45 \begin{align*} \frac{i x^{2} \sqrt [3]{\frac{b}{a}} \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{\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{a} \Gamma \left (\frac{4}{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{a} \Gamma \left (\frac{4}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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