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

Optimal. Leaf size=251 \[ \frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (x \sqrt [3]{\frac{b}{a}}+1\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}-x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}{\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}-\sqrt{3}+1\right )^2}} \sqrt{-a-b x^3}}-\frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{-a-b x^3}}{b \left (x \sqrt [3]{\frac{b}{a}}-\sqrt{3}+1\right )} \]

[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 = 0.0644166, antiderivative size = 251, 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} \[ \frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (x \sqrt [3]{\frac{b}{a}}+1\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}-x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}{\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{-\frac{x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}-\sqrt{3}+1\right )^2}} \sqrt{-a-b x^3}}-\frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{-a-b x^3}}{b \left (x \sqrt [3]{\frac{b}{a}}-\sqrt{3}+1\right )} \]

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 = 0.0338203, size = 92, normalized size = 0.37 \[ \frac{x \sqrt{\frac{b x^3}{a}+1} \left (2 \left (1+\sqrt{3}\right ) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right )+x \sqrt [3]{\frac{b}{a}} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )\right )}{2 \sqrt{-a-b x^3}} \]

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*Hyperge
ometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[-a - b*x^3])

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Maple [B]  time = 0.012, size = 1013, normalized size = 4. \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*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))-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)*Elliptic
F(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{\sqrt{-b x^{3} - a} x \left (\frac{b}{a}\right )^{\frac{1}{3}} + \sqrt{-b x^{3} - a}{\left (\sqrt{3} + 1\right )}}{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(-(sqrt(-b*x^3 - a)*x*(b/a)^(1/3) + sqrt(-b*x^3 - a)*(sqrt(3) + 1))/(b*x^3 + a), x)

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Sympy [A]  time = 2.27895, size = 131, normalized size = 0.52 \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} e^{i \pi }}{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} e^{i \pi }}{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} e^{i \pi }}{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*exp_polar(I*pi)/a)/(3*sqrt(a)*gamma(5/3)) - s
qrt(3)*I*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(a)*gamma(4/3)) - I*x*gamma(1
/3)*hyper((1/3, 1/2), (4/3,), b*x**3*exp_polar(I*pi)/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)