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

Optimal. Leaf size=241 \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{a+b x^3}}{b \left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (x \sqrt [3]{\frac{b}{a}}+1\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}-x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}{\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{\frac{x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )^2}} \sqrt{a+b x^3}} \]

[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 = 0.0692954, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031, Rules used = {1877} \[ \frac{2 \left (\frac{b}{a}\right )^{2/3} \sqrt{a+b x^3}}{b \left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (x \sqrt [3]{\frac{b}{a}}+1\right ) \sqrt{\frac{x^2 \left (\frac{b}{a}\right )^{2/3}-x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{\frac{b}{a}} x-\sqrt{3}+1}{\sqrt [3]{\frac{b}{a}} x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [3]{\frac{b}{a}} \sqrt{\frac{x \sqrt [3]{\frac{b}{a}}+1}{\left (x \sqrt [3]{\frac{b}{a}}+\sqrt{3}+1\right )^2}} \sqrt{a+b x^3}} \]

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 1877

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] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[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 = 0.0511168, size = 89, normalized size = 0.37 \[ \frac{x \sqrt{\frac{b x^3}{a}+1} \left (x \sqrt [3]{\frac{b}{a}} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )-2 \left (\sqrt{3}-1\right ) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right )\right )}{2 \sqrt{a+b x^3}} \]

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*Hyper
geometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[a + b*x^3])

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Maple [B]  time = 0.03, size = 1004, normalized size = 4.2 \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)*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 = 2.17516, size = 124, normalized size = 0.51 \begin{align*} \frac{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} 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{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]

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)) - sqrt
(3)*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)) + x*gamma(1/3)*hyp
er((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)