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

Optimal. Leaf size=248 \[ \frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1-x \sqrt [3]{\frac {b}{a}}\right ) \sqrt {\frac {x^2 \left (\frac {b}{a}\right )^{2/3}+x \sqrt [3]{\frac {b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b}{a}} x-\sqrt {3}+1}{-\sqrt [3]{\frac {b}{a}} x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {\frac {1-x \sqrt [3]{\frac {b}{a}}}{\left (x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1\right )^2}} \sqrt {a-b x^3}}-\frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {a-b x^3}}{b \left (x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1\right )} \]

[Out]

-2*(b/a)^(2/3)*(-b*x^3+a)^(1/2)/b/(1-(b/a)^(1/3)*x+3^(1/2))+3^(1/4)*(1-(b/a)^(1/3)*x)*EllipticE((1-(b/a)^(1/3)
*x-3^(1/2))/(1-(b/a)^(1/3)*x+3^(1/2)),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((1+(b/a)^(1/3)*x+(b/a)^(2/3)*x
^2)/(1-(b/a)^(1/3)*x+3^(1/2))^2)^(1/2)/(b/a)^(1/3)/(-b*x^3+a)^(1/2)/((1-(b/a)^(1/3)*x)/(1-(b/a)^(1/3)*x+3^(1/2
))^2)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1877} \[ \frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1-x \sqrt [3]{\frac {b}{a}}\right ) \sqrt {\frac {x^2 \left (\frac {b}{a}\right )^{2/3}+x \sqrt [3]{\frac {b}{a}}+1}{\left (x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b}{a}} x-\sqrt {3}+1}{-\sqrt [3]{\frac {b}{a}} x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{\frac {b}{a}} \sqrt {\frac {1-x \sqrt [3]{\frac {b}{a}}}{\left (x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1\right )^2}} \sqrt {a-b x^3}}-\frac {2 \left (\frac {b}{a}\right )^{2/3} \sqrt {a-b x^3}}{b \left (x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1\right )} \]

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 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*}

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Mathematica [C]  time = 0.06, size = 89, normalized size = 0.36 \[ -\frac {x \sqrt {1-\frac {b x^3}{a}} \left (2 \left (\sqrt {3}-1\right ) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\frac {b x^3}{a}\right )+x \sqrt [3]{\frac {b}{a}} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\frac {b x^3}{a}\right )\right )}{2 \sqrt {a-b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-b x^{3} + a} x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {-b x^{3} + a} {\left (\sqrt {3} - 1\right )}}{b x^{3} - a}, x\right ) \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueDone

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maple [B]  time = 0.18, size = 950, normalized size = 3.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {3} - 1}{\sqrt {-b x^{3} + a}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int -\frac {\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}-1}{\sqrt {a-b\,x^3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.19, size = 129, normalized size = 0.52 \[ - \frac {x^{2} \sqrt [3]{\frac {b}{a}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} - \frac {\sqrt {3} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} + \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \]

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(2*I*pi)/a)/(3*sqrt(a)*gamma(5/3)) - s
qrt(3)*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), b*x**3*exp_polar(2*I*pi)/a)/(3*sqrt(a)*gamma(4/3)) + x*gamma(1/3
)*hyper((1/3, 1/2), (4/3,), b*x**3*exp_polar(2*I*pi)/a)/(3*sqrt(a)*gamma(4/3))

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