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)*(-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)),2*I-I*3^(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)*(1/2*6^(1/2)+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 = 251, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, 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*}

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Mathematica [C]  time = 0.04, 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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fricas [F]  time = 0.59, 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 = 0.08, size = 1013, normalized size = 4.04 \[ \text {result too large to display} \]

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*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))-2/3*I*(1/a*b)^(1/3)*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)*((-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))+(-a*b^2)^(1/3)/b*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))

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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 \[ \int \frac {\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {-b\,x^3-a}} \,d x \]

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 = 6.61, size = 131, normalized size = 0.52 \[ - \frac {i x^{2} \sqrt [3]{\frac {b}{a}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} - \frac {\sqrt {3} i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} - \frac {i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \]

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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