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

Optimal. Leaf size=549 \[ -\frac {2 \sqrt {2-\sqrt {3}} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{\frac {b}{a}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}+\frac {2 \sqrt [3]{\frac {b}{a}} \sqrt {b x^3-a}}{b^{2/3} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

[Out]

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

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Rubi [A]  time = 0.22, antiderivative size = 549, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1880, 219, 1879} \[ -\frac {2 \sqrt {2-\sqrt {3}} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{\frac {b}{a}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}+\frac {2 \sqrt [3]{\frac {b}{a}} \sqrt {b x^3-a}}{b^{2/3} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

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]

Rule 1880

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a,
3]]}, Dist[(c*r - (1 + Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 + Sqrt[3])*s + r*x)
/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && NeQ[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 {\sqrt [3]{\frac {b}{a}} \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt {-a+b x^3}} \, dx}{\sqrt [3]{b}}-\left (-1+\sqrt {3}+\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {-a+b x^3}} \, dx\\ &=\frac {2 \sqrt [3]{\frac {b}{a}} \sqrt {-a+b x^3}}{b^{2/3} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{\frac {b}{a}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {-a+b x^3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1-\sqrt {3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}}{\sqrt [3]{b}}\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {-a+b x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 90, normalized size = 0.16 \[ -\frac {x \sqrt {1-\frac {b x^3}{a}} \left (2 \left (\sqrt {3}-1\right ) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\frac {b x^3}{a}\right )+x \sqrt [3]{\frac {b}{a}} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\frac {b x^3}{a}\right )\right )}{2 \sqrt {b x^3-a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-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 = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {3} - 1}{\sqrt {b x^{3} - a}}, x\right ) \]

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(-(x*(b/a)^(1/3) + sqrt(3) - 1)/sqrt(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 = 953, normalized size = 1.74 \[ \text {result too large to display} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.15, size = 114, normalized size = 0.21 \[ \frac {i x^{2} \sqrt [3]{\frac {b}{a}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3}}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} - \frac {i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3}}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} + \frac {\sqrt {3} i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3}}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \]

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

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