3.6 \(\int \frac {1}{(2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {a-b x^3}} \, dx\)

Optimal. Leaf size=288 \[ -\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{3 \sqrt {3} \sqrt {a} \sqrt [3]{b}} \]

[Out]

-2/9*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*b^(1/3)*x)*3^(1/2)/(-b*x^3+a)^(1/2))/b^(1/3)*3^(1/2)/a^(1/2)-2/9*2^(1/3)*
(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))),I*3^(1/2)+2*I)
*(1/2*6^(1/2)+1/2*2^(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)*
3^(3/4)/a^(1/3)/b^(1/3)/(-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 = 0.33, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2134, 218, 2137, 203} \[ -\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \text {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{3 \sqrt {3} \sqrt {a} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/Sqrt[a - b*x^3]])/(3*Sqrt[3]*Sqrt[a]*b^(1/3)) - (2*
2^(1/3)*Sqrt[2 + Sqrt[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[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]])/(3*3^(1/4)*a^(1/3)*b^(1/3)*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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 218

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 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 2134

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[2/(3*c), Int[1/Sqrt[a + b*x^3], x], x
] + Dist[1/(3*c), Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 -
4*a*d^3, 0]

Rule 2137

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(2*e)/d, Subst[Int[
1/(1 + 3*a*x^2), x], x, (1 + (2*d*x)/c)/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx &=\frac {\int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx}{3\ 2^{2/3} \sqrt [3]{a}}+\frac {\sqrt [3]{2} \int \frac {1}{\sqrt {a-b x^3}} \, dx}{3 \sqrt [3]{a}}\\ &=-\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \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 )}{3 \sqrt [4]{3} \sqrt [3]{a} \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}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+3 a x^2} \, dx,x,\frac {1-\frac {\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {a-b x^3}}\right )}{3 \sqrt [3]{b}}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{3 \sqrt {3} \sqrt {a} \sqrt [3]{b}}-\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \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 )}{3 \sqrt [4]{3} \sqrt [3]{a} \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 = 0.21, size = 166, normalized size = 0.58 \[ \frac {2 i \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {b^{2/3} x^2}{a^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac {i \sqrt {3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\left (\sqrt [3]{-1}+2^{2/3}\right ) \sqrt [3]{b} \sqrt {a-b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(
2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(
1/3))*a^(1/3))]], (-1)^(1/3)])/(((-1)^(1/3) + 2^(2/3))*b^(1/3)*Sqrt[a - b*x^3])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(-b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(-b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b^{\frac {1}{3}} x +2^{\frac {2}{3}} a^{\frac {1}{3}}\right ) \sqrt {-b \,x^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{\sqrt {-b x^{3} + a} {\left (b^{\frac {1}{3}} x - 2^{\frac {2}{3}} a^{\frac {1}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2^(2/3)*a^(1/3)-b^(1/3)*x)/(-b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {a-b\,x^3}\,\left (2^{2/3}\,a^{1/3}-b^{1/3}\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{- 2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(-2**(2/3)*a**(1/3)*sqrt(a - b*x**3) + b**(1/3)*x*sqrt(a - b*x**3)), x)

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