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

Optimal. Leaf size=481 \[ -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}-\frac {2 \sqrt {a-b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

[Out]

-2*(-b*x^3+a)^(1/2)/b^(1/3)/(-b^(1/3)*x+a^(1/3)*(1+3^(1/2)))+3^(1/4)*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))),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)/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)-4*3^(1/4)*a^(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)/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 = 0.14, antiderivative size = 481, 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 = {1878, 218, 1877} \[ -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}-\frac {2 \sqrt {a-b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a - b*x^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)) + (3^(1/4)*Sqrt[2 - Sqrt[3]]*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]*Ellipt
icE[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^(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]) - (4*3^(1/4)*Sqr
t[2 + Sqrt[3]]*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[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^(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 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 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]

Rule 1878

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] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt {a-b x^3}} \, dx &=\left (2 \sqrt {3} \sqrt [3]{a}\right ) \int \frac {1}{\sqrt {a-b x^3}} \, dx+\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt {a-b x^3}} \, dx\\ &=-\frac {2 \sqrt {a-b x^3}}{\sqrt [3]{b} \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} \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 )}{\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 {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{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}} 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 [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 = 0.09, size = 91, normalized size = 0.19 \[ \frac {x \sqrt {1-\frac {b x^3}{a}} \left (2 \left (1+\sqrt {3}\right ) \sqrt [3]{a} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\frac {b x^3}{a}\right )-\sqrt [3]{b} x \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\frac {b x^3}{a}\right )\right )}{2 \sqrt {a-b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[1 - (b*x^3)/a]*(2*(1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] - b^(1/3)*x*Hyperg
eometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/(2*Sqrt[a - b*x^3])

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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-b x^{3} + a} b^{\frac {1}{3}} x - \sqrt {-b x^{3} + a} a^{\frac {1}{3}} {\left (\sqrt {3} + 1\right )}}{b x^{3} - a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.29, size = 949, normalized size = 1.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*I/b^(2/3)*3^(1/2)*(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)*b/(a*b^2)^(1/3))^(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^(1/3)*(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)*b/(a*b^2)^(1/3))^(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*a^(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)*b/(a*b^2)^(1/3))^(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 \[ -\int \frac {b^{\frac {1}{3}} x - a^{\frac {1}{3}} {\left (\sqrt {3} + 1\right )}}{\sqrt {-b x^{3} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 14.66, size = 128, normalized size = 0.27 \[ - \frac {\sqrt [3]{b} x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} + \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt [6]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {\sqrt {3} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt [6]{a} \Gamma \left (\frac {4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**(1/3)*x**2*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)) + x*gam
ma(1/3)*hyper((1/3, 1/2), (4/3,), b*x**3*exp_polar(2*I*pi)/a)/(3*a**(1/6)*gamma(4/3)) + sqrt(3)*x*gamma(1/3)*h
yper((1/3, 1/2), (4/3,), b*x**3*exp_polar(2*I*pi)/a)/(3*a**(1/6)*gamma(4/3))

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