3.54 \(\int \frac {2+3 x}{(2^{2/3}-x) \sqrt {-1+x^3}} \, dx\)

Optimal. Leaf size=176 \[ \frac {2 \left (3-2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2 \left (2+3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {x^3-1}}\right )}{3 \sqrt {3}} \]

[Out]

-2/9*(2+3*2^(2/3))*arctanh((1-2^(1/3)*x)*3^(1/2)/(x^3-1)^(1/2))*3^(1/2)+2/9*(3-2*2^(1/3))*(1-x)*EllipticF((1-x
+3^(1/2))/(1-x-3^(1/2)),2*I-I*3^(1/2))*(1/2*6^(1/2)-1/2*2^(1/2))*((x^2+x+1)/(1-x-3^(1/2))^2)^(1/2)*3^(3/4)/(x^
3-1)^(1/2)/((-1+x)/(1-x-3^(1/2))^2)^(1/2)

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Rubi [A]  time = 0.22, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2139, 219, 2137, 206} \[ \frac {2 \left (3-2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2 \left (2+3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {x^3-1}}\right )}{3 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2 + 3*2^(2/3))*ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[-1 + x^3]])/(3*Sqrt[3]) + (2*(3 - 2*2^(1/3))*Sqrt[2
- Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] -
x)], -7 + 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

Rule 206

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

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 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]

Rule 2139

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(2*d*e + c*f)/(3*c
*d), Int[1/Sqrt[a + b*x^3], x], x] + Dist[(d*e - c*f)/(3*c*d), Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x]
, 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[b*c^3 + 8*a*d^3,
0]) && NeQ[2*d*e + c*f, 0]

Rubi steps

\begin {align*} \int \frac {2+3 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx &=-\left (\frac {1}{3} \left (3-2 \sqrt [3]{2}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx\right )+\frac {1}{3} \left (3+\sqrt [3]{2}\right ) \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx\\ &=\frac {2 \left (3-2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {1}{3} \left (2 \left (2+3\ 2^{2/3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {1-\sqrt [3]{2} x}{\sqrt {-1+x^3}}\right )\\ &=-\frac {2 \left (2+3\ 2^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {-1+x^3}}\right )}{3 \sqrt {3}}+\frac {2 \left (3-2 \sqrt [3]{2}\right ) \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.33, size = 333, normalized size = 1.89 \[ \frac {2 \sqrt [6]{2} \sqrt {-\frac {i (x-1)}{\sqrt {3}+3 i}} \left (4 \sqrt {3} \left (3+\sqrt [3]{2}\right ) \sqrt {2 i x+\sqrt {3}+i} \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {2 i x+\sqrt {3}+i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )-3 i \sqrt {-2 i x+\sqrt {3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt {3}+\sqrt [3]{2} \sqrt {3}\right ) x-\sqrt [3]{2} \sqrt {3}+2 \sqrt {3}-3 i \sqrt [3]{2}-6 i\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {2 i x+\sqrt {3}+i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (i+2 i 2^{2/3}+\sqrt {3}\right ) \sqrt {2 i x+\sqrt {3}+i} \sqrt {x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-3*I)*Sqrt[-I + Sqrt[3] - (2*I)*x]*(-6*I - (3*I)*2^(1/3) +
2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt
[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 4*Sqrt[3]*(3 + 2^(1/3))*Sqrt[I + Sqrt[3] + (
2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I
)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3
] + (2*I)*x]*Sqrt[-1 + x^3])

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   catdef: division by zero

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{1,[2]%%%} / %%%{%%{[2,0]:[1,0,0,-2]%%},[2]%%%} Error: Bad Argum
ent Value

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maple [A]  time = 0.04, size = 266, normalized size = 1.51 \[ -\frac {6 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {2 \left (-2-3 \,2^{\frac {2}{3}}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*(
(x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),((
3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+2*(-2-3*2^(2/3))*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(1/2
)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/
(x^3-1)^(1/2)/(-2^(2/3)+1)*EllipticPi(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(-2^(2/3)+1),((3/
2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {3\,x+2}{\sqrt {x^3-1}\,\left (x-2^{2/3}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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