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

Optimal. Leaf size=76 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} \sqrt{a} \left (1-x \sqrt [3]{\frac{b}{a}}\right )}{\sqrt{b x^3-a}}\right )}{\sqrt{3+2 \sqrt{3}} \sqrt{a} \sqrt [3]{\frac{b}{a}}} \]

[Out]

(2*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[a]*(1 - (b/a)^(1/3)*x))/Sqrt[-a + b*x^3]])/
(Sqrt[3 + 2*Sqrt[3]]*Sqrt[a]*(b/a)^(1/3))

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Rubi [A]  time = 0.33305, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} \sqrt{a} \left (1-x \sqrt [3]{\frac{b}{a}}\right )}{\sqrt{b x^3-a}}\right )}{\sqrt{3+2 \sqrt{3}} \sqrt{a} \sqrt [3]{\frac{b}{a}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[a]*(1 - (b/a)^(1/3)*x))/Sqrt[-a + b*x^3]])/
(Sqrt[3 + 2*Sqrt[3]]*Sqrt[a]*(b/a)^(1/3))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-(b/a)**(1/3)*x-3**(1/2))/(1-(b/a)**(1/3)*x+3**(1/2))/(b*x**3-a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 5.59776, size = 836, normalized size = 11. \[ \frac{\left (26+15 \sqrt{3}\right ) a x \left (x \left (x \left (-\frac{16 \sqrt{3} a \left (\frac{b}{a}\right )^{2/3} F_1\left (1;\frac{1}{2},1;2;\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}{b \left (F_1\left (2;\frac{1}{2},2;3;\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )+\left (5+3 \sqrt{3}\right ) F_1\left (2;\frac{3}{2},1;3;\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )\right ) x^3+4 \left (5+3 \sqrt{3}\right ) a F_1\left (1;\frac{1}{2},1;2;\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}-\frac{21 b x F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}{3 b \left (F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )+\left (5+3 \sqrt{3}\right ) F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )\right ) x^3+14 \left (5+3 \sqrt{3}\right ) a F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}\right )-\frac{60 \left (3+\sqrt{3}\right ) a \sqrt [3]{\frac{b}{a}} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}{3 b \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )+\left (5+3 \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )\right ) x^3+10 \left (5+3 \sqrt{3}\right ) a F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}\right )-\frac{96 \left (1+\sqrt{3}\right ) a F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}{3 b \left (F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )+\left (5+3 \sqrt{3}\right ) F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )\right ) x^3+8 \left (5+3 \sqrt{3}\right ) a F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right )}\right )}{3 \left (5+3 \sqrt{3}\right ) \left (2 \left (5+3 \sqrt{3}\right ) a-b x^3\right ) \sqrt{b x^3-a}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((26 + 15*Sqrt[3])*a*x*((-96*(1 + Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/
a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])/(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3
, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3,
(b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1
, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])) + x*((-60*(3 + Sqrt[3])*a*(b/a
)^(1/3)*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])/(10
*(5 + 3*Sqrt[3])*a*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[
3]*a)] + 3*b*x^3*(AppellF1[5/3, 1/2, 2, 8/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3
]*a)] + (5 + 3*Sqrt[3])*AppellF1[5/3, 3/2, 1, 8/3, (b*x^3)/a, (b*x^3)/(10*a + 6*
Sqrt[3]*a)])) + x*((-16*Sqrt[3]*a*(b/a)^(2/3)*AppellF1[1, 1/2, 1, 2, (b*x^3)/a,
(b*x^3)/(10*a + 6*Sqrt[3]*a)])/(4*(5 + 3*Sqrt[3])*a*AppellF1[1, 1/2, 1, 2, (b*x^
3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + b*x^3*(AppellF1[2, 1/2, 2, 3, (b*x^3)/a, (
b*x^3)/(10*a + 6*Sqrt[3]*a)] + (5 + 3*Sqrt[3])*AppellF1[2, 3/2, 1, 3, (b*x^3)/a,
(b*x^3)/(10*a + 6*Sqrt[3]*a)])) - (21*b*x*AppellF1[4/3, 1/2, 1, 7/3, (b*x^3)/a,
(b*x^3)/(10*a + 6*Sqrt[3]*a)])/(14*(5 + 3*Sqrt[3])*a*AppellF1[4/3, 1/2, 1, 7/3,
(b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + 3*b*x^3*(AppellF1[7/3, 1/2, 2, 10/3,
(b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + (5 + 3*Sqrt[3])*AppellF1[7/3, 3/2, 1
, 10/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)]))))))/(3*(5 + 3*Sqrt[3])*(2*(5
+ 3*Sqrt[3])*a - b*x^3)*Sqrt[-a + b*x^3])

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Maple [F]  time = 0.114, size = 0, normalized size = 0. \[ \int{1 \left ( 1-\sqrt [3]{{\frac{b}{a}}}x-\sqrt{3} \right ) \left ( 1-\sqrt [3]{{\frac{b}{a}}}x+\sqrt{3} \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{3}-a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x*(b/a)^(1/3) + sqrt(3) - 1)/(sqrt(b*x^3 - a)*(x*(b/a)^(1/3) - sqrt(3) - 1)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x*(b/a)^(1/3) + sqrt(3) - 1)/(sqrt(b*x^3 - a)*(x*(b/a)^(1/3) - sqrt(3) - 1)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 13.9764, size = 0, normalized size = 0. \[ \mathrm{NaN} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

nan

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GIAC/XCAS [A]  time = 0.616158, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x*(b/a)^(1/3) + sqrt(3) - 1)/(sqrt(b*x^3 - a)*(x*(b/a)^(1/3) - sqrt(3) - 1)),x, algorithm="giac")

[Out]

sage0*x