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

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

[Out]

(-2*ArcTan[(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 = 0.313967, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt{a} \left (x \sqrt [3]{\frac{b}{a}}+1\right )}{\sqrt{-a-b x^3}}\right )}{\sqrt{2 \sqrt{3}-3} \sqrt{a} \sqrt [3]{\frac{b}{a}}} \]

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*ArcTan[(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 = 8.48211, size = 1545, normalized size = 20.33 \[ \text{result too large to display} \]

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]

(32*(26 - 15*Sqrt[3])*a^2*x*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), (b*x^3)/(-1
0*a + 6*Sqrt[3]*a)])/((-5 + 3*Sqrt[3])*Sqrt[-a - b*x^3]*(2*(-5 + 3*Sqrt[3])*a -
b*x^3)*(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))]))) - (32*Sqrt[3]*(26 - 15*Sqrt[3])*a^2*x
*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)])/((-5 +
3*Sqrt[3])*Sqrt[-a - b*x^3]*(2*(-5 + 3*Sqrt[3])*a - b*x^3)*(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))]))) - (60*(26 - 15*Sqrt[3])*a^2*(b/a)^(1/3)*x^2*AppellF1[2/3, 1/2, 1
, 5/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)])/((-5 + 3*Sqrt[3])*Sqrt[-a -
b*x^3]*(2*(-5 + 3*Sqrt[3])*a - b*x^3)*(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])*A
ppellF1[5/3, 3/2, 1, 8/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))]))) + (2
0*Sqrt[3]*(26 - 15*Sqrt[3])*a^2*(b/a)^(1/3)*x^2*AppellF1[2/3, 1/2, 1, 5/3, -((b*
x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)])/((-5 + 3*Sqrt[3])*Sqrt[-a - b*x^3]*(2*(
-5 + 3*Sqrt[3])*a - b*x^3)*(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))]))) - (16*(26 - 15*S
qrt[3])*a^2*(b/a)^(2/3)*x^3*AppellF1[1, 1/2, 1, 2, -((b*x^3)/a), (b*x^3)/(-10*a
+ 6*Sqrt[3]*a)])/(Sqrt[3]*(-5 + 3*Sqrt[3])*Sqrt[-a - b*x^3]*(2*(-5 + 3*Sqrt[3])*
a - b*x^3)*(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))]))) - (7*(26 - 15*Sqrt[3])*a*b*x^4*AppellF1[4/3, 1/
2, 1, 7/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)])/((-5 + 3*Sqrt[3])*Sqrt[
-a - b*x^3]*(2*(-5 + 3*Sqrt[3])*a - b*x^3)*(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))]))
)

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Maple [F]  time = 0.113, 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 = 12.9848, 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 = 0.599709, 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