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

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

[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 = 0.174189, 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, Rules used = {2140, 206} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} \sqrt{a} \left (x \sqrt [3]{\frac{b}{a}}+1\right )}{\sqrt{-a-b x^3}}\right )}{\sqrt{3+2 \sqrt{3}} \sqrt{a} \sqrt [3]{\frac{b}{a}}} \]

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))

Rule 2140

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[((1 + k)*e)/d, Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + ((1 + k)*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^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]

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])

Rubi steps

\begin{align*} \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 &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-\left (3+2 \sqrt{3}\right ) a x^2} \, dx,x,\frac{1+\sqrt [3]{\frac{b}{a}} x}{\sqrt{-a-b x^3}}\right )}{\sqrt [3]{\frac{b}{a}}}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} \sqrt{a} \left (1+\sqrt [3]{\frac{b}{a}} x\right )}{\sqrt{-a-b x^3}}\right )}{\sqrt{3+2 \sqrt{3}} \sqrt{a} \sqrt [3]{\frac{b}{a}}}\\ \end{align*}

Mathematica [C]  time = 0.781634, size = 670, normalized size = 8.82 \[ \frac{x \left (-\frac{3 \left (10496 \sqrt{3} a^3 F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )+18176 a^3 F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )-b x^3 \left (2 \left (5+3 \sqrt{3}\right ) a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right ) \left (8 \left (5+3 \sqrt{3}\right ) a F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )-3 b x^3 \left (F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )+\left (5+3 \sqrt{3}\right ) F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )\right )\right )\right )}{a \left (2 \left (5+3 \sqrt{3}\right ) a+b x^3\right ) \left (8 \left (5+3 \sqrt{3}\right ) a F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )-3 b x^3 \left (F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )+\left (5+3 \sqrt{3}\right ) F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )\right )\right )}-8 x^2 \left (\frac{b}{a}\right )^{2/3} \sqrt{\frac{3 b x^3}{a}+3} F_1\left (1;\frac{1}{2},1;2;-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )+12 \left (3+\sqrt{3}\right ) x \sqrt [3]{\frac{b}{a}} \sqrt{\frac{b x^3}{a}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right )\right )}{24 \left (5+3 \sqrt{3}\right ) \sqrt{-a-b x^3}} \]

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]

(x*(12*(3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*
a + 6*Sqrt[3]*a))] - 8*(b/a)^(2/3)*x^2*Sqrt[3 + (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, -((b*x^3)/a), -((b*x^3)/(1
0*a + 6*Sqrt[3]*a))] - (3*(18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))]
+ 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - b*x^3*(2*(5 +
3*Sqrt[3])*a + b*x^3)*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((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))]))))/(a*(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))])))))/(24*(5 + 3*Sqrt[3])*Sqrt[-a - b*x^3])

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{ \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 \end{align*}

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. \begin{align*} \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} \end{align*}

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, 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 [B]  time = 6.06471, size = 3345, normalized size = 44.01 \begin{align*} \text{result too large to display} \end{align*}

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, algorithm="fricas")

[Out]

[1/2*sqrt(1/3)*sqrt((2*sqrt(3) - 3)*(b/a)^(1/3)/b)*log((b^8*x^24 - 1840*a*b^7*x^21 + 67264*a^2*b^6*x^18 - 5862
4*a^3*b^5*x^15 + 504064*a^4*b^4*x^12 + 2140160*a^5*b^3*x^9 + 3100672*a^6*b^2*x^6 + 1089536*a^7*b*x^3 + 28672*a
^8 + 4*sqrt(1/3)*((3*a*b^7*x^22 - 2688*a^2*b^6*x^19 + 56952*a^3*b^5*x^16 - 93504*a^4*b^4*x^13 - 63552*a^5*b^3*
x^10 - 377856*a^6*b^2*x^7 - 314880*a^7*b*x^4 - 24576*a^8*x + 2*sqrt(3)*(a*b^7*x^22 - 764*a^2*b^6*x^19 + 16860*
a^3*b^5*x^16 - 19792*a^4*b^4*x^13 + 42368*a^5*b^3*x^10 + 104448*a^6*b^2*x^7 + 90880*a^7*b*x^4 + 7168*a^8*x))*s
qrt(-b*x^3 - a)*(b/a)^(2/3) - 2*(30*a*b^7*x^21 - 5010*a^2*b^6*x^18 + 44640*a^3*b^5*x^15 - 21360*a^4*b^4*x^12 +
79872*a^5*b^3*x^9 + 233856*a^6*b^2*x^6 + 86016*a^7*b*x^3 + 3072*a^8 + sqrt(3)*(17*a*b^7*x^21 - 2920*a^2*b^6*x
^18 + 24864*a^3*b^5*x^15 - 26576*a^4*b^4*x^12 - 56000*a^5*b^3*x^9 - 115968*a^6*b^2*x^6 - 56320*a^7*b*x^3 - 102
4*a^8))*sqrt(-b*x^3 - a)*(b/a)^(1/3) + 6*(81*a*b^7*x^20 - 4752*a^2*b^6*x^17 + 14472*a^3*b^5*x^14 - 24192*a^4*b
^4*x^11 - 39744*a^5*b^3*x^8 - 69120*a^6*b^2*x^5 - 13824*a^7*b*x^2 + sqrt(3)*(47*a*b^7*x^20 - 2724*a^2*b^6*x^17
+ 8976*a^3*b^5*x^14 - 4928*a^4*b^4*x^11 + 32448*a^5*b^3*x^8 + 37632*a^6*b^2*x^5 + 8192*a^7*b*x^2))*sqrt(-b*x^
3 - a))*sqrt((2*sqrt(3) - 3)*(b/a)^(1/3)/b) - 8*(3*a*b^7*x^23 - 1077*a^2*b^6*x^20 + 13320*a^3*b^5*x^17 - 19200
*a^4*b^4*x^14 - 111360*a^5*b^3*x^11 - 345024*a^6*b^2*x^8 - 328704*a^7*b*x^5 - 61440*a^8*x^2 + 2*sqrt(3)*(a*b^7
*x^23 - 299*a^2*b^6*x^20 + 4260*a^3*b^5*x^17 + 1520*a^4*b^4*x^14 + 26720*a^5*b^3*x^11 + 105024*a^6*b^2*x^8 + 9
3184*a^7*b*x^5 + 17920*a^8*x^2))*(b/a)^(2/3) - 32*sqrt(3)*(35*a*b^7*x^21 - 1141*a^2*b^6*x^18 + 2544*a^3*b^5*x^
15 + 6760*a^4*b^4*x^12 + 39520*a^5*b^3*x^9 + 55680*a^6*b^2*x^6 + 19712*a^7*b*x^3 + 512*a^8) + 32*(9*a*b^7*x^22
- 846*a^2*b^6*x^19 + 4617*a^3*b^5*x^16 + 5472*a^4*b^4*x^13 + 43776*a^5*b^3*x^10 + 98496*a^6*b^2*x^7 + 59328*a
^7*b*x^4 + 4608*a^8*x + sqrt(3)*(5*a*b^7*x^22 - 505*a^2*b^6*x^19 + 2130*a^3*b^5*x^16 - 4928*a^4*b^4*x^13 - 286
88*a^5*b^3*x^10 - 53760*a^6*b^2*x^7 - 35200*a^7*b*x^4 - 2560*a^8*x))*(b/a)^(1/3))/(b^8*x^24 + 80*a*b^7*x^21 +
2368*a^2*b^6*x^18 + 30080*a^3*b^5*x^15 + 121984*a^4*b^4*x^12 - 240640*a^5*b^3*x^9 + 151552*a^6*b^2*x^6 - 40960
*a^7*b*x^3 + 4096*a^8)), -sqrt(1/3)*sqrt(-(2*sqrt(3) - 3)*(b/a)^(1/3)/b)*arctan(-1/2*sqrt(1/3)*(sqrt(-b*x^3 -
a)*b*x^2 - 2*sqrt(-b*x^3 - a)*(sqrt(3)*a*x + 2*a*x)*(b/a)^(2/3) - 2*sqrt(-b*x^3 - a)*(sqrt(3)*a + a)*(b/a)^(1/
3))*sqrt(-(2*sqrt(3) - 3)*(b/a)^(1/3)/b)/(b*x^3 + a))]

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

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]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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, algorithm="giac")

[Out]

sage0*x