3.50 \(\int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{(2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x) \sqrt {-a-b x^3}} \, dx\)

Optimal. Leaf size=66 \[ \frac {2\ 2^{2/3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \]

[Out]

2/3*2^(2/3)*arctanh(a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)*3^(1/2)/(-b*x^3-a)^(1/2))/a^(1/6)/b^(1/3)*3^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2137, 206} \[ \frac {2\ 2^{2/3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*2^(2/3)*ArcTanh[(Sqrt[3]*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/Sqrt[-a - b*x^3]])/(Sqrt[3]*a^(1/6)*b^(1/3)
)

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 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]

Rubi steps

\begin {align*} \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx &=\frac {\left (2\ 2^{2/3} \sqrt [3]{a}\right ) \operatorname {Subst}\left (\int \frac {1}{1-3 a x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {-a-b x^3}}\right )}{\sqrt [3]{b}}\\ &=\frac {2\ 2^{2/3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}}\\ \end {align*}

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Mathematica [C]  time = 0.68, size = 375, normalized size = 5.68 \[ -\frac {2 \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\sqrt [3]{-1} 2^{2/3} \sqrt {3} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {b^{2/3} x^2}{a^{2/3}}-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac {i \sqrt {3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )-\frac {2 \left (\sqrt [3]{-1}+2^{2/3}\right ) \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3}}\right )}{\left (\sqrt [3]{-1}+2^{2/3}\right ) \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {-a-b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {-2 b^{\frac {1}{3}} x +2^{\frac {2}{3}} a^{\frac {1}{3}}}{\left (b^{\frac {1}{3}} x +2^{\frac {2}{3}} a^{\frac {1}{3}}\right ) \sqrt {-b \,x^{3}-a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.65, size = 103, normalized size = 1.56 \[ \frac {\sqrt {3}\,4^{1/3}\,\ln \left (\frac {{\left (\sqrt {-b\,x^3-a}+\sqrt {3}\,\sqrt {a}+2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\right )}^3\,\left (\sqrt {3}\,\sqrt {a}-\sqrt {-b\,x^3-a}+2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\right )}{{\left (2^{2/3}\,a^{1/3}+b^{1/3}\,x\right )}^6}\right )}{3\,a^{1/6}\,b^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3^(1/2)*4^(1/3)*log((((- a - b*x^3)^(1/2) + 3^(1/2)*a^(1/2) + 2^(1/3)*3^(1/2)*a^(1/6)*b^(1/3)*x)^3*(3^(1/2)*a
^(1/2) - (- a - b*x^3)^(1/2) + 2^(1/3)*3^(1/2)*a^(1/6)*b^(1/3)*x))/(2^(2/3)*a^(1/3) + b^(1/3)*x)^6))/(3*a^(1/6
)*b^(1/3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {2^{\frac {2}{3}} \sqrt [3]{a}}{2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {- a - b x^{3}} + \sqrt [3]{b} x \sqrt {- a - b x^{3}}}\right )\, dx - \int \frac {2 \sqrt [3]{b} x}{2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {- a - b x^{3}} + \sqrt [3]{b} x \sqrt {- a - b x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-2**(2/3)*a**(1/3)/(2**(2/3)*a**(1/3)*sqrt(-a - b*x**3) + b**(1/3)*x*sqrt(-a - b*x**3)), x) - Integr
al(2*b**(1/3)*x/(2**(2/3)*a**(1/3)*sqrt(-a - b*x**3) + b**(1/3)*x*sqrt(-a - b*x**3)), x)

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